Topics in Analytic Geometry

Transcription

1 Topics in Analtic Geometr 0 0. Lines 0. Introduction to Conics: Parabolas 0. Ellipses 0. Hperbolas 0.5 Rotation of Conics 0.6 Parametric Equations 0.7 Polar Coordinates 0.8 Graphs of Polar Equations 0.9 Polar Equations of Conics In Mathematics A conic is a collection of points satisfing a geometric propert. In Real Life Conics are used as models in construction, planetar orbits, radio navigation, and projectile motion. For instance, ou can use conics to model the orbits of the planets as the move about the sun. Using the techniques presented in this chapter, ou can determine the distances between the planets and the center of the sun. (See Eercises 55 6, page 796.) Mike Agliolo/Photo Researchers, Inc. IN CAREERS There are man careers that use conics and other topics in analtic geometr. Several are listed below. Home Contractor Eercise 69, page 7 Civil Engineer Eercises 7 and 7, page 70 Artist Eercise 5, page 759 Astronomer Eercises 6 and 6, page

2 76 Chapter 0 Topics in Analtic Geometr 0. LINES What ou should learn Find the inclination of a line. Find the angle between two lines. Find the distance between a point and a line. Wh ou should learn it The inclination of a line can be used to measure heights indirectl. For instance, in Eercise 70 on page 7, the inclination of a line can be used to determine the change in elevation from the base to the top of the Falls Incline Railwa in Niagara Falls, Ontario, Canada. Inclination of a Line In Section., ou learned that the graph of the linear equation m b is a nonvertical line with slope m and -intercept 0, b. There, the slope of a line was described as the rate of change in with respect to. In this section, ou will look at the slope of a line in terms of the angle of inclination of the line. Ever nonhorizontal line must intersect the -ais. The angle formed b such an intersection determines the inclination of the line, as specified in the following definition. Definition of Inclination The inclination of a nonhorizontal line is the positive angle (less than ) measured counterclockwise from the -ais to the line. (See Figure 0..) θ = 0 JTB Photo/Japan Travel Bureau/PhotoLibrar θ = Horizontal Line Vertical Line Acute Angle Obtuse Angle FIGURE 0. The inclination of a line is related to its slope in the following manner. θ θ Inclination and Slope If a nonvertical line has inclination and slope m, then m tan. For a proof of this relation between inclination and slope, see Proofs in Mathematics on page 80.

3 Section 0. Lines 77 Eample Finding the Inclination of a Line FIGURE 0. = θ = 5 Find the inclination of the line. The slope of this line is m. So, its inclination is determined from the equation tan. From Figure 0., it follows that 0 < arctan. The angle of inclination is radian or 5. Now tr Eercise 7. < This means that. Eample Finding the Inclination of a Line Find the inclination of the line 6. The slope of this line is m. So, its inclination is determined from the equation + = 6 tan. θ 6. From Figure 0., it follows that < arctan <. This means that FIGURE The angle of inclination is about.55 radians or about 6.. θ = θ θ Now tr Eercise. θ θ θ The Angle Between Two Lines Two distinct lines in a plane are either parallel or intersecting. If the intersect and are nonperpendicular, their intersection forms two pairs of opposite angles. One pair is acute and the other pair is obtuse. The smaller of these angles is called the angle between the two lines. As shown in Figure 0., ou can use the inclinations of the two lines to find the angle between the two lines. If two lines have inclinations and, where < and <, the angle between the two lines is FIGURE 0..

4 78 Chapter 0 Topics in Analtic Geometr You can use the formula for the tangent of the difference of two angles tan tan tan tan tan tan to obtain the formula for the angle between two lines. Angle Between Two Lines If two nonperpendicular lines have slopes m and m, the angle between the two lines is tan m m m m. Eample Finding the Angle Between Two Lines FIGURE = 0 θ = 0 Find the angle between the two lines. Line : 0 Line : 0 The two lines have slopes of m and m, respectivel. So, the tangent of the angle between the two lines is tan m. m m m Finall, ou can conclude that the angle is arctan as shown in Figure radians Now tr Eercise. The Distance Between a Point and a Line Finding the distance between a line and a point not on the line is an application of perpendicular lines. This distance is defined as the length of the perpendicular line segment joining the point and the line, as shown in Figure 0.6. (, ) (, ) d Distance Between a Point and a Line The distance between the point, and the line A B C 0 is d A B C A B. FIGURE 0.6 Remember that the values of A, B, and C in this distance formula correspond to the general equation of a line, A B C 0. For a proof of this formula for the distance between a point and a line, see Proofs in Mathematics on page 80.

5 Section 0. Lines 79 = + (, ) 5 FIGURE 0.7 Eample Finding the Distance Between a Point and a Line Find the distance between the point, and the line. The general form of the equation is 0. So, the distance between the point and the line is d 8.58 units. 5 The line and the point are shown in Figure 0.7. Now tr Eercise 5. Eample 5 An Application of Two Distance Formulas 6 5 B (0, ) h C (5, ) A (, 0) 5 FIGURE 0.8 Figure 0.8 shows a triangle with vertices A, 0, B0,, and C5,. a. Find the altitude h from verte B to side AC. b. Find the area of the triangle. a. To find the altitude, use the formula for the distance between line AC and the point 0,. The equation of line AC is obtained as follows. Slope: Equation: m Point-slope form Multipl each side b. 0 General form So, the distance between this line and the point 0, is Altitude h 0 7 units. b. Using the formula for the distance between two points, ou can find the length of the base AC to be b 5 0 Distance Formula 8 Simplif. 7 units. Simplif. Finall, the area of the triangle in Figure 0.8 is A bh 7 7 square units. Now tr Eercise 59. Formula for the area of a triangle Substitute for b and h. Simplif.

6 70 Chapter 0 Topics in Analtic Geometr 0. EXERCISES VOCABULARY: Fill in the blanks. See for worked-out solutions to odd-numbered eercises.. The of a nonhorizontal line is the positive angle (less than ) measured counterclockwise from the -ais to the line.. If a nonvertical line has inclination and slope m, then m.. If two nonperpendicular lines have slopes m and m, the angle between the two lines is tan.. The distance between the point, and the line A B C 0 is given b d. SKILLS AND APPLICATIONS In Eercises 5, find the slope of the line with inclination θ = 6 θ = ,, 0,,, 6, 6,, 0, 8, 8,,, 0, 0, 0 0, 00, 50, radians radians. radians. radians.7.88 θ = In Eercises 8, find the inclination degrees) of the line with a slope of m.. m. m 5. m 6. m 7. m 8. m 5 θ = (in radians and In Eercises 7 6, find the inclination degrees) of the line (in radians and In Eercises 7 6, find the angle (in radians and degrees) between the lines θ θ In Eercises 9 6, find the inclination (in radians and degrees) of the line passing through the points. 9.,, 0, 0.,, 0,

7 Section 0. Lines θ θ Point 5., 5., 55. 6, 56., 57. 0, 8 58., Line In Eercises 59 6, the points represent the vertices of a triangle. (a) Draw triangle ABC in the coordinate plane, (b) find the altitude from verte B of the triangle to side AC, and (c) find the area of the triangle. 59. A 0, 0, B,, C, A 0, 0, B, 5, C 5, 6. A B,, C 5, 0,, 6. A, 5, B, 0, C 6, In Eercises 6 and 6, find the distance between the parallel lines ANGLE MEASUREMENT In Eercises 7 50, find the slope of each side of the triangle and use the slopes to find the measures of the interior angles (, 8) (, 5) (, 5) (, ) (, ) (6, ) ROAD GRADE A straight road rises with an inclination of 0.0 radian from the horizontal (see figure). Find the slope of the road and the change in elevation over a two-mile stretch of the road (, ) (, 0) (, ) (, ) (, ) (, ) mi 0. radian In Eercises 5 58, find the distance between the point and the line. Point 5. 0, , 0 Line ROAD GRADE A straight road rises with an inclination of 0.0 radian from the horizontal. Find the slope of the road and the change in elevation over a one-mile stretch of the road.

8 7 Chapter 0 Topics in Analtic Geometr 67. PITCH OF A ROOF A roof has a rise of feet for ever horizontal change of 5 feet (see figure). Find the inclination of the roof. 68. CONVEYOR DESIGN A moving conveor is built so that it rises meter for each meters of horizontal travel. (a) Draw a diagram that gives a visual representation of the problem. (b) Find the inclination of the conveor. (c) The conveor runs between two floors in a factor. The distance between the floors is 5 meters. Find the length of the conveor. 69. TRUSS Find the angles and shown in the drawing of the roof truss. 70. The Falls Incline Railwa in Niagara Falls, Ontario, Canada is an inclined railwa that was designed to carr people from the Cit of Niagara Falls to Queen Victoria Park. The railwa is approimatel 70 feet long with a 6% uphill grade (see figure). θ 5 ft ft 6 ft 70 ft α β 9 ft Not drawn to scale 6 ft 6 ft (c) Using the origin of a rectangular coordinate sstem as the base of the inclined plane, find the equation of the line that models the railwa track. (d) Sketch a graph of the equation ou found in part (c). EXPLORATION TRUE OR FALSE? In Eercises 7 and 7, determine whether the statement is true or false. Justif our answer. 7. A line that has an inclination greater than radians has a negative slope. 7. To find the angle between two lines whose angles of inclination and are known, substitute and for m and m, respectivel, in the formula for the angle between two lines. 7. Consider a line with slope m and -intercept 0,. (a) Write the distance d between the origin and the line as a function of m. (b) Graph the function in part (a). (c) Find the slope that ields the maimum distance between the origin and the line. (d) Find the asmptote of the graph in part (b) and interpret its meaning in the contet of the problem. 7. CAPSTONE Discuss wh the inclination of a line can be an angle that is larger than, but the angle between two lines cannot be larger than. Decide whether the following statement is true or false: The inclination of a line is the angle between the line and the -ais. Eplain. 75. Consider a line with slope m and -intercept 0,. (a) Write the distance d between the point, and the line as a function of m. (b) Graph the function in part (a). (c) Find the slope that ields the maimum distance between the point and the line. (d) Is it possible for the distance to be 0? If so, what is the slope of the line that ields a distance of 0? (e) Find the asmptote of the graph in part (b) and interpret its meaning in the contet of the problem. (a) Find the inclination of the railwa. (b) Find the change in elevation from the base to the top of the railwa.

9 Section 0. Introduction to Conics: Parabolas 7 0. INTRODUCTION TO CONICS: PARABOLAS What ou should learn Recognize a conic as the intersection of a plane and a double-napped cone. Write equations of parabolas in standard form and graph parabolas. Use the reflective propert of parabolas to solve real-life problems. Wh ou should learn it Parabolas can be used to model and solve man tpes of real-life problems. For instance, in Eercise 7 on page 79, a parabola is used to model the cables of the Golden Gate Bridge. Conics Conic sections were discovered during the classical Greek period, 600 to 00 B.C. The earl Greeks were concerned largel with the geometric properties of conics. It was not until the 7th centur that the broad applicabilit of conics became apparent and plaed a prominent role in the earl development of calculus. A conic section (or simpl conic) is the intersection of a plane and a doublenapped cone. Notice in Figure 0.9 that in the formation of the four basic conics, the intersecting plane does not pass through the verte of the cone. When the plane does pass through the verte, the resulting figure is a degenerate conic, as shown in Figure 0.0. Cosmo Condina/The Image Bank/ Gett Images Circle Ellipse Parabola Hperbola FIGURE 0.9 Basic Conics Point Line Two Intersecting FIGURE 0.0 Degenerate Conics Lines There are several was to approach the stud of conics. You could begin b defining conics in terms of the intersections of planes and cones, as the Greeks did, or ou could define them algebraicall, in terms of the general second-degree equation A B C D E F 0. However, ou will stud a third approach, in which each of the conics is defined as a locus (collection) of points satisfing a geometric propert. For eample, in Section., ou learned that a circle is defined as the collection of all points, that are equidistant from a fied point h, k. This leads to the standard form of the equation of a circle h k r. Equation of circle

10 7 Chapter 0 Topics in Analtic Geometr Parabolas In Section., ou learned that the graph of the quadratic function f a b c is a parabola that opens upward or downward. The following definition of a parabola is more general in the sense that it is independent of the orientation of the parabola. Definition of Parabola A parabola is the set of all points, in a plane that are equidistant from a fied line (directri) and a fied point (focus) not on the line. Directri FIGURE 0. Focus Verte Parabola d d d d The midpoint between the focus and the directri is called the verte, and the line passing through the focus and the verte is called the ais of the parabola. Note in Figure 0. that a parabola is smmetric with respect to its ais. Using the definition of a parabola, ou can derive the following standard form of the equation of a parabola whose directri is parallel to the -ais or to the -ais. Standard Equation of a Parabola The standard form of the equation of a parabola with verte at h, k is as follows. h p k, p 0 Vertical ais, directri: k p k p h, p 0 Horizontal ais, directri: h p The focus lies on the ais p units (directed distance) from the verte. If the verte is at the origin 0, 0, the equation takes one of the following forms. p Vertical ais p Horizontal ais See Figure 0.. For a proof of the standard form of the equation of a parabola, see Proofs in Mathematics on page 805. p > 0 Verte: ( h, k) Ais: = h Focus: ( hk, + p) Directri: = k p p < 0 Ais: = h Directri: = k p Verte: (h, k) Focus: (h, k + p) Directri: = h p p > 0 Verte: ( h, k) Focus: ( h + p, k) Ais: =k= Focus: (h + p, k) Directri: = h p p < 0 Verte: (h, k) Ais: = k (a) h p k Vertical ais: p > 0 FIGURE 0. (b) h p k Vertical ais: p < 0 (c) k p h Horizontal ais: p > 0 (d) k p h Horizontal ais: p < 0

11 Section 0. Introduction to Conics: Parabolas 75 TECHNOLOGY Use a graphing utilit to confirm the equation found in Eample. In order to graph the equation, ou ma have to use two separate equations: and 8 8. Upper part Lower part Eample Verte at the Origin Find the standard equation of the parabola with verte at the origin and focus, 0. The ais of the parabola is horizontal, passing through 0, 0 and, 0, as shown in Figure 0.. Verte = 8 Focus (, 0) (0, 0) FIGURE 0. The standard form is p, where h 0, k 0, and p. So, the equation is 8. Now tr Eercise. Eample Finding the Focus of a Parabola The technique of completing the square is used to write the equation in Eample in standard form. You can review completing the square in Appendi A.5. Verte (, ) Focus(, ) = + FIGURE 0. Find the focus of the parabola given b. To find the focus, convert to standard form b completing the square. Write original equation. Multipl each side b. Add to each side. Complete the square. Combine like terms. Standard form Comparing this equation with h p k ou can conclude that h, k, and p. Because p is negative, the parabola opens downward, as shown in Figure 0.. So, the focus of the parabola is h, k p,. Now tr Eercise.

12 76 Chapter 0 Topics in Analtic Geometr Eample Finding the Standard Equation of a Parabola 8 6 FIGURE 0.5 Light source at focus FIGURE 0.6 Focus Focus ( ) = ( ) Focus (, ) Verte (, ) 6 8 Ais Parabolic reflector: Light is reflected in parallel ras. α Ais P Find the standard form of the equation of the parabola with verte, and focus,. Then write the quadratic form of the equation. Because the ais of the parabola is vertical, passing through, consider the equation h p k where h, k, and p. So, the standard form is. You can obtain the more common quadratic form as follows. 6 Write original equation. Multipl. Add to each side. The graph of this parabola is shown in Figure 0.5. Application 6 Now tr Eercise 55. Divide each side b. A line segment that passes through the focus of a parabola and has endpoints on the parabola is called a focal chord. The specific focal chord perpendicular to the ais of the parabola is called the latus rectum. Parabolas occur in a wide variet of applications. For instance, a parabolic reflector can be formed b revolving a parabola around its ais. The resulting surface has the propert that all incoming ras parallel to the ais are reflected through the focus of the parabola. This is the principle behind the construction of the parabolic mirrors used in reflecting telescopes. Conversel, the light ras emanating from the focus of a parabolic reflector used in a flashlight are all parallel to one another, as shown in Figure 0.6. A line is tangent to a parabola at a point on the parabola if the line intersects, but does not cross, the parabola at the point. Tangent lines to parabolas have special properties related to the use of parabolas in constructing reflective surfaces. and,, FIGURE 0.7 α Tangent line Reflective Propert of a Parabola The tangent line to a parabola at a point P makes equal angles with the following two lines (see Figure 0.7).. The line passing through P and the focus. The ais of the parabola

13 Section 0. Introduction to Conics: Parabolas 77 = 0, d α FIGURE 0.8 ( ) d α (0, b) TECHNOLOGY (, ) Use a graphing utilit to confirm the result of Eample. B graphing and in the same viewing window, ou should be able to see that the line touches the parabola at the point,. Eample Finding the Tangent Line at a Point on a Parabola Find the equation of the tangent line to the parabola given b at the point,. For this parabola, p and the focus is 0,, as shown in Figure 0.8. You can find the -intercept 0, b of the tangent line b equating the lengths of the two sides of the isosceles triangle shown in Figure 0.8: and d b d 0 5. Note that d rather than b b. The order of subtraction for the distance is important because the distance must be positive. Setting d d produces b 5 So, the slope of the tangent line is m b. 0 and the equation of the tangent line in slope-intercept form is. Now tr Eercise 65. You can review techniques for writing linear equations in Section.. CLASSROOM DISCUSSION Satellite Dishes Cross sections of satellite dishes are parabolic in shape. Use the figure shown to write a paragraph eplaining wh satellite dishes are parabolic. Amplifier Dish reflector Cable to radio or TV

14 78 Chapter 0 Topics in Analtic Geometr 0. EXERCISES VOCABULARY: Fill in the blanks.. A is the intersection of a plane and a double-napped cone. See for worked-out solutions to odd-numbered eercises.. When a plane passes through the verte of a double-napped cone, the intersection is a.. A collection of points satisfing a geometric propert can also be referred to as a of points.. A is defined as the set of all points, in a plane that are equidistant from a fied line, called the, and a fied point, called the, not on the line. 5. The line that passes through the focus and the verte of a parabola is called the of the parabola. 6. The of a parabola is the midpoint between the focus and the directri. 7. A line segment that passes through the focus of a parabola and has endpoints on the parabola is called a. 8. A line is to a parabola at a point on the parabola if the line intersects, but does not cross, the parabola at the point. SKILLS AND APPLICATIONS In Eercises 9, describe in words how a plane could intersect with the double-napped cone shown to form the conic section. (e) (f) 6 9. Circle 0. Ellipse. Parabola. Hperbola In Eercises 8, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (b) (c) (d) In Eercises 9, find the standard form of the equation of the parabola with the given characteristic(s) and verte at the origin (, 6) 8 (, 6). Focus:. Focus:,0. Focus:, 0. Focus: 0, 5. Directri: 6. Directri: 7. Directri: 8. Directri: 9. Vertical ais and passes through the point, 6 0. Vertical ais and passes through the point,. Horizontal ais and passes through the point, 5. Horizontal ais and passes through the point, 8 8

15 Section 0. Introduction to Conics: Parabolas 79 In Eercises 6, find the verte, focus, and directri of the parabola, and sketch its graph In Eercises 7 50, find the verte, focus, and directri of the parabola. Use a graphing utilit to graph the parabola In Eercises 5 60, find the standard form of the equation of the parabola with the given characteristics (, 0) (.5, ) (, ) (, 0) 8 6 (0, ) Verte:, ; focus: 6, 56. Verte:, ; focus:, Verte: 0, ; directri: 58. Verte:, ; directri: 59. Focus:, ; directri: 60. Focus: 0, 0; directri: 8 8 (, ) In Eercises 6 and 6, change the equation of the parabola so that its graph matches the description ; upper half of parabola 6. ; lower half of parabola 8 (0, 0) (5, ) In Eercises 6 and 6, the equations of a parabola and a tangent line to the parabola are given. Use a graphing utilit to graph both equations in the same viewing window. Determine the coordinates of the point of tangenc. Parabola Tangent Line 0 0 In Eercises 65 68, find an equation of the tangent line to the parabola at the given point, and find the -intercept of the line. 65.,, 8 66.,, 9 67.,, 68.,, REVENUE The revenue R (in dollars) generated b the sale of units of a patio furniture set is given b 06 R,05. 5 Use a graphing utilit to graph the function and approimate the number of sales that will maimize revenue. 70. REVENUE The revenue R (in dollars) generated b the sale of units of a digital camera is given b 5 5 R 5,55. 7 Use a graphing utilit to graph the function and approimate the number of sales that will maimize revenue. 7. SUSPENSION BRIDGE Each cable of the Golden Gate Bridge is suspended (in the shape of a parabola) between two towers that are 80 meters apart. The top of each tower is 5 meters above the roadwa. The cables touch the roadwa midwa between the towers. (a) Draw a sketch of the bridge. Locate the origin of a rectangular coordinate sstem at the center of the roadwa. Label the coordinates of the known points. (b) Write an equation that models the cables. (c) Complete the table b finding the height of the suspension cables over the roadwa at a distance of meters from the center of the bridge. Distance, Height,

16 70 Chapter 0 Topics in Analtic Geometr 7. SATELLITE DISH The receiver in a parabolic satellite dish is.5 feet from the verte and is located at the focus (see figure). Write an equation for a cross section of the reflector. (Assume that the dish is directed upward and the verte is at the origin.) Receiver.5 ft 75. BEAM DEFLECTION A simpl supported beam is meters long and has a load at the center (see figure). The deflection of the beam at its center is centimeters. Assume that the shape of the deflected beam is parabolic. (a) Write an equation of the parabola. (Assume that the origin is at the center of the deflected beam.) (b) How far from the center of the beam is the deflection equal to centimeter? cm 7. ROAD DESIGN Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road that is feet wide is 0. foot higher in the center than it is on the sides (see figure). Cross section of road surface (a) Find an equation of the parabola that models the road surface. (Assume that the origin is at the center of the road.) (b) How far from the center of the road is the road surface 0. foot lower than in the middle? 7. HIGHWAY DESIGN Highwa engineers design a parabolic curve for an entrance ramp from a straight street to an interstate highwa (see figure). Find an equation of the parabola Interstate ft (000, 800) ft Not drawn to scale 76. BEAM DEFLECTION Repeat Eercise 75 if the length of the beam is 6 meters and the deflection of the beam at the center is centimeters. 77. FLUID FLOW Water is flowing from a horizontal pipe 8 feet above the ground. The falling stream of water has the shape of a parabola whose verte 0, 8 is at the end of the pipe (see figure). The stream of water strikes the ground at the point 0, 0. Find the equation of the path taken b the water ft (, 6) (, 6) FIGURE FOR 77 FIGURE FOR 78 m 6 (0, 6) LATTICE ARCH A parabolic lattice arch is 6 feet high at the verte. At a height of 6 feet, the width of the lattice arch is feet (see figure). How wide is the lattice arch at ground level? 79. SATELLITE ORBIT A satellite in a 00-mile-high circular orbit around Earth has a velocit of approimatel 7,500 miles per hour. If this velocit is multiplied b, the satellite will have the minimum velocit necessar to escape Earth s gravit and it will follow a parabolic path with the center of Earth as the focus (see figure on the net page). Not drawn to scale 800 (000, 800) Street

17 Section 0. Circular orbit 00 miles Parabolic path 7 Introduction to Conics: Parabolas 85. Let, be the coordinates of a point on the parabola p. The equation of the line tangent to the parabola at the point is. p What is the slope of the tangent line? Not drawn to scale FIGURE FOR 79 (a) Find the escape velocit of the satellite. (b) Find an equation of the parabolic path of the satellite (assume that the radius of Earth is 000 miles). 80. PATH OF A SOFTBALL The path of a softball is modeled b , where the coordinates and are measured in feet, with 0 corresponding to the position from which the ball was thrown. (a) Use a graphing utilit to graph the trajector of the softball. (b) Use the trace feature of the graphing utilit to approimate the highest point and the range of the trajector. PROJECTILE MOTION In Eercises 8 and 8, consider the path of a projectile projected horizontall with a velocit of v feet per second at a height of s feet, where the model for the path is ⴝ ⴚ v ⴚ s CAPSTONE Eplain what each of the following equations represents, and how equations (a) and (b) are equivalent. (a) a h k, a 0 (b) h p k, p 0 (c) k p h, p GRAPHICAL REASONING p. (a) Use a graphing utilit to graph the parabola for p, p, p, and p. Describe the effect on the graph when p increases. (b) Locate the focus for each parabola in part (a). (c) For each parabola in part (a), find the length of the latus rectum (see figure). How can the length of the latus rectum be determined directl from the standard form of the equation of the parabola? Latus rectum Focus = p In this model (in which air resistance is disregarded), is the height (in feet) of the projectile and is the horizontal distance (in feet) the projectile travels. 8. A ball is thrown from the top of a 00-foot tower with a velocit of 8 feet per second. (a) Find the equation of the parabolic path. (b) How far does the ball travel horizontall before striking the ground? 8. A cargo plane is fling at an altitude of 0,000 feet and a speed of 50 miles per hour. A suppl crate is dropped from the plane. How man feet will the crate travel horizontall before it hits the ground? Consider the parabola (d) Eplain how the result of part (c) can be used as a sketching aid when graphing parabolas. 88. GEOMETRY The area of the shaded region in the 8 figure is A p b. = p =b EXPLORATION TRUE OR FALSE? In Eercises 8 and 8, determine whether the statement is true or false. Justif our answer. 8. It is possible for a parabola to intersect its directri. 8. If the verte and focus of a parabola are on a horizontal line, then the directri of the parabola is vertical. (a) Find the area when p and b. (b) Give a geometric eplanation of wh the area approaches 0 as p approaches 0.

18 7 Chapter 0 Topics in Analtic Geometr 0. ELLIPSES What ou should learn Write equations of ellipses in standard form and graph ellipses. Use properties of ellipses to model and solve real-life problems. Find eccentricities of ellipses. Wh ou should learn it Ellipses can be used to model and solve man tpes of real-life problems. For instance, in Eercise 65 on page 79, an ellipse is used to model the orbit of Halle s comet. Introduction The second tpe of conic is called an ellipse, and is defined as follows. Definition of Ellipse An ellipse is the set of all points, in a plane, the sum of whose distances from two distinct fied points (foci) is constant. See Figure 0.9. Focus d (, ) d Focus Major ais Verte Center Minor ais Verte Harvard College Observator/Photo Researchers, Inc. d d is constant. FIGURE 0.9 FIGURE 0.0 The line through the foci intersects the ellipse at two points called vertices. The chord joining the vertices is the major ais, and its midpoint is the center of the ellipse. The chord perpendicular to the major ais at the center is the minor ais of the ellipse. See Figure 0.0. You can visualize the definition of an ellipse b imagining two thumbtacks placed at the foci, as shown in Figure 0.. If the ends of a fied length of string are fastened to the thumbtacks and the string is drawn taut with a pencil, the path traced b the pencil will be an ellipse. (, ) FIGURE 0. b + c b + c ( h, k) b To derive the standard form of the equation of an ellipse, consider the ellipse in Figure 0. with the following points: center, h, k; vertices, h ± a, k; foci, h ± c, k. Note that the center is the midpoint of the segment joining the foci. The sum of the distances from an point on the ellipse to the two foci is constant. Using a verte point, this constant sum is b + c = a b + c = a FIGURE 0. c a a c a c a Length of major ais or simpl the length of the major ais. Now, if ou let, be an point on the ellipse, the sum of the distances between, and the two foci must also be a.

19 Section 0. Ellipses 7 That is, h c k h c k a which, after epanding and regrouping, reduces to a c h a k a a c. Finall, in Figure 0., ou can see that b a c which implies that the equation of the ellipse is b h a k a b h a k b. You would obtain a similar equation in the derivation b starting with a vertical major ais. Both results are summarized as follows. Consider the equation of the ellipse h a k b. If ou let a b, then the equation can be rewritten as h k a which is the standard form of the equation of a circle with radius r a (see Section.). Geometricall, when a b for an ellipse, the major and minor aes are of equal length, and so the graph is a circle. Standard Equation of an Ellipse The standard form of the equation of an ellipse, with center h, k and major and minor aes of lengths a and b, respectivel, where 0 < b < a, is h h Figure 0. shows both the horizontal and vertical orientations for an ellipse. a b Major ais is horizontal. Major ais is vertical. The foci lie on the major ais, c units from the center, with c a b. If the center is at the origin 0, 0, the equation takes one of the following forms. a b k b k a. Major ais is horizontal. b a Major ais is vertical. ( h) ( k) + = a b ( h) ( k) + = b a (h, k) b (h, k) a a b Major ais is horizontal. FIGURE 0. Major ais is vertical.

20 7 Chapter 0 Topics in Analtic Geometr Eample Finding the Standard Equation of an Ellipse FIGURE 0. (0, ) (, ) (, ) a = b = 5 Find the standard form of the equation of the ellipse having foci at 0, and, and a major ais of length 6, as shown in Figure 0.. Because the foci occur at 0, and,, the center of the ellipse is, ) and the distance from the center to one of the foci is c. Because a 6, ou know that a. Now, from c a b, ou have b a c 5. Because the major ais is horizontal, the standard equation is. 5 This equation simplifies to 9 5. Now tr Eercise. Eample Sketching an Ellipse ( + ) ( ) + = ( 5, ) (, ) (, ) (, ) (, ) ( +, ) 5 (, 0) FIGURE 0.5 Sketch the ellipse given b Begin b writing the original equation in standard form. In the fourth step, note that 9 and are added to both sides of the equation when completing the squares Write original equation. Group terms Factor out of -terms. Write in completed square form. Divide each side b. Write in standard form. From this standard form, it follows that the center is h, k,. Because the denominator of the -term is a, the endpoints of the major ais lie two units to the right and left of the center. Similarl, because the denominator of the -term is b, the endpoints of the minor ais lie one unit up and down from the center. Now, from c a b, ou have c. So, the foci of the ellipse are, and,. The ellipse is shown in Figure 0.5. Now tr Eercise 7.

21 Section 0. Ellipses 75 Eample Analzing an Ellipse ( ) ( + ) + = (, + ( (, ), FIGURE 0.6 Verte ( ( Focus Center Focus Verte (, ) (, 6) Find the center, vertices, and foci of the ellipse B completing the square, ou can write the original equation in standard form Write original equation. Group terms. The major ais is vertical, where h, k, a, b, and c a b 6. So, ou have the following Center:, Vertices:, 6 Foci: The graph of the ellipse is shown in Figure 0.6. Now tr Eercise 5. 6, Factor out of -terms. Write in completed square form. Divide each side b 6. Write in standard form.,, TECHNOLOGY You can use a graphing utilit to graph an ellipse b graphing the upper and lower portions in the same viewing window. For instance, to graph the ellipse in Eample, first solve for to get and Use a viewing window in which 6 9 and 7. You should obtain the graph shown below

22 76 Chapter 0 Topics in Analtic Geometr Application Ellipses have man practical and aesthetic uses. For instance, machine gears, supporting arches, and acoustic designs often involve elliptical shapes. The orbits of satellites and planets are also ellipses. Eample investigates the elliptical orbit of the moon about Earth. Eample An Application Involving an Elliptical Orbit 767,60 km Moon The moon travels about Earth in an elliptical orbit with Earth at one focus, as shown in Figure 0.7. The major and minor aes of the orbit have lengths of 768,800 kilometers and 767,60 kilometers, respectivel. Find the greatest and smallest distances (the apogee and perigee, respectivel) from Earth s center to the moon s center. Earth Perigee FIGURE ,800 km Apogee Because a 768,800 and b 767,60, ou have a 8,00 and b 8,80 which implies that c a b 8,00 8,80,08. So, the greatest distance between the center of Earth and the center of the moon is WARNING / CAUTION Note in Eample and Figure 0.7 that Earth is not the center of the moon s orbit. a c 8,00,08 05,508 kilometers and the smallest distance is a c 8,00,08 6,9 kilometers. Now tr Eercise 65. Eccentricit One of the reasons it was difficult for earl astronomers to detect that the orbits of the planets are ellipses is that the foci of the planetar orbits are relativel close to their centers, and so the orbits are nearl circular. To measure the ovalness of an ellipse, ou can use the concept of eccentricit. Definition of Eccentricit The eccentricit e of an ellipse is given b the ratio e c a. Note that 0 < e < for ever ellipse.

23 Section 0. Ellipses 77 To see how this ratio is used to describe the shape of an ellipse, note that because the foci of an ellipse are located along the major ais between the vertices and the center, it follows that 0 < c < a. For an ellipse that is nearl circular, the foci are close to the center and the ratio ca is small, as shown in Figure 0.8. On the other hand, for an elongated ellipse, the foci are close to the vertices and the ratio ca is close to, as shown in Figure 0.9. Foci Foci e = c a c e is small. e = c a c e is close to. a FIGURE 0.8 FIGURE 0.9 a NASA The time it takes Saturn to orbit the sun is about 9. Earth ears. The orbit of the moon has an eccentricit of e 0.059, and the eccentricities of the eight planetar orbits are as follows. Mercur: e Jupiter: e 0.08 Venus: e Saturn: e 0.05 Earth: e Uranus: e 0.07 Mars: e 0.09 Neptune: e CLASSROOM DISCUSSION Ellipses and Circles a. Show that the equation of an ellipse can be written as h a k. a e b. For the equation in part (a), let a, h, and k, and use a graphing utilit to graph the ellipse for e 0.95, e 0.75, e 0.5, e 0.5, and e 0.. Discuss the changes in the shape of the ellipse as e approaches 0. c. Make a conjecture about the shape of the graph in part (b) when e 0. What is the equation of this ellipse? What is another name for an ellipse with an eccentricit of 0?

24 78 Chapter 0 Topics in Analtic Geometr 0. EXERCISES VOCABULARY: Fill in the blanks. See for worked-out solutions to odd-numbered eercises.. An is the set of all points, in a plane, the sum of whose distances from two distinct fied points, called, is constant.. The chord joining the vertices of an ellipse is called the, and its midpoint is the of the ellipse.. The chord perpendicular to the major ais at the center of the ellipse is called the of the ellipse.. The concept of is used to measure the ovalness of an ellipse. SKILLS AND APPLICATIONS In Eercises 5 0, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (b) (c) (e) (d) (f) In Eercises 8, find the standard form of the equation of the ellipse with the given characteristics and center at the origin Vertices: ±7, 0; foci: ±, 0. Vertices: 0, ±8; foci: 0, ± 5. Foci: ±5, 0; major ais of length 6. Foci: ±, 0; major ais of length 0 7. Vertices: 0, ±5; passes through the point, 8. Vertical major ais; passes through the points 0, 6 and, 0 In Eercises 9 8, find the standard form of the equation of the ellipse with the given characteristics (, ) 6 (, 6) (, 0) 5 8 (0, ) (, 0) (, 0) 8 (0, ) (, ) (, 0) 5 6 (0, ) (, ) (, ). Vertices: 0,, 8, ; minor ais of length. Foci: 0, 0,, 0; major ais of length 6. Foci: 0, 0, 0, 8; major ais of length 6. Center:, ; verte:, ; minor ais of length 5. Center: 0, ; a c; vertices:,,, 6. Center:, ; a c; foci:,, 5, ( 0, ) (, 0) (, 0) ( 0, )

25 Section 0. Ellipses Vertices: 0,,, ; endpoints of the minor ais:,,, 8. Vertices: 5, 0, 5, ; endpoints of the minor ais:, 6, 9, 6 In Eercises 9 5, identif the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricit of the conic (if applicable), and sketch its graph In Eercises 5 56, use a graphing utilit to graph the ellipse. Find the center, foci, and vertices. (Recall that it ma be necessar to solve the equation for and obtain two equations.) In Eercises 57 60, find the eccentricit of the ellipse Find an equation of the ellipse with vertices ±5, 0 and eccentricit e Find an equation of the ellipse with vertices 0, ±8 and eccentricit e. 6. ARCHITECTURE A semielliptical arch over a tunnel for a one-wa road through a mountain has a major ais of 50 feet and a height at the center of 0 feet. (a) Draw a rectangular coordinate sstem on a sketch of the tunnel with the center of the road entering the tunnel at the origin. Identif the coordinates of the known points. (b) Find an equation of the semielliptical arch. (c) You are driving a moving truck that has a width of 8 feet and a height of 9 feet. Will the moving truck clear the opening of the arch? 6. ARCHITECTURE A fireplace arch is to be constructed in the shape of a semiellipse. The opening is to have a height of feet at the center and a width of 6 feet along the base (see figure). The contractor draws the outline of the ellipse using tacks as described at the beginning of this section. Determine the required positions of the tacks and the length of the string. 65. COMET ORBIT Halle s comet has an elliptical orbit, with the sun at one focus. The eccentricit of the orbit is approimatel The length of the major ais of the orbit is approimatel 5.88 astronomical units. (An astronomical unit is about 9 million miles.) (a) Find an equation of the orbit. Place the center of the orbit at the origin, and place the major ais on the -ais. (b) Use a graphing utilit to graph the equation of the orbit. (c) Find the greatest (aphelion) and smallest (perihelion) distances from the sun s center to the comet s center.

26 750 Chapter 0 Topics in Analtic Geometr 66. SATELLITE ORBIT The first artificial satellite to orbit Earth was Sputnik I (launched b the former Soviet Union in 957). Its highest point above Earth s surface was 97 kilometers, and its lowest point was 8 kilometers (see figure). The center of Earth was at one focus of the elliptical orbit, and the radius of Earth is 678 kilometers. Find the eccentricit of the orbit. EXPLORATION TRUE OR FALSE? In Eercises 7 and 7, determine whether the statement is true or false. Justif our answer. 7. The graph of 0 is an ellipse. 7. It is easier to distinguish the graph of an ellipse from the graph of a circle if the eccentricit of the ellipse is large (close to ). 75. Consider the ellipse, a b 0. a b Focus 97 km 8 km 67. MOTION OF A PENDULUM The relation between the velocit (in radians per second) of a pendulum and its angular displacement from the vertical can be modeled b a semiellipse. A -centimeter pendulum crests 0 when the angular displacement is 0. radian and 0. radian. When the pendulum is at equilibrium 0, the velocit is.6 radians per second. (a) Find an equation that models the motion of the pendulum. Place the center at the origin. (b) Graph the equation from part (a). (c) Which half of the ellipse models the motion of the pendulum? 68. GEOMETRY A line segment through a focus of an ellipse with endpoints on the ellipse and perpendicular to the major ais is called a latus rectum of the ellipse. Therefore, an ellipse has two latera recta. Knowing the length of the latera recta is helpful in sketching an ellipse because it ields other points on the curve (see figure). Show that the length of each latus rectum is b a. Latera recta F A (d) Use a graphing utilit to graph the area function and use the graph to support our conjecture in part (c). 76. THINK ABOUT IT At the beginning of this section it was noted that an ellipse can be drawn using two thumbtacks, a string of fied length (greater than the distance between the two tacks), and a pencil. If the ends of the string are fastened at the tacks and the string is drawn taut with a pencil, the path traced b the pencil is an ellipse. (a) What is the length of the string in terms of a? (b) Eplain wh the path is an ellipse. 77. THINK ABOUT IT Find the equation of an ellipse such that for an point on the ellipse, the sum of the distances from the point, and 0, is CAPSTONE Describe the relationship between circles and ellipses. How are the similar? How do the differ? a F In Eercises 69 7, sketch the graph of the ellipse, using latera recta (see Eercise 68). (a) The area of the ellipse is given b A ab. Write the area of the ellipse as a function of a. (b) Find the equation of an ellipse with an area of 6 square centimeters. (c) Complete the table using our equation from part (a), and make a conjecture about the shape of the ellipse with maimum area. 79. PROOF Show that a b c for the ellipse a b where a > 0, b > 0, and the distance from the center of the ellipse 0, 0 to a focus is c.

27 Section 0. Hperbolas HYPERBOLAS What ou should learn Write equations of hperbolas in standard form. Find asmptotes of and graph hperbolas. Use properties of hperbolas to solve real-life problems. Classif conics from their general equations. Wh ou should learn it Hperbolas can be used to model and solve man tpes of real-life problems. For instance, in Eercise 5 on page 759, hperbolas are used in long distance radio navigation for aircraft and ships. Introduction The third tpe of conic is called a hperbola. The definition of a hperbola is similar to that of an ellipse. The difference is that for an ellipse the sum of the distances between the foci and a point on the ellipse is fied, whereas for a hperbola the difference of the distances between the foci and a point on the hperbola is fied. Definition of Hperbola A hperbola is the set of all points, in a plane, the difference of whose distances from two distinct fied points (foci) is a positive constant. See Figure 0.0. Focus d (, ) d Focus Branch Verte Center c a Transverse ais Verte Branch d d is a positive constant. FIGURE 0.0 FIGURE 0. U.S. Nav, William Lipski/AP Photo The graph of a hperbola has two disconnected branches. The line through the two foci intersects the hperbola at its two vertices. The line segment connecting the vertices is the transverse ais, and the midpoint of the transverse ais is the center of the hperbola. See Figure 0.. The development of the standard form of the equation of a hperbola is similar to that of an ellipse. Note in the definition below that a, b, and c are related differentl for hperbolas than for ellipses. Standard Equation of a Hperbola The standard form of the equation of a hperbola with center h, k is h a k a Transverse ais is horizontal. Transverse ais is vertical. The vertices are a units from the center, and the foci are c units from the center. Moreover, c a b. If the center of the hperbola is at the origin 0, 0, the equation takes one of the following forms. a b k b h b. Transverse ais is horizontal. a b Transverse ais is vertical.

28 75 Chapter 0 Topics in Analtic Geometr Figure 0. shows both the horizontal and vertical orientations for a hperbola. ( h) ( k) = a b ( k) ( h) = a b ( hk, + c) ( h c, k) ( hk, ) ( h + c, k) ( hk, ) Transverse ais is horizontal. FIGURE 0. ( hk, c) Transverse ais is vertical. Eample Finding the Standard Equation of a Hperbola When finding the standard form of the equation of an conic, it is helpful to sketch a graph of the conic with the given characteristics. Find the standard form of the equation of the hperbola with foci, and 5, and vertices 0, and,. B the Midpoint Formula, the center of the hperbola occurs at the point,. Furthermore, c 5 and a, and it follows that b c a 9 5. So, the hperbola has a horizontal transverse ais and the standard form of the equation is 5. This equation simplifies to See Figure ( ) ( ) = ( 5 ( 5 (0, ) (, ) (, ) (, ) (5, ) FIGURE 0. Now tr Eercise 5.

29 Section 0. Hperbolas 75 Conjugate ais (h, k + b) Asmptote Asmptotes of a Hperbola Each hperbola has two asmptotes that intersect at the center of the hperbola, as shown in Figure 0.. The asmptotes pass through the vertices of a rectangle of dimensions a b b, with its center at h, k. The line segment of length b joining h, k b and h, k b or h b, k and h b, k is the conjugate ais of the hperbola. FIGURE 0. (h, k) (h, k b) (h a, k) (h + a, k) Asmptote Asmptotes of a Hperbola The equations of the asmptotes of a hperbola are k ± b h a k ± a h. b Transverse ais is horizontal. Transverse ais is vertical. Eample Using Asmptotes to Sketch a Hperbola Sketch the hperbola whose equation is 6. Algebraic Divide each side of the original equation b 6, and rewrite the equation in standard form. Write in standard form. From this, ou can conclude that a, b, and the transverse ais is horizontal. So, the vertices occur at, 0 and, 0, and the endpoints of the conjugate ais occur at 0, and 0,. Using these four points, ou are able to sketch the rectangle shown in Figure 0.5. Now, from c a b, ou have c 0 5. So, the foci of the hperbola are 5, 0 and 5, 0. Finall, b drawing the asmptotes through the corners of this rectangle, ou can complete the sketch shown in Figure 0.6. Note that the asmptotes are and. Graphical Solve the equation of the hperbola for as follows. 6 6 ± 6 Then use a graphing utilit to graph 6 and 6 in the same viewing window. Be sure to use a square setting. From the graph in Figure 0.7, ou can see that the transverse ais is horizontal. You can use the zoom and trace features to approimate the vertices to be, 0 and, 0. 6 = 6 8 (0, ) = 6 6 (, 0) 6 (, 0) 6 (0, ) ( 5, 0) 6 6 ( 5, 0) 6 = FIGURE 0.7 FIGURE 0.5 FIGURE 0.6 Now tr Eercise.

30 75 Chapter 0 Topics in Analtic Geometr Eample Finding the Asmptotes of a Hperbola Sketch the hperbola given b and find the equations of its asmptotes and the foci Write original equation. Group terms. Factor from -terms. Add to each side. Write in completed square form. (, 7) 5 (, ) ( + ) = (, 0) ( ) 5 (, ) (, 7 ) FIGURE 0.8 Divide each side b. Write in standard form. From this equation ou can conclude that the hperbola has a vertical transverse ais, centered at, 0, has vertices, and,, and has a conjugate ais with endpoints, 0 and, 0. To sketch the hperbola, draw a rectangle through these four points. The asmptotes are the lines passing through the corners of the rectangle. Using a and b, ou can conclude that the equations of the asmptotes are and Finall, ou can determine the foci b using the equation c a b. So, ou have c 7, and the foci are,7 and, 7. The hperbola is shown in Figure 0.8. Now tr Eercise 9.. TECHNOLOGY You can use a graphing utilit to graph a hperbola b graphing the upper and lower portions in the same viewing window. For instance, to graph the hperbola in Eample, first solve for to get and. Use a viewing window in which 9 9 and 6 6. You should obtain the graph shown below. Notice that the graphing utilit does not draw the asmptotes. However, if ou trace along the branches, ou will see that the values of the hperbola approach the asmptotes

31 Section 0. Hperbolas 755 = 8 Eample Using Asmptotes to Find the Standard Equation 6 6 (, ) (, 5) Find the standard form of the equation of the hperbola having vertices, 5 and, and having asmptotes 8 and as shown in Figure 0.9. B the Midpoint Formula, the center of the hperbola is,. Furthermore, the hperbola has a vertical transverse ais with a. From the original equations, ou can determine the slopes of the asmptotes to be FIGURE 0.9 = + m and m a a b b and, because a, ou can conclude a b b b. So, the standard form of the equation is. Now tr Eercise. As with ellipses, the eccentricit of a hperbola is e c a Eccentricit and because c > a, it follows that e >. If the eccentricit is large, the branches of the hperbola are nearl flat, as shown in Figure 0.0. If the eccentricit is close to, the branches of the hperbola are more narrow, as shown in Figure 0.. e is large. e is close to. Verte Focus Verte Focus e = c a c e = c a a c a FIGURE 0.0 FIGURE 0.

32 756 Chapter 0 Topics in Analtic Geometr Applications The following application was developed during World War II. It shows how the properties of hperbolas can be used in radar and other detection sstems. Eample 5 An Application Involving Hperbolas Two microphones, mile apart, record an eplosion. Microphone A receives the sound seconds before microphone B. Where did the eplosion occur? (Assume sound travels at 00 feet per second.) 000 Assuming sound travels at 00 feet per second, ou know that the eplosion took place 00 feet farther from B than from A, as shown in Figure 0.. The locus of all points that are 00 feet closer to A than to B is one branch of the hperbola 00 B A where a b c c a c a and FIGURE 0. c = ( c a) = 580 a So, b c a ,759,600, and ou can conclude that the eplosion occurred somewhere on the right branch of the hperbola,0,000. 5,759,600 Now tr Eercise 5. Hperbolic orbit Verte Elliptical orbit Sun p Parabolic orbit FIGURE 0. Another interesting application of conic sections involves the orbits of comets in our solar sstem. Of the 60 comets identified prior to 970, 5 have elliptical orbits, 95 have parabolic orbits, and 70 have hperbolic orbits. The center of the sun is a focus of each of these orbits, and each orbit has a verte at the point where the comet is closest to the sun, as shown in Figure 0.. Undoubtedl, there have been man comets with parabolic or hperbolic orbits that were not identified. We onl get to see such comets once. Comets with elliptical orbits, such as Halle s comet, are the onl ones that remain in our solar sstem. If p is the distance between the verte and the focus (in meters), and v is the velocit of the comet at the verte (in meters per second), then the tpe of orbit is determined as follows.. Ellipse: v < GMp. Parabola: v GMp. Hperbola: v > GMp In each of these relations, M kilograms (the mass of the sun) and G cubic meter per kilogram-second squared (the universal gravitational constant).

33 Section 0. Hperbolas 757 General Equations of Conics Classifing a Conic from Its General Equation The graph of A C D E F 0 is one of the following.. Circle:. Parabola: A C AC 0 A 0 or C 0, but not both.. Ellipse: AC > 0 A and C have like signs.. Hperbola: AC < 0 A and C have unlike signs. The test above is valid if the graph is a conic. The test does not appl to equations such as, whose graph is not a conic. Eample 6 Classifing Conics from General Equations Classif the graph of each equation. a b c. 0 d. 8 0 HISTORICAL NOTE The Granger Collection Caroline Herschel (750 88) was the first woman to be credited with detecting a new comet. During her long life, this English astronomer discovered a total of eight new comets. a. For the equation 9 5 0, ou have AC 0 0. Parabola So, the graph is a parabola. b. For the equation 8 6 0, ou have AC < 0. Hperbola So, the graph is a hperbola. c. For the equation 0, ou have AC > 0. Ellipse So, the graph is an ellipse. d. For the equation 8 0, ou have A C. Circle So, the graph is a circle. Now tr Eercise 6. CLASSROOM DISCUSSION Sketching Conics Sketch each of the conics described in Eample 6. Write a paragraph describing the procedures that allow ou to sketch the conics efficientl.

34 758 Chapter 0 Topics in Analtic Geometr 0. EXERCISES VOCABULARY: Fill in the blanks. See for worked-out solutions to odd-numbered eercises.. A is the set of all points, in a plane, the difference of whose distances from two distinct fied points, called, is a positive constant.. The graph of a hperbola has two disconnected parts called.. The line segment connecting the vertices of a hperbola is called the, and the midpoint of the line segment is the of the hperbola.. Each hperbola has two that intersect at the center of the hperbola. SKILLS AND APPLICATIONS In Eercises 5 8, match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (b) (c) 8 8 (d) In Eercises 9, find the center, vertices, foci, and the equations of the asmptotes of the hperbola, and sketch its graph using the asmptotes as an aid In Eercises 8, find the center, vertices, foci, and the equations of the asmptotes of the hperbola. Use a graphing utilit to graph the hperbola and its asmptotes In Eercises 9, find the standard form of the equation of the hperbola with the given characteristics and center at the origin. 9. Vertices: 0, ±; foci: 0, ± 0. Vertices: ±, 0; foci: ±6, 0. Vertices: ±, 0; asmptotes: ±5. Vertices: 0, ±; asmptotes: ±. Foci: 0, ±8; asmptotes: ±. Foci: ±0, 0; asmptotes: ± In Eercises 5 6, find the standard form of the equation of the hperbola with the given characteristics. 5. Vertices:, 0, 6, 0; foci: 0, 0, 8, 0 6. Vertices:,,, ; foci:, 6,, 6 7. Vertices:,,, 9; foci:, 0,, 0 8. Vertices:,,, ); foci:,,,

35 Section 0. Hperbolas Vertices:,,, ; passes through the point 0, 5 0. Vertices:,,, ; passes through the point 5,. Vertices: 0,, 0, 0; passes through the point 5,. Vertices:,,, ; passes through the point 0,5. Vertices:,,, ; asmptotes:,. Vertices:, 0,, 6; asmptotes: 6, 5. Vertices: 0,, 6, ; asmptotes:, 6. Vertices:, 0,, ; asmptotes:, In Eercises 7 50, write the standard form of the equation of the hperbola (, 0) (, ) (, 5) (0, ) 8 8 (0, ) (, ) (, ) 6 (0, 0) (5, ) 5. ART A sculpture has a hperbolic cross section (see figure). (, ) (, 0) (, ) 8 ( 8, ) (, ) (, 0) 8 (, ) 6 (, ) (, 0) (0, ) (, 0) (a) Write an equation that models the curved sides of the sculpture. (b) Each unit in the coordinate plane represents foot. Find the width of the sculpture at a height of 5 feet. 5. SOUND LOCATION You and a friend live miles apart (on the same east-west street) and are talking on the phone. You hear a clap of thunder from lightning in a storm, and 8 seconds later our friend hears the thunder. Find an equation that gives the possible places where the lightning could have occurred. (Assume that the coordinate sstem is measured in feet and that sound travels at 00 feet per second.) 5. SOUND LOCATION Three listening stations located at 00, 0, 00, 00, and 00, 0 monitor an eplosion. The last two stations detect the eplosion second and seconds after the first, respectivel. Determine the coordinates of the eplosion. (Assume that the coordinate sstem is measured in feet and that sound travels at 00 feet per second.) 5. LORAN Long distance radio navigation for aircraft and ships uses snchronized pulses transmitted b widel separated transmitting stations. These pulses travel at the speed of light (86,000 miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on a hperbola having the transmitting stations as foci. Assume that two stations, 00 miles apart, are positioned on the rectangular coordinate sstem at points with coordinates 50, 0 and 50, 0, and that a ship is traveling on a hperbolic path with coordinates, 75 (see figure) Station Station Not drawn to scale Ba (a) Find the -coordinate of the position of the ship if the time difference between the pulses from the transmitting stations is 000 microseconds (0.00 second). (b) Determine the distance between the ship and station when the ship reaches the shore. (c) The ship wants to enter a ba located between the two stations. The ba is 0 miles from station. What should be the time difference between the pulses? (d) The ship is 60 miles offshore when the time difference in part (c) is obtained. What is the position of the ship?

36 760 Chapter 0 Topics in Analtic Geometr 55. PENDULUM The base for a pendulum of a clock has the shape of a hperbola (see figure). TRUE OR FALSE? In Eercises 7 76, determine whether the statement is true or false. Justif our answer. (, 9) (, 9) (, 0) 8 (, 9) (, 0) 8 (, 9) (a) Write an equation of the cross section of the base. (b) Each unit in the coordinate plane represents foot. Find the width of the base of the pendulum inches from the bottom. 56. HYPERBOLIC MIRROR A hperbolic mirror (used in some telescopes) has the propert that a light ra directed at a focus will be reflected to the other focus. The focus of a hperbolic mirror (see figure) has coordinates, 0. Find the verte of the mirror if the mount at the top edge of the mirror has coordinates,. (, ) (, 0) EXPLORATION 7. In the standard form of the equation of a hperbola, the larger the ratio of b to a, the larger the eccentricit of the hperbola. 7. In the standard form of the equation of a hperbola, the trivial solution of two intersecting lines occurs when b If D 0 and E 0, then the graph of D E 0 is a hperbola., where a b a, b > 0, intersect at right angles, then a b. 76. If the asmptotes of the hperbola 77. Consider a hperbola centered at the origin with a horizontal transverse ais. Use the definition of a hperbola to derive its standard form. 78. WRITING Eplain how the central rectangle of a hperbola can be used to sketch its asmptotes. 79. THINK ABOUT IT Change the equation of the hperbola so that its graph is the bottom half of the hperbola (, 0) 80. CAPSTONE In Eercises 57 7, classif the graph of the equation as a circle, a parabola, an ellipse, or a hperbola Given the hperbolas and 9 6 describe an common characteristics that the hperbolas share, as well as an differences in the graphs of the hperbolas. Verif our results b using a graphing utilit to graph each of the hperbolas in the same viewing window. 8. A circle and a parabola can have 0,,,, or points of intersection. Sketch the circle given b. Discuss how this circle could intersect a parabola with an equation of the form C. Then find the values of C for each of the five cases described below. Use a graphing utilit to verif our results. (a) No points of intersection (b) One point of intersection (c) Two points of intersection (d) Three points of intersection (e) Four points of intersection

37 Section 0.5 Rotation of Conics ROTATION OF CONICS What ou should learn Rotate the coordinate aes to einate the -term in equations of conics. Use the discriminant to classif conics. Wh ou should learn it As illustrated in Eercises 6 on page 767, rotation of the coordinate aes can help ou identif the graph of a general second-degree equation. Rotation In the preceding section, ou learned that the equation of a conic with aes parallel to one of the coordinate aes has a standard form that can be written in the general form A C D E F 0. Horizontal or vertical ais In this section, ou will stud the equations of conics whose aes are rotated so that the are not parallel to either the -ais or the -ais. The general equation for such conics contains an -term. A B C D E F 0 Equation in -plane To einate this -term, ou can use a procedure called rotation of aes. The objective is to rotate the - and -aes until the are parallel to the aes of the conic. The rotated aes are denoted as the -ais and the -ais, as shown in Figure 0.. θ FIGURE 0. After the rotation, the equation of the conic in the new A C D E F 0. -plane will have the form Equation in -plane Because this equation has no -term, ou can obtain a standard form b completing the square. The following theorem identifies how much to rotate the aes to einate the -term and also the equations for determining the new coefficients A, C, D, E, and F. Rotation of Aes to Einate an -Term The general second-degree equation A B C D E F 0 can be rewritten as A C D E F 0 b rotating the coordinate aes through an angle, where cot A C B. The coefficients of the new equation are obtained b making the substitutions cos sin and sin cos.

38 76 Chapter 0 Topics in Analtic Geometr WARNING / CAUTION Remember that the substitutions and were developed to einate the -term in the rotated sstem. You can use this as a check on our work. In other words, if our final equation contains an -term, ou know that ou have made a mistake. cos sin sin cos Eample Rotation of Aes for a Hperbola Write the equation 0 in standard form. Because A 0, B, and C 0, ou have which implies that and cot A C B cos sin sin cos. 0 ( ) ( ) = ( ( = 0 Vertices: In -sstem:, 0,, 0 In -sstem:,,, FIGURE 0.5 ( ( The equation in the equation sstem is obtained b substituting these epressions in the Write in standard form. In the -sstem, this is a hperbola centered at the origin with vertices at ±, 0, as shown in Figure 0.5. To find the coordinates of the vertices in the -sstem, substitute the coordinates ±, 0 in the equations and This substitution ields the vertices, and, in the -sstem. Note also that the asmptotes of the hperbola have equations which correspond to the original - and -aes. Now tr Eercise.. ±,

39 Section 0.5 Rotation of Conics 76 Eample Rotation of Aes for an Ellipse Sketch the graph of = 0 Vertices: In -sstem: ±, 0 In -sstem:,,, FIGURE 0.6 ( ) ( ) + = Because A 7, B 6, and C, ou have which implies that The equation in the -sstem is obtained b making the substitutions and in the original equation. So, ou have which simplifies to Write in standard form. This is the equation of an ellipse centered at the origin with vertices ±, 0 in the -sstem, as shown in Figure 0.6. cot A C B cos 6 sin 6 sin 6 cos Now tr Eercise 9.

40 76 Chapter 0 Topics in Analtic Geometr Eample Rotation of Aes for a Parabola 5 θ FIGURE 0.7 Sketch the graph of Because A, B, and C, ou have Using this information, draw a right triangle as shown in Figure 0.7. From the figure, ou can see that cos 5. To find the values of sin and cos, ou can use the half-angle formulas in the forms So, cot A C B sin cos and sin cos cos cos Consequentl, ou use the substitutions cos sin cos cos = 0 ( + ) = ( )( 5) Verte: In -sstem: 5, In -sstem: FIGURE , 6 55 θ 6.6 sin cos Substituting these epressions in the original equation, ou have which simplifies as follows Group terms. Write in completed square form. Write in standard form. The graph of this equation is a parabola with verte 5,. Its ais is parallel to the -ais in the -sstem, and because sin 5, as shown in Figure Now tr Eercise ,

41 Section 0.5 Rotation of Conics 765 Invariants Under Rotation In the rotation of aes theorem listed at the beginning of this section, note that the constant term is the same in both equations, Such quantities are invariant under rotation. The net theorem lists some other rotation invariants. F F. Rotation Invariants The rotation of the coordinate aes through an angle that transforms the equation A B C D E F 0 into the form A C D E F 0 has the following rotation invariants.. F F. A C A C. B AC B AC WARNING / CAUTION If there is an -term in the equation of a conic, ou should realize then that the conic is rotated. Before rotating the aes, ou should use the discriminant to classif the conic. You can use the results of this theorem to classif the graph of a second-degree equation with an -term in much the same wa ou do for a second-degree equation without an -term. Note that because the invariant B AC reduces to B AC AC. Discriminant This quantit is called the discriminant of the equation A B C D E F 0. Now, from the classification procedure given in Section 0., ou know that the sign of determines the tpe of graph for the equation AC B 0, A C D E F 0. Consequentl, the sign of B AC will determine the tpe of graph for the original equation, as given in the following classification. Classification of Conics b the Discriminant The graph of the equation A B C D E F 0 is, ecept in degenerate cases, determined b its discriminant as follows.. Ellipse or circle: B AC < 0. Parabola: B AC 0. Hperbola: B AC > 0 For eample, in the general equation ou have A, B 7, and C 5. So the discriminant is B AC Because < 0, the graph of the equation is an ellipse or a circle.

42 766 Chapter 0 Topics in Analtic Geometr Eample Rotation and Graphing Utilities For each equation, classif the graph of the equation, use the Quadratic Formula to solve for, and then use a graphing utilit to graph the equation. a. b. c a. Because B AC 9 6 < 0, the graph is a circle or an ellipse. Solve for as follows. 0 Write original equation. 0 Quadratic form a b c 0 ± ± 6 7 Graph both of the equations to obtain the ellipse shown in Figure Top half of ellipse Bottom half of ellipse FIGURE 0.9 b. Because B AC 6 6 0, the graph is a parabola Write original equation Quadratic form a b c 0 6 ± Graphing both of the equations to obtain the parabola shown in Figure c. Because B AC 6 8 > 0, the graph is a hperbola. 0 0 FIGURE Write original equation. Quadratic form a b c 0 8 ± The graphs of these two equations ield the hperbola shown in Figure 0.5. Now tr Eercise. 5 5 CLASSROOM DISCUSSION FIGURE Classifing a Graph as a Hperbola In Section.6, it was mentioned that the graph of f / is a hperbola. Use the techniques in this section to verif this, and justif each step. Compare our results with those of another student.

43 Section 0.5 Rotation of Conics EXERCISES VOCABULARY: Fill in the blanks. See for worked-out solutions to odd-numbered eercises.. The procedure used to einate the -term in a general second-degree equation is called of.. After rotating the coordinate aes through an angle, the general second-degree equation in the new -plane will have the form.. Quantities that are equal in both the original equation of a conic and the equation of the rotated conic are.. The quantit B AC is called the of the equation A B C D E F 0. SKILLS AND APPLICATIONS In Eercises 5, the -coordinate sstem has been rotated degrees from the -coordinate sstem. The coordinates of a point in the -coordinate sstem are given. Find the coordinates of the point in the rotated coordinate sstem. 90, 0, 0,, 5,, 60,, In Eercises 6, rotate the aes to einate the -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of aes In Eercises 7 6, use a graphing utilit to graph the conic. Determine the angle through which the aes are rotated. Eplain how ou used the graphing utilit to obtain the graph ,, 0,, 5,, 60,, In Eercises 7, match the graph with its equation. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (b) (c) (e) (d) (f)

44 Chapter 0 Topics in Analtic Geometr In Eercises 50, (a) use the discriminant to classif the graph, (b) use the Quadratic Formula to solve for, and (c) use a graphing utilit to graph the equation In Eercises 5 56, sketch (if possible) the graph of the degenerate conic In Eercises 57 70, find an points of intersection of the graphs algebraicall and then verif using a graphing utilit EXPLORATION TRUE OR FALSE? In Eercises 7 and 7, determine whether the statement is true or false. Justif our answer. 7. The graph of the equation k where k is an constant less than, is a hperbola. 7. After a rotation of aes is used to einate the -term from an equation of the form A B C D E F 0 the coefficients of the - and -terms remain A and C, respectivel. 7. Show that the equation r is invariant under rotation of aes. 7. CAPSTONE 6 Consider the equation (a) Without calculating, eplain how to rewrite the equation so that it does not have an -term. (b) Eplain how to identif the graph of the equation. 75. Find the lengths of the major and minor aes of the ellipse graphed in Eercise.

45 Section 0.6 Parametric Equations PARAMETRIC EQUATIONS Jed Jacobsohn/Gett Images What ou should learn Evaluate sets of parametric equations for given values of the parameter. Sketch curves that are represented b sets of parametric equations. Rewrite sets of parametric equations as single rectangular equations b einating the parameter. Find sets of parametric equations for graphs. Wh ou should learn it Parametric equations are useful for modeling the path of an object. For instance, in Eercise 6 on page 775, ou will use a set of parametric equations to model the path of a baseball. Plane Curves Up to this point ou have been representing a graph b a single equation involving the two variables and. In this section, ou will stud situations in which it is useful to introduce a third variable to represent a curve in the plane. To see the usefulness of this procedure, consider the path followed b an object that is propelled into the air at an angle of 5. If the initial velocit of the object is 8 feet per second, it can be shown that the object follows the parabolic path 7 Rectangular equation as shown in Figure 0.5. However, this equation does not tell the whole stor. Although it does tell ou where the object has been, it does not tell ou when the object was at a given point, on the path. To determine this time, ou can introduce a third variable t, called a parameter. It is possible to write both and as functions of t to obtain the parametric equations t Parametric equation for 6t t. Parametric equation for From this set of equations ou can determine that at time t 0, the object is at the point 0, 0. Similarl, at time t, the object is at the point, 6, and so on, as shown in Figure 0.5. Rectangular equation: = + t = 8 7 (6, 8) Parametric equations: 9 t = (7, 0) = t (0, 0) = 6 t + t t = Curvilinear Motion: Two Variables for Position, One Variable for Time FIGURE 0.5 For this particular motion problem, and are continuous functions of t, and the resulting path is a plane curve. (Recall that a continuous function is one whose graph can be traced without lifting the pencil from the paper.) Definition of Plane Curve If f and g are continuous functions of t on an interval I, the set of ordered pairs ft, gt is a plane curve C. The equations ft and gt are parametric equations for C, and t is the parameter.

46 770 Chapter 0 Topics in Analtic Geometr Sketching a Plane Curve When sketching a curve represented b a pair of parametric equations, ou still plot points in the -plane. Each set of coordinates, is determined from a value chosen for the parameter t. Plotting the resulting points in the order of increasing values of t traces the curve in a specific direction. This is called the orientation of the curve. Eample Sketching a Curve WARNING / CAUTION When using a value of t to find, be sure to use the same value of t to find the corresponding value of. Organizing our results in a table, as shown in Eample, can be helpful. Sketch the curve given b the parametric equations t and t, t. Using values of t in the specified interval, the parametric equations ield the points, shown in the table. t 0 t = t = 0 t = FIGURE 0.5 t = t = 0 t = FIGURE t = = t = t t = 6 t = t = = t = t t t = 6 t = t B plotting these points in the order of increasing t, ou obtain the curve C shown in Figure 0.5. Note that the arrows on the curve indicate its orientation as t increases from to. So, if a particle were moving on this curve, it would start at 0, and then move along the curve to the point 5,. Now tr Eercises 5(a) and (b). Note that the graph shown in Figure 0.5 does not define as a function of. This points out one benefit of parametric equations the can be used to represent graphs that are more general than graphs of functions. It often happens that two different sets of parametric equations have the same graph. For eample, the set of parametric equations t and t, t has the same graph as the set given in Eample. However, b comparing the values of t in Figures 0.5 and 0.5, ou can see that this second graph is traced out more rapidl (considering t as time) than the first graph. So, in applications, different parametric representations can be used to represent various speeds at which objects travel along a given path.

47 Section 0.6 Parametric Equations 77 Einating the Parameter Eample uses simple point plotting to sketch the curve. This tedious process can sometimes be simplified b finding a rectangular equation (in and ) that has the same graph. This process is called einating the parameter. Parametric equations Solve for t in one equation. Substitute in other equation. Rectangular equation t t t Now ou can recognize that the equation represents a parabola with a horizontal ais and verte at, 0. When converting equations from parametric to rectangular form, ou ma need to alter the domain of the rectangular equation so that its graph matches the graph of the parametric equations. Such a situation is demonstrated in Eample. Eample Einating the Parameter Parametric equations: =, = t t + t + t = 0 FIGURE 0.55 t = t = 0.75 Sketch the curve represented b the equations and b einating the parameter and adjusting the domain of the resulting rectangular equation. Solving for t in the equation for produces which implies that t Now, substituting in the equation for, ou obtain the rectangular equation t t. t t t From this rectangular equation, ou can recognize that the curve is a parabola that opens downward and has its verte at 0,. Also, this rectangular equation is defined for all values of, but from the parametric equation for ou can see that the curve is defined onl when t >. This implies that ou should restrict the domain of to positive values, as shown in Figure Now tr Eercise 5(c). t t.

48 77 Chapter 0 Topics in Analtic Geometr To einate the parameter in equations involving trigonometric functions, tr using identities such as or sin sec cos tan as shown in Eample. It is not necessar for the parameter in a set of parametric equations to represent time. The net eample uses an angle as the parameter. Eample Sketch the curve represented b cos Einating an Angle Parameter and b einating the parameter. sin, Begin b solving for cos and sin in the equations. 0 θ = θ = FIGURE 0.56 θ = θ = 0 = cos θ = sin θ cos Use the identit sin cos sin 9 and sin Solve for cos and cos 6 to form an equation involving onl and. Pthagorean identit Substitute for cos and for sin. Rectangular equation From this rectangular equation, ou can see that the graph is an ellipse centered at 0, 0, with vertices 0, and 0, and minor ais of length b 6, as shown in Figure Note that the elliptic curve is traced out counterclockwise as varies from 0 to. Now tr Eercise 7. sin. In Eamples and, it is important to realize that einating the parameter is primaril an aid to curve sketching. If the parametric equations represent the path of a moving object, the graph alone is not sufficient to describe the object s motion. You still need the parametric equations to tell ou the position, direction, and speed at a given time. Finding Parametric Equations for a Graph You have been studing techniques for sketching the graph represented b a set of parametric equations. Now consider the reverse problem that is, how can ou find a set of parametric equations for a given graph or a given phsical description? From the discussion following Eample, ou know that such a representation is not unique. That is, the equations t and t, t produced the same graph as the equations t and t, t. This is further demonstrated in Eample.

49 Section 0.6 Parametric Equations 77 = t = t t t = t = 0 t = Eample Finding Parametric Equations for a Graph Find a set of parametric equations to represent the graph of, following parameters. a. t b. t using the t = t = a. Letting t, ou obtain the parametric equations t and t. b. Letting t, ou obtain the parametric equations FIGURE 0.57 t and t t t. In Figure 0.57, note how the resulting curve is oriented b the increasing values of t. For part (a), the curve would have the opposite orientation. Now tr Eercise 5. Eample 5 Parametric Equations for a Ccloid Describe the ccloid traced out b a point P on the circumference of a circle of radius a as the circle rolls along a straight line in a plane. As the parameter, let be the measure of the circle s rotation, and let the point P, begin at the origin. When P is at the origin; when P is at a maimum point a, a; and when P is back on the -ais at a, 0. From Figure 0.58, ou can see that APC 80. So, ou have sin sin80 0,, sinapc AC a BD a, In Eample 5, PD represents the arc of the circle between points P and D. cos cos80 which implies that BD a sin and Because the circle rolls along the -ais, ou know that OD PD AP a cos. a. Furthermore, because BA DC a, ou have OD BD a a sin So, the parametric equations are cosapc AP a and a sin BA AP a a cos. and a cos. TECHNOLOGY a P = (, ) ( a, a) Ccloid: = a( θ sin θ), = a( cos θ) ( a, a) You can use a graphing utilit in parametric mode to obtain a graph similar to Figure 0.58 b graphing the following equations. X T T sin T Y T cos T a O A FIGURE 0.58 B θ C D a Now tr Eercise 67. (a, 0) a (a, 0)

50 77 Chapter 0 Topics in Analtic Geometr 0.6 EXERCISES See for worked-out solutions to odd-numbered eercises. VOCABULARY: Fill in the blanks.. If f and g are continuous functions of t on an interval I, the set of ordered pairs f t, gt is a C.. The of a curve is the direction in which the curve is traced out for increasing values of the parameter.. The process of converting a set of parametric equations to a corresponding rectangular equation is called the.. A curve traced b a point on the circumference of a circle as the circle rolls along a straight line in a plane is called a. SKILLS AND APPLICATIONS 5. Consider the parametric equations t and t. (a) Create a table of - and -values using t 0,,,, and. (b) Plot the points, generated in part (a), and sketch a graph of the parametric equations. (c) Find the rectangular equation b einating the parameter. Sketch its graph. How do the graphs differ? 6. Consider the parametric equations cos and sin. (a) Create a table of - and -values using, 0,, and. (b) Plot the points, generated in part (a), and sketch a graph of the parametric equations. (c) Find the rectangular equation b einating the parameter. Sketch its graph. How do the graphs differ?, In Eercises 7 6, (a) sketch the curve represented b the parametric equations (indicate the orientation of the curve) and (b) einate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation if necessar. 7. t 8. t t t 9. t 0. t t t. t. t t t. t. t t t t t 5. t 6. t t t 7. cos 8. cos sin sin 9. 6 sin 0. cos 6 cos sin. cos. 5 cos sin 6 sin. e t. e t e t e t 5. t 6. ln t ln t t In Eercises 7 and 8, determine how the plane curves differ from each other. 7. (a) t (b) cos t cos (c) e t (d) e t e t e t 8. (a) t (b) t t t (c) sin t (d) e t sin t e t In Eercises 9, einate the parameter and obtain the standard form of the rectangular equation. 9. Line through, and, : t, t 0. Circle: h r cos, k r sin. Ellipse: h a cos, k b sin. Hperbola: h a sec, k b tan In Eercises 0, use the results of Eercises 9 to find a set of parametric equations for the line or conic.. Line: passes through 0, 0 and, 6. Line: passes through, and 6, 5. Circle: center:, ; radius: 6. Circle: center: 5, ; radius:

51 Section 0.6 Parametric Equations Ellipse: vertices: ±5, 0; foci: ±, 0 8. Ellipse: vertices:, 7,, ; foci: (, 5,, 9. Hperbola: vertices: ±, 0; foci: ±5, 0 0. Hperbola: vertices: ±, 0; foci: ±, 0 In Eercises 8, find a set of parametric equations for the rectangular equation using (a) t and (b) t In Eercises 9 56, use a graphing utilit to graph the curve represented b the parametric equations. 9. Ccloid: sin, cos 50. Ccloid:, cos 5. Prolate ccloid: 5. Prolate ccloid: sin, cos cos 5. Hpoccloid: cos, sin 5. Curtate ccloid: 8 sin, 8 cos 55. Witch of Agnesi: cot, sin 56. Folium of Descartes: In Eercises 57 60, match the parametric equations with the correct graph and describe the domain and range. [The graphs are labeled (a), (b), (c), and (d).] (a) (b) sin sin, t t, t t 57. Lissajous curve: cos, sin 58. Evolute of ellipse: cos, 6 sin 59. Involute of circle: cos sin 60. Serpentine curve: cot, sin cos PROJECTILE MOTION A projectile is launched at a height of h feet above the ground at an angle of with the horizontal. The initial velocit is v 0 feet per second, and the path of the projectile is modeled b the parametric equations v 0 cos t and In Eercises 6 and 6, use a graphing utilit to graph the paths of a projectile launched from ground level at each value of and v 0. For each case, use the graph to approimate the maimum height and the range of the projectile. 6. (a) v 0 88 feet per second (b) v 0 feet per second (c) v 0 88 feet per second (d) v 0 feet per second 6. (a) v 0 50 feet per second (b) v 0 0 feet per second (c) v 0 50 feet per second (d) v 0 0 feet per second 60, 60, 5, 5, 5, 5, 0, 0, 6. SPORTS The center field fence in Yankee Stadium is 7 feet high and 08 feet from home plate. A baseball is hit at a point feet above the ground. It leaves the bat at an angle of degrees with the horizontal at a speed of 00 miles per hour (see figure). ft θ h v 0 sin t 6t. 7 ft 08 ft sin cos Not drawn to scale (c) (d) (a) Write a set of parametric equations that model the path of the baseball. (b) Use a graphing utilit to graph the path of the baseball when Is the hit a home run? 5. (c) Use the graphing utilit to graph the path of the baseball when Is the hit a home run?. (d) Find the minimum angle required for the hit to be a home run.

52 776 Chapter 0 Topics in Analtic Geometr 6. SPORTS An archer releases an arrow from a bow at a point 5 feet above the ground. The arrow leaves the bow at an angle of 5 with the horizontal and at an initial speed of 5 feet per second. (a) Write a set of parametric equations that model the path of the arrow. (b) Assuming the ground is level, find the distance the arrow travels before it hits the ground. (Ignore air resistance.) (c) Use a graphing utilit to graph the path of the arrow and approimate its maimum height. (d) Find the total time the arrow is in the air. 65. PROJECTILE MOTION Einate the parameter t from the parametric equations v0 cos t and h v0 sin t 66. PATH OF A PROJECTILE The path of a projectile is given b the rectangular equation (a) Use the result of Eercise 65 to find h, v0, and. Find the parametric equations of the path. (b) Use a graphing utilit to graph the rectangular equation for the path of the projectile. Confirm our answer in part (a) b sketching the curve represented b the parametric equations. (c) Use the graphing utilit to approimate the maimum height of the projectile and its range. 67. CURTATE CYCLOID A wheel of radius a units rolls along a straight line without slipping. The curve traced b a point P that is b units from the center b < a is called a curtate ccloid (see figure). Use the angle shown in the figure to find a set of parametric equations for the curve. ( a, a + b) P b θ (0, a b) a a θ (, ) EXPLORATION TRUE OR FALSE? In Eercises 69 and 70, determine whether the statement is true or false. Justif our answer. 6 sec tan h. v0 a 6t for the motion of a projectile to show that the rectangular equation is 68. EPICYCLOID A circle of radius one unit rolls around the outside of a circle of radius two units without slipping. The curve traced b a point on the circumference of the smaller circle is called an epiccloid (see figure). Use the angle shown in the figure to find a set of parametric equations for the curve. a 69. The two sets of parametric equations t, t and t, 9t have the same rectangular equation. 70. If is a function of t, and is a function of t, then must be a function of. 7. WRITING Write a short paragraph eplaining wh parametric equations are useful. 7. WRITING Eplain the process of sketching a plane curve given b parametric equations. What is meant b the orientation of the curve? 7. Use a graphing utilit set in parametric mode to enter the parametric equations from Eample. Over what values should ou let t var to obtain the graph shown in Figure 0.55? 7. CAPSTONE Consider the parametric equations 8 cos t and 8 sin t. (a) Describe the curve represented b the parametric equations. (b) How does the curve represented b the parametric equations 8 cos t and 8 sin t 6 compare with the curve described in part (a)? (c) How does the original curve change when cosine and sine are interchanged?

53 Section 0.7 Polar Coordinates POLAR COORDINATES What ou should learn Plot points on the polar coordinate sstem. Convert points from rectangular to polar form and vice versa. Convert equations from rectangular to polar form and vice versa. Wh ou should learn it Polar coordinates offer a different mathematical perspective on graphing. For instance, in Eercises 5 8 on page 78, ou are asked to find multiple representations of polar coordinates. Introduction So far, ou have been representing graphs of equations as collections of points, on the rectangular coordinate sstem, where and represent the directed distances from the coordinate aes to the point,. In this section, ou will stud a different sstem called the polar coordinate sstem. To form the polar coordinate sstem in the plane, fi a point O, called the pole (or origin), and construct from O an initial ra called the polar ais, as shown in Figure Then each point P in the plane can be assigned polar coordinates r, as follows.. r directed distance from O to P. directed angle, counterclockwise from polar ais to segment OP O r = directed distance θ FIGURE 0.59 P= ( r, θ) = directed angle Polar ais Eample Plotting Points on the Polar Coordinate Sstem a. The point r,, lies two units from the pole on the terminal side of the angle as shown in Figure b. The point r,, 6 lies three units from the pole on the terminal side of the angle as shown in Figure 0.6. c. The point r,, 6 coincides with the point, 6, as shown in Figure 0.6., 6, θ =, ( ) 0 FIGURE θ = 6 FIGURE 0.6 Now tr Eercise 7. ( ), 6 0 FIGURE 0.6 ( ), 6 θ = 6

54 778 Chapter 0 Topics in Analtic Geometr In rectangular coordinates, each point, has a unique representation. This is not true for polar coordinates. For instance, the coordinates r, and r, represent the same point, as illustrated in Eample. Another wa to obtain multiple representations of a point is to use negative values for r. Because r is a directed distance, the coordinates r, and r, represent the same point. In general, the point r, can be represented as r, r, ± n or r, r, where n is an integer. Moreover, the pole is represented b 0,, where is an angle. ± n Eample Multiple Representations of Points 0 ( ), θ = (, ) = (, 5 ) = (, 7 ) = (, ) =... FIGURE 0.6 Plot the point, and find three additional polar representations of this point, using < <. The point is shown in Figure 0.6. Three other representations are as follows., 5,, 7,,, Now tr Eercise. Add to. Replace r b r; subtract Replace r b r; add to. from. Coordinate Conversion (r, θ) (, ) To establish the relationship between polar and rectangular coordinates, let the polar ais coincide with the positive -ais and the pole with the origin, as shown in Figure 0.6. Because, lies on a circle of radius r, it follows that r. Moreover, for r > 0, the definitions of the trigonometric functions impl that r tan, cos r, and sin r. Pole θ (Origin) FIGURE 0.6 Polar ais (-ais) If r < 0, ou can show that the same relationships hold. Coordinate Conversion The polar coordinates r, are related to the rectangular coordinates, as follows. Polar-to-Rectangular Rectangular-to-Polar r cos tan r sin r

55 Section 0.7 Polar Coordinates 779 ( r, θ) = (, ) (, ) = (, 0) FIGURE 0.65 ( ) ( r, θ) =, 6 ( ) (, ) =, Eample Polar-to-Rectangular Conversion Convert each point to rectangular coordinates. a., b., a. For the point r,,, ou have the following. r cos cos r sin sin 0 The rectangular coordinates are,, 0. (See Figure 0.65.) b. For the point r,, ou have the following. cos 6 sin 6 6 6, The rectangular coordinates are Now tr Eercise.,,. (, ) = (, ) (r, θ) =, FIGURE 0.66 ( ) (, ) = (0, ) FIGURE 0.67 ( ) (r, θ) =, 0 0 Eample Rectangular-to-Polar Conversion Convert each point to polar coordinates. a., b. 0, a. For the second-quadrant point,,, ou have tan. Because lies in the same quadrant as,, use positive r. r So, one set of polar coordinates is r,,, as shown in Figure b. Because the point, 0, lies on the positive -ais, choose and r. This implies that one set of polar coordinates is Figure Now tr Eercise. r,,, as shown in

56 780 Chapter 0 Topics in Analtic Geometr 0 FIGURE FIGURE Equation Conversion B comparing Eamples and, ou can see that point conversion from the polar to the rectangular sstem is straightforward, whereas point conversion from the rectangular to the polar sstem is more involved. For equations, the opposite is true. To convert a rectangular equation to polar form, ou simpl replace b r cos and b r sin. For instance, the rectangular equation can be written in polar form as follows. Rectangular equation r sin r cos Polar equation r sec Simplest form On the other hand, converting a polar equation to rectangular form requires considerable ingenuit. Eample 5 demonstrates several polar-to-rectangular conversions that enable ou to sketch the graphs of some polar equations. Eample 5 Converting Polar Equations to Rectangular Form Describe the graph of each polar equation and find the corresponding rectangular equation. a. r b. c. r sec a. The graph of the polar equation r consists of all points that are two units from the pole. In other words, this graph is a circle centered at the origin with a radius of, as shown in Figure You can confirm this b converting to rectangular form, using the relationship r. r Polar equation Rectangular equation b. The graph of the polar equation consists of all points on the line that makes an angle of with the positive polar ais, as shown in Figure To convert to rectangular form, make use of the relationship tan Polar equation Rectangular equation c. The graph of the polar equation r sec is not evident b simple inspection, so convert to rectangular form b using the relationship r cos. r sec Polar equation tan r tan Rectangular equation Now ou see that the graph is a vertical line, as shown in Figure Now tr Eercise 09. r cos. FIGURE 0.70

57 Section 0.7 Polar Coordinates EXERCISES VOCABULARY: Fill in the blanks.. The origin of the polar coordinate sstem is called the. See for worked-out solutions to odd-numbered eercises.. For the point r,, r is the from O to P and is the, counterclockwise from the polar ais to the line segment OP.. To plot the point r,, use the coordinate sstem.. The polar coordinates r, are related to the rectangular coordinates, as follows: tan r SKILLS AND APPLICATIONS In Eercises 5 8, plot the point given in polar coordinates and find two additional polar representations of the point, using < < , 6., 6 7., 8., 9., 0. In Eercises 9 8, a point in polar coordinates is given. Convert the point to rectangular coordinates. 9., 0.,.,., 6. 0, 7. 0, ,.6 6.,.7 7., ,.6 ( r, θ) =,., 5. 0, ( ) 0 (r, θ) =, 5 ( ).,., 5 0 5, ( r, θ) =, ( ) (r, θ) = (0, ) , 76 6., ,. 8., 5.76 In Eercises 9 6, use a graphing utilit to find the rectangular coordinates of the point given in polar coordinates. Round our results to two decimal places. 9., 9 0., 9..5,.. 8.5,.5..5, , , ,. In Eercises 7 5, a point in rectangular coordinates is given. Convert the point to polar coordinates. 7., 8., 9., 0.,. 6, 0., 0. 0, 5. 0, 5 5., 6., 7., 8., 9., 50., 5. 6, , 5. 5, 5. 7, 5 In Eercises 55 6, use a graphing utilit to find one set of polar coordinates for the point given in rectangular coordinates. 55., 56., 57. 5, 58. 7, 59., 60. 5, 6. 5, , , 7 9, In Eercises 65 8, convert the rectangular equation to polar form. Assume a >

58 Chapter a a 0 Topics in Analtic Geometr a a a 0 In Eercises 85 08, convert the polar equation to rectangular form r sin r cos 6 r r csc r sec r cos r sin r sin 05. r sin 07. r 6 sin r cos r 5 sin r 0 r csc r sec r sin r cos r cos 06. r cos r cos sin In Eercises 09 8, describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph r 6 6 r sin r 6 cos r sec r 8 r cos r sin r csc EXPLORATION TRUE OR FALSE? In Eercises 9 and 0, determine whether the statement is true or false. Justif our answer. 9. If n for some integer n, then r, and r, represent the same point on the polar coordinate sstem. 0. If r r, then r, and r, represent the same point on the polar coordinate sstem.. Convert the polar equation r h cos k sin to rectangular form and verif that it is the equation of a circle. Find the radius of the circle and the rectangular coordinates of the center of the circle.. Convert the polar equation r cos sin to rectangular form and identif the graph.. THINK ABOUT IT (a) Show that the distance between the points r, and r, is r r rr cos. (b) Describe the positions of the points relative to each other for. Simplif the Distance Formula for this case. Is the simplification what ou epected? Eplain. (c) Simplif the Distance Formula for 90. Is the simplification what ou epected? Eplain. (d) Choose two points on the polar coordinate sstem and find the distance between them. Then choose different polar representations of the same two points and appl the Distance Formula again. Discuss the result.. GRAPHICAL REASONING (a) Set the window format of our graphing utilit on rectangular coordinates and locate the cursor at an position off the coordinate aes. Move the cursor horizontall and observe an changes in the displaed coordinates of the points. Eplain the changes in the coordinates. Now repeat the process moving the cursor verticall. (b) Set the window format of our graphing utilit on polar coordinates and locate the cursor at an position off the coordinate aes. Move the cursor horizontall and observe an changes in the displaed coordinates of the points. Eplain the changes in the coordinates. Now repeat the process moving the cursor verticall. (c) Eplain wh the results of parts (a) and (b) are not the same. 5. GRAPHICAL REASONING (a) Use a graphing utilit in polar mode to graph the equation r. (b) Use the trace feature to move the cursor around the circle. Can ou locate the point, 5? (c) Can ou find other polar representations of the point, 5? If so, eplain how ou did it. 6. CAPSTONE In the rectangular coordinate sstem, each point, has a unique representation. Eplain wh this is not true for a point r, in the polar coordinate sstem.

59 Section 0.8 Graphs of Polar Equations GRAPHS OF POLAR EQUATIONS What ou should learn Graph polar equations b point plotting. Use smmetr to sketch graphs of polar equations. Use zeros and maimum r-values to sketch graphs of polar equations. Recognize special polar graphs. Wh ou should learn it Equations of several common figures are simpler in polar form than in rectangular form. For instance, Eercise on page 789 shows the graph of a circle and its polar equation. Introduction In previous chapters, ou learned how to sketch graphs on rectangular coordinate sstems. You began with the basic point-plotting method. Then ou used sketching aids such as smmetr, intercepts, asmptotes, periods, and shifts to further investigate the natures of graphs. This section approaches curve sketching on the polar coordinate sstem similarl, beginning with a demonstration of point plotting. Eample Graphing a Polar Equation b Point Plotting Sketch the graph of the polar equation r sin. The sine function is periodic, so ou can get a full range of r-values b considering values of in the interval 0, as shown in the following table r If ou plot these points as shown in Figure 0.7, it appears that the graph is a circle of radius whose center is at the point, 0,. Circle: r = sin θ 0 FIGURE 0.7 Now tr Eercise 7. You can confirm the graph in Figure 0.7 b converting the polar equation to rectangular form and then sketching the graph of the rectangular equation. You can also use a graphing utilit set to polar mode and graph the polar equation or set the graphing utilit to parametric mode and graph a parametric representation.

60 78 Chapter 0 Topics in Analtic Geometr Smmetr In Figure 0.7 on the preceding page, note that as increases from 0 to the graph is traced out twice. Moreover, note that the graph is smmetric with respect to the line Had ou known about this smmetr and retracing ahead of time, ou could have used fewer points. Smmetr with respect to the line is one of three important tpes of smmetr to consider in polar curve sketching. (See Figure 0.7.). ( r, θ) (r, θ) θ (r, θ) θ 0 Smmetr with Respect to the Line FIGURE 0.7 (r, θ) θ 0 θ (r, θ) ( r, θ) Smmetr with Respect to the Polar Ais + θ (r, θ) θ 0 ( r, θ) (r, + θ) Smmetr with Respect to the Pole Note in Eample that cos cos. This is because the cosine function is even. Recall from Section. that the cosine function is even and the sine function is odd. That is, sin sin. Tests for Smmetr in Polar Coordinates The graph of a polar equation is smmetric with respect to the following if the given substitution ields an equivalent equation.. The line Replace r, b r, or r,.. The polar ais: Replace r, b r, or r,.. The pole: Replace r, b r, or r,. : Eample Using Smmetr to Sketch a Polar Graph Use smmetr to sketch the graph of r cos. r = + cos θ 0 5 Replacing r, b r, produces r cos cos. cos cos So, ou can conclude that the curve is smmetric with respect to the polar ais. Plotting the points in the table and using polar ais smmetr, ou obtain the graph shown in Figure 0.7. This graph is called a açon. FIGURE r 5 Now tr Eercise.

61 Section 0.8 Graphs of Polar Equations Spiral of Archimedes: r = θ +, θ 0 FIGURE 0.7 The three tests for smmetr in polar coordinates listed on page 78 are sufficient to guarantee smmetr, but the are not necessar. For instance, Figure 0.7 shows the graph of r to be smmetric with respect to the line and et the tests on page 78 fail to indicate smmetr because neither of the following replacements ields an equivalent equation. Original Equation Replacement New Equation r r The equations discussed in Eamples and are of the form r sin f sin r, b r, r, b r, and, r r r cos gcos., The graph of the first equation is smmetric with respect to the line and the graph of the second equation is smmetric with respect to the polar ais. This observation can be generalized to ield the following tests. Quick Tests for Smmetr in Polar Coordinates. The graph of r f sin is smmetric with respect to the line. The graph of r gcos is smmetric with respect to the polar ais.. Zeros and Maimum r-values Two additional aids to graphing of polar equations involve knowing the -values for which r is maimum and knowing the -values for which r 0. For instance, in Eample, the maimum value of r for r sin is r, and this occurs when as shown in Figure 0.7. Moreover, r 0 when, 0. Eample Sketching a Polar Graph Sketch the graph of r cos. From the equation r cos, ou can obtain the following Smmetr: Maimum value of : r when Zero of r: r With respect to the polar ais r 0 when The table shows several -values in the interval 0,. B plotting the corresponding points, ou can sketch the graph shown in Figure Limaçon: r = cos θ FIGURE r Note how the negative r-values determine the inner loop of the graph in Figure This graph, like the one in Figure 0.7, is a açon. Now tr Eercise

62 786 Chapter 0 Topics in Analtic Geometr Some curves reach their zeros and maimum r-values at more than one point, as shown in Eample. Eample Sketching a Polar Graph Sketch the graph of r cos. Smmetr: r With respect to the polar ais r 0,,, Maimum value of : when or Zeros of r: r 0 when or,, 5 0,,, 6,, r 0 0 B plotting these points and using the specified smmetr, zeros, and maimum values, ou can obtain the graph shown in Figure This graph is called a rose curve, and each of the loops on the graph is called a petal of the rose curve. Note how the entire curve is generated as increases from 0 to TECHNOLOGY Use a graphing utilit in polar mode to verif the graph of r cos shown in Figure FIGURE Now tr Eercise

63 Section 0.8 Graphs of Polar Equations 787 Special Polar Graphs Several important tpes of graphs have equations that are simpler in polar form than in rectangular form. For eample, the circle r sin in Eample has the more complicated rectangular equation. Several other tpes of graphs that have simple polar equations are shown below. Limaçons r a ± b cos r a ± b sin a > 0, b > a a < a a b < b b < b Limaçon with Cardioid Dimpled Conve inner loop (heart-shaped) açon açon Rose Curves n petals if n is odd, n petals if n is even n. n = 0 n = 0 a 0 a 0 a a n = 5 n = r a cos n r a cos n r a sin n r a sin n Rose curve Rose curve Rose curve Rose curve Circles and Lemniscates 0 a 0 a 0 0 a a r a cos r a sin r a sin r a cos Circle Circle Lemniscate Lemniscate

64 788 Chapter 0 Topics in Analtic Geometr Eample 5 Sketching a Rose Curve, θ = θ = (, ) (, 0) 0 r = cos θ FIGURE 0.77 ( ) ( ), Sketch the graph of r cos. Tpe of curve: Rose curve with n petals Smmetr: With respect to polar ais, the line and the pole, Maimum value of : when Zeros of r: r r r 0 when 0,,,, Using this information together with the additional points shown in the following table, ou obtain the graph shown in Figure r 6 0 Now tr Eercise. Eample 6 Sketching a Lemniscate Sketch the graph of r 9 sin. 0 ( ), r = 9 sin θ FIGURE 0.78 ( ), Tpe of curve: Smmetr: Lemniscate With respect to the pole Maimum value of : when Zeros of r: r 0 when If sin < 0, this equation has no solution points. So, ou restrict the values of to those for which sin 0. 0 r or r 0, Moreover, using smmetr, ou need to consider onl the first of these two intervals. B finding a few additional points (see table below), ou can obtain the graph shown in Figure r ±sin 0 ± ± ± 0 Now tr Eercise 7.

65 Section 0.8 Graphs of Polar Equations EXERCISES VOCABULARY: Fill in the blanks.. The graph of r f sin is smmetric with respect to the line.. The graph of r gcos is smmetric with respect to the.. The equation r cos represents a.. The equation r cos represents a. 5. The equation r sin represents a. 6. The equation r sin represents a. SKILLS AND APPLICATIONS See for worked-out solutions to odd-numbered eercises. In Eercises 7, identif the tpe of polar graph r = 5 cos θ r = ( cos θ ).. r = sin θ 0 0 r = cosθ r = 5 5 sin θ r = 6 cos θ 0 In Eercises 8, sketch the graph of the polar equation using smmetr, zeros, maimum r-values, and an other additional points.. r. r 7 5. r 6. r 7. r sin 8. r cos 9. r cos 0. r sin. r sin. r cos. r 6 sin. r sin 5. r sin 6. r cos 7. r cos 8. r cos 9. r 5 sin 0. r cos. r 6 cos. r sin. r sec. r 5 csc 5. 6 r 6. r sin sin cos 7. r 9 cos 8. r sin cos In Eercises 9 58, use a graphing utilit to graph the polar equation. Describe our viewing window. 9. r r 5 In Eercises 8, test for smmetr with respect to the polar ais, and the pole.. r cos. r 9 cos 5. r 6. r sin cos 7. r 6 cos 8. r 5 sin /, In Eercises 9, find the maimum value of and an zeros of r. 9. r 0 0 sin 0. r 6 cos. r cos. r sin r 5. r 5 5. r r 8 cos 5. r cos 55. r sin 56. r cos 57. r 8 sin cos 58. r csc 5 In Eercises 59 6, use a graphing utilit to graph the polar equation. Find an interval for for which the graph is traced onl once. 59. r 8 cos 60. r 5 cos 5 6. r cos 6. r sin

66 790 Chapter 0 6. r 6 sin Topics in Analtic Geometr 6. r (c) Show that if the graph is rotated counterclockwise radians about the pole, the equation of the rotated graph is r f cos. In Eercises 65 68, use a graphing utilit to graph the polar equation and show that the indicated line is an asmptote of the graph. Name of Graph 65. Conchoid 66. Conchoid 67. Hperbolic spiral 68. Strophoid Polar Equation r sec r csc r r cos sec Asmptote EXPLORATION TRUE OR FALSE? In Eercises 69 and 70, determine whether the statement is true or false. Justif our answer. 69. In the polar coordinate sstem, if a graph that has smmetr with respect to the polar ais were folded on the line 0, the portion of the graph above the polar ais would coincide with the portion of the graph below the polar ais. 70. In the polar coordinate sstem, if a graph that has smmetr with respect to the pole were folded on the line, the portion of the graph on one side of the fold would coincide with the portion of the graph on the other side of the fold. 7. Sketch the graph of r 6 cos over each interval. Describe the part of the graph obtained in each case. (a) 0 (b) (c) (d) 7. GRAPHICAL REASONING Use a graphing utilit to graph the polar equation r 6 cos for (a) 0, (b), and (c). Use the graphs to describe the effect of the angle. Write the equation as a function of sin for part (c). 7. The graph of r f is rotated about the pole through an angle. Show that the equation of the rotated graph is r f. 7. Consider the graph of r f sin. (a) Show that if the graph is rotated counterclockwise radians about the pole, the equation of the rotated graph is r f cos. (b) Show that if the graph is rotated counterclockwise radians about the pole, the equation of the rotated graph is r f sin. In Eercises 75 78, use the results of Eercises 7 and Write an equation for the açon r sin after it has been rotated through the given angle. (a) (b) (c) (d) 76. Write an equation for the rose curve r sin after it has been rotated through the given angle. (a) (b) (c) (d) Sketch the graph of each equation. (a) r sin (b) r sin 78. Sketch the graph of each equation. (a) r sec (c) r sec (d) r sec (b) r sec 79. THINK ABOUT IT How man petals do the rose curves given b r cos and r sin have? Determine the numbers of petals for the curves given b r cos n and r sin n, where n is a positive integer. 80. Use a graphing utilit to graph and identif r k sin for k 0,,, and. 8. Consider the equation r sin k. (a) Use a graphing utilit to graph the equation for k.5. Find the interval for over which the graph is traced onl once. (b) Use a graphing utilit to graph the equation for k.5. Find the interval for over which the graph is traced onl once. (c) Is it possible to find an interval for over which the graph is traced onl once for an rational number k? Eplain. 8. CAPSTONE Write a brief paragraph that describes wh some polar curves have equations that are simpler in polar form than in rectangular form. Besides a circle, give an eample of a curve that is simpler in polar form than in rectangular form. Give an eample of a curve that is simpler in rectangular form than in polar form.

67 Section 0.9 Polar Equations of Conics POLAR EQUATIONS OF CONICS What ou should learn Define conics in terms of eccentricit. Write and graph equations of conics in polar form. Use equations of conics in polar form to model real-life problems. Wh ou should learn it The orbits of planets and satellites can be modeled with polar equations. For instance, in Eercise 65 on page 796, a polar equation is used to model the orbit of a satellite. Alternative Definition of Conic In Sections 0. and 0., ou learned that the rectangular equations of ellipses and hperbolas take simple forms when the origin lies at their centers. As it happens, there are man important applications of conics in which it is more convenient to use one of the foci as the origin. In this section, ou will learn that polar equations of conics take simple forms if one of the foci lies at the pole. To begin, consider the following alternative definition of conic that uses the concept of eccentricit. Alternative Definition of Conic The locus of a point in the plane that moves so that its distance from a fied point (focus) is in a constant ratio to its distance from a fied line (directri) is a conic. The constant ratio is the eccentricit of the conic and is denoted b e. Moreover, the conic is an ellipse if e <, a parabola if e, and a hperbola if e >. (See Figure 0.79.) In Figure 0.79, note that for each tpe of conic, the focus is at the pole. Directri Q P Directri Directri Q P F = (0, 0) 0 Q P 0 F = (0, 0) P Q 0 F = (0, 0) Corbis Ellipse: 0 < e < Parabola: e Hperbola e > PF PF PF PF > PQ < PQ PQ FIGURE 0.79 PQ Polar Equations of Conics The benefit of locating a focus of a conic at the pole is that the equation of the conic takes on a simpler form. For a proof of the polar equations of conics, see Proofs in Mathematics on page 806. Polar Equations of Conics The graph of a polar equation of the form ep ep. r or. r ± e cos ± e sin is a conic, where e > 0 is the eccentricit and p is the distance between the focus (pole) and the directri.

68 79 Chapter 0 Topics in Analtic Geometr Equations of the form r Vertical directri correspond to conics with a vertical directri and smmetr with respect to the polar ais. Equations of the form r Horizontal directri correspond to conics with a horizontal directri and smmetr with respect to the line Moreover, the converse is also true that is, an conic with a focus at the pole and having a horizontal or vertical directri can be represented b one of these equations.. ep gcos ± e cos ep gsin ± e sin Eample Identifing a Conic from Its Equation Identif the tpe of conic represented b the equation r Algebraic To identif the tpe of conic, rewrite the equation in the form r ep ± e cos. r 5 cos 5 cos Write original equation. Divide numerator and denominator b. Because e <, ou can conclude that the graph is an ellipse. 5. cos Graphical You can start sketching the graph b plotting points from to Because the equation is of the form r gcos, the graph of r is smmetric with respect to the polar ais. So, ou can complete the sketch, as shown in Figure From this, ou can conclude that the graph is an ellipse.. r = 5 cos θ (, ) (5, 0) FIGURE 0.80 Now tr Eercise 5. For the ellipse in Figure 0.80, the major ais is horizontal and the vertices lie at 5, 0 and,. So, the length of the major ais is a 8. To find the length of the minor ais, ou can use the equations e ca and b a c to conclude that b a c a ea a e. Ellipse Because e ou have b 9, 5, which implies that b 5 5. So, the length of the minor ais is b 65. A similar analsis for hperbolas ields b c a ea a a e. Hperbola

69 Section 0.9 Polar Equations of Conics 79 Eample Sketching a Conic from Its Polar Equation 6, ( ) (, ) 8 r = + 5 sin θ FIGURE Identif the conic r and sketch its graph. 5 sin Dividing the numerator and denominator b, ou have r. 5 sin Because e 5 >, the graph is a hperbola. The transverse ais of the hperbola lies on the line and the vertices occur at, and 6,. Because the length of the transverse ais is, ou can see that a 6. To find b, write, b a e 6 5 So, b 8. Finall, ou can use a and b to determine that the asmptotes of the hperbola are 0 ±. The graph is shown in Figure 0.8. Now tr Eercise. 6. TECHNOLOGY Use a graphing utilit set in polar mode to verif the four orientations shown at the right. Remember that e must be positive, but p can be positive or negative. In the net eample, ou are asked to find a polar equation of a specified conic. To do this, let p be the distance between the pole and the directri.. Horizontal directri above the pole:. Horizontal directri below the pole: ep. Vertical directri to the right of the pole: r e cos. Vertical directri to the left of the pole: r r r ep e sin ep e sin ep e cos Eample Finding the Polar Equation of a Conic Find the polar equation of the parabola whose focus is the pole and whose directri is the line. Directri: = FIGURE 0.8 (0, 0) r = + sin θ 0 From Figure 0.8, ou can see that the directri is horizontal and above the pole, so ou can choose an equation of the form r Moreover, because the eccentricit of a parabola is e and the distance between the pole and the directri is p, ou have the equation r ep. e sin sin. Now tr Eercise 9.

70 79 Chapter 0 Topics in Analtic Geometr Applications Kepler s Laws (listed below), named after the German astronomer Johannes Kepler (57 60), can be used to describe the orbits of the planets about the sun.. Each planet moves in an elliptical orbit with the sun at one focus.. A ra from the sun to the planet sweeps out equal areas of the ellipse in equal times.. The square of the period (the time it takes for a planet to orbit the sun) is proportional to the cube of the mean distance between the planet and the sun. Although Kepler simpl stated these laws on the basis of observation, the were later validated b Isaac Newton (6 77). In fact, Newton was able to show that each law can be deduced from a set of universal laws of motion and gravitation that govern the movement of all heavenl bodies, including comets and satellites. This is illustrated in the net eample, which involves the comet named after the English mathematician and phsicist Edmund Halle (656 7). If ou use Earth as a reference with a period of ear and a distance of astronomical unit (an astronomical unit is defined as the mean distance between Earth and the sun, or about 9 million miles), the proportionalit constant in Kepler s third law is. For eample, because Mars has a mean distance to the sun of d.5 astronomical units, its period P is given b d P. So, the period of Mars is P.88 ears. Eample Halle s Comet 0 Earth FIGURE 0.8 Sun Halles comet Halle s comet has an elliptical orbit with an eccentricit of e The length of the major ais of the orbit is approimatel 5.88 astronomical units. Find a polar equation for the orbit. How close does Halle s comet come to the sun? Using a vertical ais, as shown in Figure 0.8, choose an equation of the form r ep e sin. Because the vertices of the ellipse occur when and ou can determine the length of the major ais to be the sum of the r-values of the vertices. That is,, a 0.967p 0.967p 9.79p So, p.0 and ep Using this value of ep in the equation, ou have r where r is measured in astronomical units. To find the closest point to the sun (the focus), substitute in this equation to obtain r sin sin 0.59 astronomical unit 55,000,000 miles. Now tr Eercise 6.

71 Section 0.9 Polar Equations of Conics VOCABULARY EXERCISES See for worked-out solutions to odd-numbered eercises. In Eercises, fill in the blanks.. The locus of a point in the plane that moves so that its distance from a fied point (focus) is in a constant ratio to its distance from a fied line (directri) is a.. The constant ratio is the of the conic and is denoted b. ep. An equation of the form r has a directri to the of the pole. e cos. Match the conic with its eccentricit. (a) e < (b) e (c) e > (i) parabola (ii) hperbola (iii) ellipse SKILLS AND APPLICATIONS In Eercises 5 8, write the polar equation of the conic for e, e 0.5, and e.5. Identif the conic for each equation. Verif our answers with a graphing utilit. 5. e e r 6. r e cos e cos 7. e e r 8. r e sin e sin In Eercises 9, match the polar equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (c) (e) (b) (d) (f) r cos 0. r. r sin. r. r sin. r In Eercises 5 8, identif the conic and sketch its graph r 6. r cos sin r 8. r sin cos 9. r 0. r cos sin. 6 9 r. r sin cos. 5 r. r sin cos 5. r 6. r 6 cos 6 sin 7. r 8. r cos sin In Eercises 9, use a graphing utilit to graph the polar equation. Identif the graph r 0. r sin sin. r. r cos cos. r. r 7 sin cos cos cos sin

72 796 Chapter 0 Topics in Analtic Geometr In Eercises 5 8, use a graphing utilit to graph the rotated conic. 5. r cos (See Eercise 5.) 6. r sin (See Eercise 0.) 7. 6 r sin 6 (See Eercise.) 8. 5 r cos (See Eercise.) In Eercises 9 5, find a polar equation of the conic with its focus at the pole. Conic Eccentricit Directri 9. Parabola 0. Parabola. Ellipse. Ellipse. Hperbola. Hperbola Conic e e e e e e Verte or Vertices 5. Parabola 6. Parabola 7. Parabola 8. Parabola 9. Ellipse 50. Ellipse 5. Ellipse 5. Hperbola 5. Hperbola 5. Hperbola, 8, 0 5, 0,, 0, 0,,,, 0, 0,,, 0, 8, 0,, 9,,,, 55. PLANETARY MOTION The planets travel in elliptical orbits with the sun at one focus. Assume that the focus is at the pole, the major ais lies on the polar ais, and the length of the major ais is a (see figure). Show that the polar equation of the orbit is r a e e cos, where e is the eccentricit. Planet r Sun θ a PLANETARY MOTION Use the result of Eercise 55 to show that the minimum distance ( perihelion distance) from the sun to the planet is r a e and the maimum distance ( aphelion distance) is r a e. PLANETARY MOTION In Eercises 57 6, use the results of Eercises 55 and 56 to find the polar equation of the planet s orbit and the perihelion and aphelion distances. 57. Earth a miles, e Saturn a kilometers, e Venus a kilometers, e Mercur a miles, e Mars a miles, e Jupiter a kilometers, e ASTRONOMY The comet Encke has an elliptical orbit with an eccentricit of e The length of the major ais of the orbit is approimatel. astronomical units. Find a polar equation for the orbit. How close does the comet come to the sun? 6. ASTRONOMY The comet Hale-Bopp has an elliptical orbit with an eccentricit of e The length of the major ais of the orbit is approimatel 500 astronomical units. Find a polar equation for the orbit. How close does the comet come to the sun? 65. SATELLITE TRACKING A satellite in a 00-mile-high circular orbit around Earth has a velocit of approimatel 7,500 miles per hour. If this velocit is multiplied b, the satellite will have the minimum velocit necessar to escape Earth s gravit and will follow a parabolic path with the center of Earth as the focus (see figure). 00 miles Circular orbit (a) Find a polar equation of the parabolic path of the satellite (assume the radius of Earth is 000 miles). (b) Use a graphing utilit to graph the equation ou found in part (a). (c) Find the distance between the surface of the Earth and the satellite when 0. Parabolic path (d) Find the distance between the surface of Earth and the satellite when Not drawn to scale

73 Section 0.9 Polar Equations of Conics ROMAN COLISEUM The Roman Coliseum is an elliptical amphitheater measuring approimatel 88 meters long and 56 meters wide. (a) Find an equation to model the coliseum that is of the form (b) Find a polar equation to model the coliseum. (Assume e and p 5.98.) (c) Use a graphing utilit to graph the equations ou found in parts (a) and (b). Are the graphs the same? Wh or wh not? (d) In part (c), did ou prefer graphing the rectangular equation or the polar equation? Eplain. EXPLORATION TRUE OR FALSE? In Eercises 67 70, determine whether the statement is true or false. Justif our answer. 67. For a given value of e > over the interval to the graph of r is the same as the graph of r 68. The graph of r has a horizontal directri above the pole. 69. The conic represented b the following equation is an ellipse. 70. The conic represented b the following equation is a parabola. r 7. WRITING Eplain how the graph of each conic differs 5 from the graph of r. (See Eercise 7.) sin (a) a, r b. e e cos e. e cos sin 6 r 9 cos 6 cos 5 cos (b) r 5 sin 0 (c) (d) 7. Show that the polar equation of the ellipse a b is 7. Show that the polar equation of the hperbola a b is In Eercises 75 80, use the results of Eercises 7 and 7 to write the polar form of the equation of the conic Hperbola One focus: 5, Vertices:,,, 80. Ellipse One focus:, 0 Vertices: 5, 0, 5, 8. Consider the polar equation r (a) Identif the conic without graphing the equation. (b) Without graphing the following polar equations, describe how each differs from the given polar equation. (c) Use a graphing utilit to verif our results in part (b). 8. The equation r r. 0. cos r 5 cos 0. cos ep ± e sin r r r b e cos b e cos r 0. sin is the equation of an ellipse with e <. What happens to the lengths of both the major ais and the minor ais when the value of e remains fied and the value of p changes? Use an eample to eplain our reasoning. 5 sin 7. CAPSTONE In our own words, define the term eccentricit and eplain how it can be used to classif conics

74 798 Chapter 0 Topics in Analtic Geometr Section 0.5 Section 0. Section 0. Section 0. Section 0. 0 CHAPTER SUMMARY What Did You Learn? Eplanation/Eamples Review Eercises Find the inclination of a line (p. 76). Find the angle between two lines (p. 77). Find the distance between a point and a line (p. 78). Recognize a conic as the intersection of a plane and a double-napped cone (p. 7). Write equations of parabolas in standard form and graph parabolas (p. 7). Use the reflective propert of parabolas to solve real-life problems (p. 76). Write equations of ellipses in standard form and graph ellipses (p. 7). Use properties of ellipses to model and solve real-life problems (p. 76). Find eccentricities (p. 76). Write equations of hperbolas in standard form (p. 75) and find asmptotes of and graph hperbolas (p. 75). Use properties of hperbolas to solve real-life problems (p. 756). Classif conics from their general equations (p. 757). Rotate the coordinate aes to einate the -term in equations of conics (p. 76). Use the discriminant to classif conics (p. 765). If a nonvertical line has inclination and slope m, then m tan. If two nonperpendicular lines have slopes m and m, the angle between the lines is tan The distance between the point, and the line A B C 0 is In the formation of the four basic conics, the intersecting plane does not pass through the verte of the cone. (See Figure 0.9.) The standard form of the equation of a parabola with verte at h, k is h p k, p 0 (vertical ais), or k p h, p 0 (horizontal ais). The tangent line to a parabola at a point P makes equal angles with () the line passing through P and the focus and () the ais of the parabola. Horizontal Major Ais h a k b Vertical Major Ais The properties of ellipses can be used to find distances from Earth s center to the moon s center in its orbit. (See Eample.) The eccentricit e of an ellipse is given b e ca. Horizontal Transverse Ais h k a b Asmptotes k ± ba h m m m m. d A B C A B. h b Vertical Transverse Ais k h a b Asmptotes k ± ab h The properties of hperbolas can be used in radar and other detection sstems. (See Eample 5.) The graph of A C D E F 0 is a circle if A C, a parabola if AC 0, an ellipse if AC > 0, and a hperbola if AC < 0. The equation A B C D E F 0 can be rewritten as A C D E F 0 b rotating the coordinate aes through an angle, where cot A CB. The graph of A B C D E F 0 is, ecept in degenerate cases, an ellipse or a circle if B AC < 0, a parabola if B AC 0, and a hperbola if B AC > 0. k a 5 8 9, 0, , ,

75 Chapter Summar 799 Section 0.9 Section 0.8 Section 0.7 Section 0.6 What Did You Learn? Eplanation/Eamples Review Eercises Evaluate sets of parametric equations for given values of the parameter (p. 769). Sketch curves that are represented b sets of parametric equations (p. 770). Rewrite sets of parametric equations as single rectangular equations b einating the parameter (p. 77). Find sets of parametric equations for graphs (p. 77). Plot points on the polar coordinate sstem (p. 777). Convert points (p. 778) and equations (p. 780) from rectangular to polar form and vice versa. Use point plotting (p. 78) and smmetr (p. 78) to sketch graphs of polar equations. Use zeros and maimum r-values to sketch graphs of polar equations (p. 785). Recognize special polar graphs (p. 787). Define conics in terms of eccentricit (p. 79). Write and graph equations of conics in polar form (p. 79). Use equations of conics in polar form to model real-life problems (p. 79). If f and g are continuous functions of t on an interval I, the 5, 5 set of ordered pairs ft, gt is a plane curve C. The equations ft and gt are parametric equations for C, and t is the parameter. Sketching a curve represented b parametric equations requires plotting points in the -plane. Each set of coordinates, is determined from a value chosen for t. To einate the parameter in a pair of parametric equations, solve for t in one equation and substitute that value of t into the other equation. The result is the corresponding rectangular equation. When finding a set of parametric equations for a given graph, remember that the parametric equations are not unique. O r = directed distance Polar Coordinates r, and Rectangular Coordinates, Polar-to-Rectangular: r cos, r sin Rectangular-to-Polar: tan r To convert a rectangular equation to polar form, replace b r cos and b r sin. Converting from a polar equation to rectangular form is more comple. Graphing a polar equation b point plotting is similar to graphing a rectangular equation. A polar graph is smmetric with respect to the following if the given substitution ields an equivalent equation.. Line : Replace r, b r, or r,.. Polar ais: Replace r, b r, or r,.. Pole: Replace r, b r, or r,. Two additional aids to graphing polar equations involve knowing the -values for which r is maimum and knowing the -values for which r 0. θ = directed angle P = (r, θ) Polar ais, Several tpes of graphs, such as açons, rose curves, circles, and lemniscates, have equations that are simpler in polar form than in rectangular form. (See page 787.) The eccentricit of a conic is denoted b e. ellipse: e < parabola: e hperbola: e > The graph of a polar equation of the form () r ep ± e cos or () r ep ± e sin is a conic, where e > 0 is the eccentricit and p is the distance between the focus (pole) and the directri. Equations of conics in polar form can be used to model the orbit of Halle s comet. (See Eample.) ,

76 800 Chapter 0 Topics in Analtic Geometr 0 REVIEW EXERCISES See for worked-out solutions to odd-numbered eercises. 0. In Eercises, find the inclination (in radians and degrees) of the line with the given characteristics.. Passes through the points, and, 5. Passes through the points, and, 7. Equation:. Equation: 5 7 (, 0) (0, ) (, 0).5 cm In Eercises 5 8, find the angle between the lines. (in radians and degrees) In Eercises 9 and 0, find the distance between the point and the line. Point 9. 5, 0. 0, Line In Eercises and, state what tpe of conic is formed b the intersection of the plane and the double-napped cone... In Eercises 6, find the standard form of the equation of the parabola with the given characteristics. Then graph the parabola.. Verte: 0, 0. Verte:, 0 Focus:, 0 Focus: 0, 0 5. Verte: 0, 6. Verte:, Directri: Directri: 0 In Eercises 7 and 8, find an equation of the tangent line to the parabola at the given point, and find the -intercept of the line. 7.,, 8.,, 8 9. ARCHITECTURE A parabolic archwa is meters high at the verte. At a height of 0 meters, the width of the archwa is 8 meters (see figure). How wide is the archwa at ground level? FIGURE FOR 9 FIGURE FOR 0 0. FLASHLIGHT The light bulb in a flashlight is at the focus of its parabolic reflector,.5 centimeters from the verte of the reflector (see figure). Write an equation of a cross section of the flashlight s reflector with its focus on the positive -ais and its verte at the origin. 0. In Eercises, find the standard form of the equation of the ellipse with the given characteristics. Then graph the ellipse.. Vertices:, 0, 8, 0; foci: 0, 0, 6, 0. Vertices:,,, 7; foci:,,, 6. Vertices: 0,,, ; endpoints of the minor ais:, 0,,. Vertices:,,, ; endpoints of the minor ais: 6, 5,, 5 5. ARCHITECTURE A semielliptical archwa is to be formed over the entrance to an estate. The arch is to be set on pillars that are 0 feet apart and is to have a height (atop the pillars) of feet. Where should the foci be placed in order to sketch the arch? 6. WADING POOL You are building a wading pool that is in the shape of an ellipse. Your plans give an equation for the elliptical shape of the pool measured in feet as Find the longest distance across the pool, the shortest distance, and the distance between the foci. In Eercises 7 0, find the center, vertices, foci, and eccentricit of the ellipse

77 Review Eercises In Eercises, find the standard form of the equation of the hperbola with the given characteristics.. Vertices: 0, ±; foci: 0, ±. Vertices:,,, ; foci:,,,. Foci: 0, 0, 8, 0; asmptotes: ±. Foci:, ±; asmptotes: ± In Eercises 5 8, find the center, vertices, foci, and the equations of the asmptotes of the hperbola, and sketch its graph using the asmptotes as an aid LORAN Radio transmitting station A is located 00 miles east of transmitting station B. A ship is in an area to the north and 0 miles west of station A. Snchronized radio pulses transmitted at 86,000 miles per second b the two stations are received second sooner from station A than from station B. How far north is the ship? 0. LOCATING AN EXPLOSION Two of our friends live miles apart and on the same east-west street, and ou live halfwa between them. You are having a threewa phone conversation when ou hear an eplosion. Si seconds later, our friend to the east hears the eplosion, and our friend to the west hears it 8 seconds after ou do. Find equations of two hperbolas that would locate the eplosion. (Assume that the coordinate sstem is measured in feet and that sound travels at 00 feet per second.) In Eercises, classif the graph of the equation as a circle, a parabola, an ellipse, or a hperbola In Eercises 5 8, rotate the aes to einate the -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of aes In Eercises 9 5, (a) use the discriminant to classif the graph, (b) use the Quadratic Formula to solve for, and (c) use a graphing utilit to graph the equation In Eercises 5 and 5, (a) create a table of - and -values for the parametric equations using t,, 0,, and, and (b) plot the points, generated in part (a) and sketch a graph of the parametric equations. 5. t and 7 t 5. and 6 t t In Eercises 55 60, (a) sketch the curve represented b the parametric equations (indicate the orientation of the curve) and (b) einate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessar. (c) Verif our result with a graphing utilit. 55. t 56. t t t 57. t 58. t t t 59. cos 60. cos sin 5 sin 6. Find a parametric representation of the line that passes through the points, and 9, Find a parametric representation of the circle with center 5, and radius Find a parametric representation of the ellipse with center,, major ais horizontal and eight units in length, and minor ais si units in length. 6. Find a parametric representation of the hperbola with vertices 0, ± and foci 0, ± In Eercises 65 68, plot the point given in polar coordinates and find two additional polar representations of the point, using 65., < < , 67. 7,.9 68.,.6

78 80 Chapter 0 Topics in Analtic Geometr In Eercises 69 7, a point in polar coordinates is given. Convert the point to rectangular coordinates. 69., , 7. 0, In Eercises 7 76, a point in rectangular coordinates is given. Convert the point to polar coordinates. 7. 0, 7. 5,5 75., 6 76., In Eercises 77 8, convert the rectangular equation to polar form In Eercises 8 88, convert the polar equation to rectangular form. 8. r 5 8. r 85. r cos 86. r 8 sin 87. r sin 88. r cos 5, 0.8 In Eercises 89 98, determine the smmetr of the r, maimum value of r, and an zeros of r. Then sketch the graph of the polar equation (plot additional points if necessar). 89. r r 9. r sin 9. r cos 5 9. r cos 9. r cos 95. r 6 sin 96. r 5 5 cos 97. r cos 98. r cos In Eercises 99 0, identif the tpe of polar graph and use a graphing utilit to graph the equation. 99. r cos 00. r 5 cos 0. r 8 cos 0. r sin 0.9 In Eercises 0 06, identif the conic and sketch its graph r 0. r sin sin r 06. r 5 cos 5 cos In Eercises 07 0, find a polar equation of the conic with its focus at the pole. 07. Parabola Verte:, 08. Parabola Verte:, 09. Ellipse Vertices: 5, 0,, 0. Hperbola Vertices:, 0, 7, 0. EXPLORER 8 On November 7, 96, the United States launched Eplorer 8. Its low and high points above the surface of Earth were 9 miles and,800 miles, respectivel. The center of Earth was at one focus of the orbit (see figure). Find the polar equation of the orbit and find the distance between the surface of Earth (assume Earth has a radius of 000 miles) and the satellite when. ASTEROID An asteroid takes a parabolic path with Earth as its focus. It is about 6,000,000 miles from Earth at its closest approach. Write the polar equation of the path of the asteroid with its verte at Find the distance between the asteroid and Earth when. EXPLORATION TRUE OR FALSE? In Eercises 5, determine whether the statement is true or false. Justif our answer.. The graph of is a hperbola.. Onl one set of parametric equations can represent the line. 5. There is a unique polar coordinate representation of each point in the plane. 6. Consider an ellipse with the major ais horizontal and 0 units in length. The number b in the standard form of the equation of the ellipse must be less than what real number? Eplain the change in the shape of the ellipse as b approaches this number. 7. What is the relationship between the graphs of the rectangular and polar equations? (a) 5, r 5 (b) Eplorer 8 r Earth 0,. a 0.

79 Chapter Test 80 0 CHAPTER TEST See for worked-out solutions to odd-numbered eercises. Take this test as ou would take a test in class. When ou are finished, check our work against the answers given in the back of the book.. Find the inclination of the line Find the angle between the lines 0 and Find the distance between the point 7, 5 and the line 5. In Eercises 7, classif the conic and write the equation in standard form. Identif the center, vertices, foci, and asmptotes (if applicable). Then sketch the graph of the conic Find the standard form of the equation of the parabola with verte,, with a vertical ais, and passing through the point, Find the standard form of the equation of the hperbola with foci 0, 0 and 0, and asmptotes ±. 0. (a) Determine the number of degrees the ais must be rotated to einate the -term of the conic (b) Graph the conic from part (a) and use a graphing utilit to confirm our result.. Sketch the curve represented b the parametric equations cos and sin. Einate the parameter and write the corresponding rectangular equation.. Find a set of parametric equations of the line passing through the points, and 6,. (There are man correct answers.) 5. Convert the polar coordinate, to rectangular form. 6. Convert the rectangular coordinate, to polar form and find two additional polar representations of this point. 5. Convert the rectangular equation 0 to polar form. In Eercises 6 9, sketch the graph of the polar equation. Identif the tpe of graph. 6. r 7. r cos sin 8. r sin 9. r sin 0. Find a polar equation of the ellipse with focus at the pole, eccentricit e, and directri.. A straight road rises with an inclination of 0.5 radian from the horizontal. Find the slope of the road and the change in elevation over a one-mile stretch of the road.. A baseball is hit at a point feet above the ground toward the left field fence. The fence is 0 feet high and 75 feet from home plate. The path of the baseball can be modeled b the parametric equations 5 cos t and 5 sin t 6t. Will the baseball go over the fence if it is hit at an angle of Will the baseball go over the fence if 5? 0?

80 PROOFS IN MATHEMATICS Inclination and Slope (p. 76) If a nonvertical line has inclination and slope m, then m tan. (, 0) (, ) θ Proof If m 0, the line is horizontal and So, the result is true for horizontal lines because m 0 tan 0. If the line has a positive slope, it will intersect the -ais. Label this point, 0, as shown in the figure. If, is a second point on the line, the slope is m 0 tan. 0. The case in which the line has a negative slope can be proved in a similar manner. Distance Between a Point and a Line (p. 78), The distance between the point and the line A B C 0 is d A B C. A B (, ) Proof For simplicit, assume that the given line is neither horizontal nor vertical (see figure). B writing the equation A B C 0 in slope-intercept form A B C B (, ) d A C = B B ou can see that the line has a slope of m AB. So, the slope of the line passing through, and perpendicular to the given line is BA, and its equation is BA. These two lines intersect at the point,, where BB A AC A B and Finall, the distance between, and, is d B AB AC A B A A B C B A B C A B A B C A B. AB A BC A B. AB A BC A B 80

81 p > 0 Verte: ( hk, ) Ais: = h Focus: ( hk, + p) (, ) Directri: = k p Parabola with vertical ais Directri: = h p p > 0 Parabolic Paths There are man natural occurrences of parabolas in real life. For instance, the famous astronomer Galileo discovered in the 7th centur that an object that is projected upward and obliquel to the pull of gravit travels in a parabolic path. Eamples of this are the center of gravit of a jumping dolphin and the path of water molecules in a drinking fountain. (, ) Verte: ( hk, ) Focus: ( h + p, k) Parabola with horizontal ais Ais: =k Standard Equation of a Parabola (p. 7) The standard form of the equation of a parabola with verte at h, k is as follows. h p k, p 0 Vertical ais, directri: k p k p h, p 0 Horizontal ais, directri: h p The focus lies on the ais p units (directed distance) from the verte. If the verte is at the origin 0, 0, the equation takes one of the following forms. p p Proof Vertical ais Horizontal ais For the case in which the directri is parallel to the -ais and the focus lies above the verte, as shown in the top figure, if, is an point on the parabola, then, b definition, it is equidistant from the focus h, k p and the directri k p. So, ou have h k p k p h k p k p h k p k p k p k p h k p k pk p k p k pk p h p pk p pk h p k. For the case in which the directri is parallel to the -ais and the focus lies to the right of the verte, as shown in the bottom figure, if, is an point on the parabola, then, b definition, it is equidistant from the focus h p, k and the directri h p. So, ou have h p k h p h p k h p h p h p k h p h p h p h ph p k h p h ph p p ph k p ph k p h. Note that if a parabola is centered at the origin, then the two equations above would simplif to p and p, respectivel. 805

82 Polar Equations of Conics (p. 79) The graph of a polar equation of the form. r ep ± e cos or ep. r ± e sin is a conic, where e > 0 is the eccentricit and p is the distance between the focus (pole) and the directri. P= ( r, θ) r = r cos θ θ p F = (0, 0) Directri Q 0 Proof ep A proof for r with p > 0 is shown here. The proofs of the other cases e cos are similar. In the figure, consider a vertical directri, p units to the right of the focus F 0, 0. If P r, is a point on the graph of r ep e cos the distance between P and the directri is PQ p p r cos p ep e cos cos p e cos e cos p e cos r e. Moreover, because the distance between P and the pole is simpl PF r, the ratio of PF to PQ is PF PQ r r e e e and, b definition, the graph of the equation must be a conic. 806

83 PROBLEM SOLVING This collection of thought-provoking and challenging eercises further eplores and epands upon concepts learned in this chapter.. Several mountain cbers are located in a mountain pass between two peaks. The angles of elevation to the two peaks are 0.8 radian and.0 radians. A range finder shows that the distances to the peaks are 50 feet and 6700 feet, respectivel (see figure). 6. A tour boat travels between two islands that are miles apart (see figure). For a trip between the islands, there is enough fuel for a 0-mile trip. Island Island mi Not drawn to scale (a) Find the angle between the two lines of sight to the peaks. (b) Approimate the amount of vertical cb that is necessar to reach the summit of each peak.. Statuar Hall is an elliptical room in the United States Capitol in Washington D.C. The room is also called the Whispering Galler because a person standing at one focus of the room can hear even a whisper spoken b a person standing at the other focus. This occurs because an sound that is emitted from one focus of an ellipse will reflect off the side of the ellipse to the other focus. Statuar Hall is 6 feet wide and 97 feet long. (a) Find an equation that models the shape of the room. (b) How far apart are the two foci? (c) What is the area of the floor of the room? (The area of an ellipse is A ab. ). Find the equation(s) of all parabolas that have the -ais as the ais of smmetr and focus at the origin.. Find the area of the square inscribed in the ellipse below. FIGURE FOR FIGURE FOR 5 5. The involute of a circle is described b the endpoint P of a string that is held taut as it is unwound from a spool (see figure). The spool does not rotate. Show that rcos + = a b sin 6700 ft.0 radians 50 ft 0.8 radian rsin cos is a parametric representation of the involute of a circle. r θ P (a) Eplain wh the region in which the boat can travel is bounded b an ellipse. (b) Let 0, 0 represent the center of the ellipse. Find the coordinates of each island. (c) The boat travels from one island, straight past the other island to the verte of the ellipse, and back to the second island. How man miles does the boat travel? Use our answer to find the coordinates of the verte. (d) Use the results from parts (b) and (c) to write an equation of the ellipse that bounds the region in which the boat can travel. 7. Find an equation of the hperbola such that for an point on the hperbola, the difference between its distances from the points, and 0, is Prove that the graph of the equation A C D E F 0 is one of the following (ecept in degenerate cases). Conic Condition (a) Circle A C (b) Parabola A 0 or C 0 (but not both) (c) Ellipse AC > 0 (d) Hperbola AC < 0 9. The following sets of parametric equations model projectile motion. v 0 cos t v 0 sin t v 0 cos t h v 0 sin t 6t (a) Under what circumstances would ou use each model? (b) Einate the parameter for each set of equations. (c) In which case is the path of the moving object not affected b a change in the velocit v? Eplain. 807

84 0. As t increases, the ellipse given b the parametric equations cos t and sin t is traced out counterclockwise. Find a parametric representation for which the same ellipse is traced out clockwise.. A hpoccloid has the parametric equations a b cos t b cos a b t b and a b sin t b sin a b t b. Use a graphing utilit to graph the hpoccloid for each value of a and b. Describe each graph. (a) a, b (b) a, b (c) a, b (d) a 0, b (e) a, b (f) a, b. The curve given b the parametric equations t t and is called a strophoid. (a) Find a rectangular equation of the strophoid. (b) Find a polar equation of the strophoid. (c) Use a graphing utilit to graph the strophoid.. The rose curves described in this chapter are of the form r a cos n or where n is a positive integer that is greater than or equal to. Use a graphing utilit to graph r a cos n and r a sin n for some noninteger values of n. Describe the graphs.. What conic section is represented b the polar equation r a sin b cos? 5. The graph of the polar equation r e cos cos sin 5 is called the butterfl curve, as shown in the figure. t t t r a sin n r = e cos θ cos θ + sin ( 5 θ ( (a) The graph shown was produced using 0. Does this show the entire graph? Eplain our reasoning. (b) Approimate the maimum r-value of the graph. Does this value change if ou use 0 instead of 0? Eplain. 6. Use a graphing utilit to graph the polar equation r cos 5 n cos for 0 for the integers n 5 to n 5. As ou graph these equations, ou should see the graph change shape from a heart to a bell. Write a short paragraph eplaining what values of n produce the heart portion of the curve and what values of n produce the bell portion. 7. The planets travel in elliptical orbits with the sun at one focus. The polar equation of the orbit of a planet with one focus at the pole and major ais of length a (see figure) is r e a e cos where e is the eccentricit. The minimum distance (perihelion) from the sun to a planet is r a e and the maimum distance (aphelion) is r a e. For the planet Neptune, a kilometers and e For the dwarf planet Pluto, a kilometers and e Planet r Sun θ (a) Find the polar equation of the orbit of each planet. (b) Find the perihelion and aphelion distances for each planet. (c) Use a graphing utilit to graph the equations of the orbits of Neptune and Pluto in the same viewing window. (d) Is Pluto ever closer to the sun than Neptune? Until recentl, Pluto was considered the ninth planet. Wh was Pluto called the ninth planet and Neptune the eighth planet? (e) Do the orbits of Neptune and Pluto intersect? Will Neptune and Pluto ever collide? Wh or wh not? a 0 808

85 Analtic Geometr in Three Dimensions. The Three-Dimensional Coordinate Sstem. Vectors in Space. The Cross Product of Two Vectors. Lines and Planes in Space In Mathematics A three-dimensional coordinate sstem is formed b passing a z-ais perpendicular to both the - and -aes at the origin. When the concept of vectors is etended to three-dimensional space, the are denoted b ordered triples v v, v, v. In Real Life The concepts discussed in this chapter have man applications in phsics and engineering. For instance, vectors can be used to find the angle between two adjacent sides of a grain elevator chute. (See Eercise 6, page 89.) George Ostertag/PhotoLibrar IN CAREERS There are man careers that use topics in analtic geometr in three dimensions. Several are listed below. Architect Eercise 77, page 86 Geographer Eercise 78, page 86 Cclist Eercises 6 and 6, page 80 Consumer Research Analst Eercise 6, page

86 80 Chapter Analtic Geometr in Three Dimensions. THE THREE-DIMENSIONAL COORDINATE SYSTEM What ou should learn Plot points in the three-dimensional coordinate sstem. Find distances between points in space and find midpoints of line segments joining points in space. Write equations of spheres in standard form and find traces of surfaces in space. Wh ou should learn it The three-dimensional coordinate sstem can be used to graph equations that model surfaces in space, such as the spherical shape of Earth, as shown in Eercise 78 on page 86. The Three-Dimensional Coordinate Sstem Recall that the Cartesian plane is determined b two perpendicular number lines called the -ais and the -ais. These aes, together with their point of intersection (the origin), allow ou to develop a two-dimensional coordinate sstem for identifing points in a plane. To identif a point in space, ou must introduce a third dimension to the model. The geometr of this three-dimensional model is called solid analtic geometr. You can construct a three-dimensional coordinate sstem b passing a z-ais perpendicular to both the - and -aes at the origin. Figure. shows the positive portion of each coordinate ais. Taken as pairs, the aes determine three coordinate planes: the -plane, the z-plane, and the z-plane. These three coordinate planes separate the three-dimensional coordinate sstem into eight octants. The first octant is the one in which all three coordinates are positive. In this three-dimensional sstem, a point P in space is determined b an ordered triple,, z, where,, and z are as follows. directed distance from z-plane to P directed distance from z-plane to P z directed distance from -plane to P z z NASA z-plane z-plane (,, z) -plane FIGURE. FIGURE. A three-dimensional coordinate sstem can have either a left-handed or a right-handed orientation. In this tet, ou will work eclusivel with right-handed sstems, as illustrated in Figure.. In a right-handed sstem, Octants II, III, and IV are found b rotating counterclockwise around the positive z-ais. Octant V is verticall below Octant I. Octants VI, VII, and VIII are then found b rotating counterclockwise around the negative z-ais. See Figure.. z z z z z z z z Octant I Octant II Octant III Octant IV Octant V Octant VI Octant VII Octant VIII FIGURE.

87 Section. The Three-Dimensional Coordinate Sstem 8 (,, ) 6 FIGURE. z (,, ) (,, 0) (, 6, ) 6 Eample Plotting Points in Space Plot each point in space. a.,, b., 6, c.,, 0 d.,, To plot the point,,, notice that,, and z. To help visualize the point, locate the point, in the -plane (denoted b a cross in Figure.). The point,, lies three units above the cross. The other three points are also shown in Figure.. Now tr Eercise. z (,, z ) The Distance and Midpoint Formulas Man of the formulas established for the two-dimensional coordinate sstem can be etended to three dimensions. For eample, to find the distance between two points in space, ou can use the Pthagorean Theorem twice, as shown in Figure.5. Note that a, b, and c z z. (,, z ) (,, z ) a a b + b (,, z ) Distance Formula in Space The distance between the points,, z and,, z given b the Distance Formula in Space is d z z. d = (,, z ) FIGURE.5 z a+ b+ c a + (,, z ) c b (,, z ) Eample Finding the Distance Between Two Points in Space Find the distance between, 0, and,,. d z z Distance Formula in Space 0 Substitute. 6 5 Simplif. Simplif. Now tr Eercise 7. Notice the similarit between the Distance Formulas in the plane and in space. The Midpoint Formulas in the plane and in space are also similar. Midpoint Formula in Space The midpoint of the line segment joining the points,, z and,, z given b the Midpoint Formula in Space is,, z z.

88 8 Chapter Analtic Geometr in Three Dimensions Eample Using the Midpoint Formula in Space Find the midpoint of the line segment joining 5,, and 0,,. Using the Midpoint Formula in Space, the midpoint is 5 0,, 5,, 7 as shown in Figure.6. z (5,, ) Midpoint: ( 5,, 7 ) (0,, ) FIGURE.6 FIGURE.7 z (,, z) r ( hk,, j) Sphere: radius r; center ( hk,, j) Now tr Eercise 5. The Equation of a Sphere A sphere with center h, k, j and radius r is defined as the set of all points,, z such that the distance between,, z and h, k, j is r, as shown in Figure.7. Using the Distance Formula, this condition can be written as h k z j r. B squaring each side of this equation, ou obtain the standard equation of a sphere. Standard Equation of a Sphere The standard equation of a sphere with center h, k, j and radius r is given b h k z j r. Notice the similarit of this formula to the equation of a circle in the plane. h k z j r Equation of sphere in space h k r Equation of circle in the plane As is true with the equation of a circle, the equation of a sphere is simplified when the center lies at the origin. In this case, the equation is z r. Sphere with center at origin

89 Section. The Three-Dimensional Coordinate Sstem 8 Eample Finding the Equation of a Sphere Find the standard equation of the sphere with center,, and radius. Does this sphere intersect the -plane? h k z j r Standard equation z Substitute. From the graph shown in Figure.8, ou can see that the center of the sphere lies three units above the -plane. Because the sphere has a radius of, ou can conclude that it does intersect the -plane at the point,, 0. z 5 5 FIGURE.8 (,, ) r = 6 7 (,, 0) Now tr Eercise 5. Eample 5 Finding the Center and Radius of a Sphere Find the center and radius of the sphere given b z 6z 8 0. z To obtain the standard equation of this sphere, complete the square as follows. (,, ) r = 6 FIGURE.9 6 z 6z 8 0 z 6z 8 z 6z z 6 So, the center of the sphere is,,, and its radius is 6. See Figure.9. Now tr Eercise 6. Note in Eample 5 that the points satisfing the equation of the sphere are surface points, not interior points. In general, the collection of points satisfing an equation involving,, and z is called a surface in space.

90 8 Chapter Analtic Geometr in Three Dimensions Finding the intersection of a surface with one of the three coordinate planes (or with a plane parallel to one of the three coordinate planes) helps one visualize the surface. Such an intersection is called a trace of the surface. For eample, the -trace of a surface consists of all points that are common to both the surface and the -plane. Similarl, the z-trace of a surface consists of all points that are common to both the surface and the z-plane. Eample 6 Finding a Trace of a Surface Sketch the -trace of the sphere given b z 5. To find the -trace of this surface, use the fact that ever point in the -plane has a z-coordinate of zero. B substituting z 0 into the original equation, the resulting equation will represent the intersection of the surface with the -plane. -trace: ( ) + ( ) = z z 5 Write original equation. 0 5 Substitute 0 for z. 6 5 Simplif. 9 Subtract 6 from each side. Equation of circle You can see that the -trace is a circle of radius, as shown in Figure.0. Now tr Eercise 7. Sphere: ( ) + ( ) + ( z + ) = 5 FIGURE.0 TECHNOLOGY Most three-dimensional graphing utilities and computer algebra sstems represent surfaces b sketching several traces of the surface. The traces are usuall taken in equall spaced parallel planes. To graph an equation involving,, and z with a three-dimensional function grapher, ou must first set the graphing mode to three-dimensional and solve the equation for z. After entering the equation, ou need to specif a rectangular viewing cube (the three-dimensional analog of a viewing window). For instance, to graph the top half of the sphere from Eample 6, solve the equation for z to obtain the solutions z ± 5. The equation z 5 represents the top half of the sphere. Enter this equation, as shown in Figure.. Net, use the viewing cube shown in Figure.. Finall, ou can displa the graph, as shown in Figure.. FIGURE. FIGURE. FIGURE.

91 Section. The Three-Dimensional Coordinate Sstem 85. EXERCISES VOCABULARY: Fill in the blanks. See for worked-out solutions to odd-numbered eercises.. A coordinate sstem can be formed b passing a z-ais perpendicular to both the -ais and the -ais at the origin.. The three coordinate planes of a three-dimensional coordinate sstem are the, the, and the.. The coordinate planes of a three-dimensional coordinate sstem separate the coordinate sstem into eight.. The distance between the points,, z and,, z can be found using the in Space. 5. The midpoint of the line segment joining the points,, z and,, z given b the Midpoint Formula in Space is. 6. A is the set of all points,, z such that the distance between,, z and a fied point h, k, j is r. 7. A in is the collection of points satisfing an equation involving,, and z. 8. The intersection of a surface with one of the three coordinate planes is called a of the surface. SKILLS AND APPLICATIONS In Eercises 9 and 0, approimate the coordinates of the points. 9. z 0. In Eercises 6, plot each point in the same threedimensional coordinate sstem.. (a),,. (a), 0, 0 (b),, (b),,. (a),, 0. (a) 0,, (b),, (b), 0, 5. (a),, 5 6. (a) 5,, (b),, (b) 5,, B A C In Eercises 7 0, find the coordinates of the point. 7. The point is located three units behind the z-plane, four units to the right of the z-plane, and five units above the -plane. 8. The point is located seven units in front of the z-plane, two units to the left of the z-plane, and one unit below the -plane. 9. The point is located on the -ais, eight units in front of the z-plane. 0. The point is located in the z-plane, one unit to the right of the z-plane, and si units above the -plane. z B A C In Eercises 6, determine the octant(s) in which,, z is located so that the condition(s) is (are) satisfied.. > 0, < 0, z > 0. < 0, > 0, z < 0. z > 0. < 0 5. < 0 6. z > 0 In Eercises 7 6, find the distance between the points. 7. 0, 0, 0, 5,, 6 8., 0, 0, 7, 0, 9.,, 5, 7,, 8 0.,, 5, 8,, 6.,,, 6, 0, 9.,, 7,,, 7. 0,, 0,, 0, 0.,, 0, 0, 6, 5. 6, 9,,,, 5 6., 0, 6, 8, 8, 0 In Eercises 7 0, find the lengths of the sides of the right triangle with the indicated vertices. Show that these lengths satisf the Pthagorean Theorem. 7. 0, 0,,, 5,, 0,, 0 8.,,,,,,, 5, , 0, 0,,,,,, 0., 0,,,,,, 0, In Eercises, find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither..,,, 5,,,,,. 5,,, 7,,,, 5,.,,, 8,,,,,.,,,, 0, 0,, 6,

92 86 Chapter Analtic Geometr in Three Dimensions In Eercises 5 5, find the midpoint of the line segment joining the points , 0, 0,,,, 5,,,,, 6, 0,,,, 5,,, 7, 5,, 5, 6,, 7 0,, 5,,, 7, 8, 0, 7,, 9, 5,, 9,, In Eercises 5 60, find the standard form of the equation of the sphere with the given characteristics Center:,, ; radius: Center:,, ; radius: Center: 5, 0, ; radius: 6 Center:,, ; radius: 5 Center:, 7, 5 ; diameter: 0 Center: 0, 5, 9 ; diameter: 8 Endpoints of a diameter:, 0, 0, 0, 0, 6 Endpoints of a diameter:, 0, 0, 0, 5, 0 In Eercises 6 70, find the center and radius of the sphere z 6 0 z 9 0 z 6z 0 0 z z 8z 9 0 z 8 6z z 8 6 7z 7 0 z 6 z z z 8z 0 In Eercises 7 7, sketch the graph of the equation and sketch the specified trace z 6; z-trace z 5; z-trace z 9; z-trace z ; -trace In Eercises 75 and 76, use a three-dimensional graphing utilit to graph the sphere. 75. z 6 8 0z z 6 8z ARCHITECTURE A spherical building has a diameter of 05 feet. The center of the building is placed at the origin of a three-dimensional coordinate sstem. What is the equation of the sphere? 78. GEOGRAPHY Assume that Earth is a sphere with a radius of 000 miles. The center of Earth is placed at the origin of a three-dimensional coordinate sstem. (a) What is the equation of the sphere? (b) Lines of longitude that run north-south could be represented b what trace(s)? What shape would each of these traces form? (c) Lines of latitude that run east-west could be represented b what trace(s)? What shape would each of these traces form? EXPLORATION TRUE OR FALSE? In Eercises 79 and 80, determine whether the statement is true or false. Justif our answer. 79. In the ordered triple,, z that represents point P in space, is the directed distance from the -plane to P. 80. The surface consisting of all points,, z in space that are the same distance r from the point h, j, k has a circle as its -trace. 8. THINK ABOUT IT What is the z-coordinate of an point in the -plane? What is the -coordinate of an point in the z-plane? What is the -coordinate of an point in the z-plane? 8. CAPSTONE Find the equation of the sphere that has the points,, 6 and,, as endpoints of a diameter. Eplain how this problem gives ou a chance to use these formulas: the Distance Formula in Space, the Midpoint Formula in Space, and the standard equation of a sphere. 8. A sphere intersects the z-plane. Describe the trace. 8. A plane intersects the -plane. Describe the trace. 85. A line segment has,, z as one endpoint and m, m, zm as its midpoint. Find the other endpoint,, z of the line segment in terms of,, z, m, m, and zm. 86. Use the result of Eercise 85 to find the coordinates of the endpoint of a line segment if the coordinates of the other endpoint and the midpoint are, 0, and 5, 8, 7, respectivel.

93 Section. Vectors in Space 87. VECTORS IN SPACE What ou should learn Find the component forms of the unit vectors in the same direction of, the magnitudes of, the dot products of, and the angles between vectors in space. Determine whether vectors in space are parallel or orthogonal. Use vectors in space to solve real-life problems. Wh ou should learn it Vectors in space can be used to represent man phsical forces, such as tension in the cables used to support auditorium lights, as shown in Eercise 60 on page 8. Vectors in Space Phsical forces and velocities are not confined to the plane, so it is natural to etend the concept of vectors from two-dimensional space to three-dimensional space. In space, vectors are denoted b ordered triples v v, v, v. Component form The zero vector is denoted b 0 0, 0, 0. Using the unit vectors i, 0, 0, j 0,, 0, and k 0, 0, in the direction of the positive z-ais, the standard unit vector notation for v is v v i v j v k Unit vector form as shown in Figure.. If v is represented b the directed line segment from Pp, p, p to Qq, q, q, as shown in Figure.5, the component form of v is produced b subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point v v, v, v q p, q p, q p. 0, 0, k i, 0, 0 z v, v, v j 0,, 0 P( p, p, p) v z Q( q, q, q) SuperStock FIGURE. FIGURE.5 Vectors in Space. Two vectors are equal if and onl if their corresponding components are equal.. The magnitude (or length) of u u, u, u is u u u u.. A unit vector u in the direction of v is u v v,. The sum of u u, u, u and v v, v, v is u v u v, u v, u v. Vector addition 5. The scalar multiple of the real number c and u u, u, u is cu cu, cu, cu. Scalar multiplication 6. The dot product of u u, u, u and v v, v, v is u v u v u v u v. v 0. Dot product

94 88 Chapter Analtic Geometr in Three Dimensions Eample Finding the Component Form of a Vector Find the component form and magnitude of the vector v having initial point,, and terminal point, 6,. Then find a unit vector in the direction of v. The component form of v is v, 6, 0,, which implies that its magnitude is v 0 8. The unit vector in the direction of v is u v v 0,, 0,, 0,,. Now tr Eercise 9. TECHNOLOGY Some graphing utilities have the capabilit to perform vector operations, such as the dot product. Consult the user s guide for our graphing utilit for specific instructions. Eample Finding the Dot Product of Two Vectors Find the dot product of 0,, and,,. 0,,,, Note that the dot product of two vectors is a real number, not a vector. Now tr Eercise. As was discussed in Section 6., the angle between two nonzero vectors is the angle, 0, between their respective standard position vectors, as shown in Figure.6. This angle can be found using the dot product. (Note that the angle between the zero vector and another vector is not defined.) v u u θ Origin FIGURE.6 v Angle Between Two Vectors If is the angle between two nonzero vectors u and v, then cos u v u v. If the dot product of two nonzero vectors is zero, the angle between the vectors is 90 (recall that cos 90 0). Such vectors are called orthogonal. For instance, the standard unit vectors i, j, and k are orthogonal to each other.

95 Section. Vectors in Space 89 z Eample Finding the Angle Between Two Vectors Find the angle between u, 0, and v,, 0. u =, 0, θ 6.9 cos u v u v, 0,,, 0, 0,,, 0 50 This implies that the angle between the two vectors is FIGURE.7 v =,, 0 arccos as shown in Figure.7. Now tr Eercise 5. Parallel Vectors u u = v w= v Recall from the definition of scalar multiplication that positive scalar multiples of a nonzero vector v have the same direction as v, whereas negative multiples have the direction opposite that of v. In general, two nonzero vectors u and v are parallel if there is some scalar c such that u cv. For eample, in Figure.8, the vectors u, v, and w are parallel because u v and w v. v Eample Parallel Vectors FIGURE.8 w Vector w has initial point,, 0 and terminal point,,. Which of the following vectors is parallel to w? a. u, 8, b. v, 8, Begin b writing w in component form. w,, 0,, a. Because u, 8,,, w ou can conclude that u is parallel to w. b. In this case, ou need to find a scalar c such that, 8, c,,. However, equating corresponding components produces c for the first two components and c for the third. So, the equation has no solution, and the vectors v and w are not parallel. Now tr Eercise 9.

96 80 Chapter Analtic Geometr in Three Dimensions You can use vectors to determine whether three points are collinear (lie on the same line). The points P, Q, and R are collinear if and onl if the vectors PQ \ and PR \ are parallel. Eample 5 Using Vectors to Determine Collinear Points Determine whether the points P,,, Q5,, 6, and R,, 0 are collinear. The component forms of PQ \ and PR \ are and PQ \ 5,, 6, 5, PR \,, 0 6, 0,. Because PR \ PQ \, ou can conclude that the are parallel. Therefore, the points P, Q, and R lie on the same line, as shown in Figure.9. R(,, 0) z PR = 6, 0, PQ =, 5, P(,, ) Q(5,, 6) FIGURE.9 Now tr Eercise 7. Eample 6 Finding the Terminal Point of a Vector The initial point of the vector v,, is P,, 6. What is the terminal point of this vector? Using the component form of the vector whose initial point is P,, 6 and whose terminal point is Qq, q, q, ou can write PQ \ q p, q p, q p q, q, q 6,,. This implies that q, q, and q 6. The solutions of these three equations are q 7, q, and q 5. So, the terminal point is Q7,, 5. Now tr Eercise 5.

97 Section. Vectors in Space 8 Application In Section 6., ou saw how to use vectors to solve an equilibrium problem in a plane. The net eample shows how to use vectors to solve an equilibrium problem in space. Eample 7 Solving an Equilibrium Problem P(, 0, 0) u FIGURE.0 z R(, 0, 0) z Q(0,, 0) v S(0,, ) w A weight of 80 pounds is supported b three ropes. As shown in Figure.0, the weight is located at S0,,. The ropes are tied to the points P, 0, 0, Q0,, 0, and R, 0, 0. Find the force (or tension) on each rope. The (downward) force of the weight is represented b the vector w 0, 0, 80. The force vectors corresponding to the ropes are as follows. 0, 0, 0 u u SP\ u SP \ u,, 0,, 0 v v SQ\ v0 SQ \ v 5 0, 5, 5 0, 0, 0 z z SR\ z SR \ z,, For the sstem to be in equilibrium, it must be true that u v z w 0 or u v z w. This ields the following sstem of linear equations. u z 000 u 5 v z 000 u 5 v z 80 Using the techniques demonstrated in Chapter 7, ou can find the solution of the sstem to be u 60.0 v 56.7 z So, the rope attached at point P has 60 pounds of tension, the rope attached at point Q has about 56.7 pounds of tension, and the rope attached at point R has 60 pounds of tension. Now tr Eercise 59.

98 8 Chapter Analtic Geometr in Three Dimensions. EXERCISES See for worked-out solutions to odd-numbered eercises. VOCABULARY: Fill in the blanks.. The vector is denoted b 0 0, 0, 0.. The standard unit vector notation for a vector v is given b.. The of a vector v is produced b subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point.. If the dot product of two nonzero vectors is zero, the angle between the vectors is 90 and the vectors are called. 5. Two nonzero vectors u and v are if there is some scalar c such that u cv. 6. The points P, Q, and R are if and onl if the vectors PQ \ and PR \ are parallel. SKILLS AND APPLICATIONS In Eercises 7 and 8, (a) find the component form of the vector v and (b) sketch the vector with its initial point at the origin (, 0, ) z (0,, ) v In Eercises 9 and 0, (a) write the component form of the vector v, (b) find the magnitude of v, and (c) find a unit vector in the direction of v. 9. Initial point: 6,, Terminal point:,, 0. Initial point: 7,, 5 Terminal point: 0, 0, In Eercises, sketch each scalar multiple of v.. v,, (a) v (b) v (c) v (d) 0v. v,, (a) v (b) v 5 (c) v (d) v. v i j k 5 (a) v (b) v (c) v (d) 0v. v i j k (a) v (b) v (c) (d) 0v In Eercises 5 8, find the vector z, given and w < > 5, 0, z u v 6. z 7u v 5 w 7. z u w 8. u v z 0 v <,, >, u <,, >, v z (,, ) v (,, 0) In Eercises 9 8, find the magnitude of v. 9. v 7, 8, 7 0. v, 0, 5. v,,. v, 0,. v i j k. v i j k 5. v i j 7k 6. v i j 6k 7. Initial point:,, Terminal point:, 0, 8. Initial point: 0,, 0 Terminal point:,, In Eercises 9 and 0, find a unit vector (a) in the direction of u and (b) in the direction opposite of u. 9. u 8i j k 0. u i 5j 0k In Eercises, find the dot product of u and v.. u,, v, 5, 8. u,, 6 v, 0,. u i 5j k v 9i j k. u j 6k v 6i j k

99 Section. In Eercises 5 8, find the angle between the vectors. 5. u 0,, v, 0, 7. u 0i 0j v j 8k 6. u,, 0 v,, 8. u 8j 0k v 0i 5k 59. TENSION The weight of a crate is 500 newtons. Find the tension in each of the supporting cables shown in the figure. z 5 cm C D 70 cm B 65 cm In Eercises 9 6, determine whether u and v are orthogonal, parallel, or neither. 9. u, 6, 5 v 8,, 0. u 0,, 6 v,,. u i j k v i 0j k 5. u i j k v i j k 0. u,, v,, 5. u 0,, v, 0, 0. u i j k v 8i j 8k 6. u i j k v i j k In Eercises 7 50, use vectors to determine whether the points are collinear ,,, 7,,,, 5,, 7,,, 8,, 0, 6, 7,,,,, 5,,, 0,,,, 5, 6,, 6, 7 In Eercises 5 5, the vector v and its initial point are given. Find the terminal point. 5. v,, 7 Initial point:, 5, 0 5. v,, Initial point: 6,, 5. v,, Initial point:,, 5. v 5,, Initial point:,, 55. Determine the values of c such that cu, where u i j k. 56. Determine the values of c such that cu, where u i j k. In Eercises 57 and 58, write the component form of v. 57. v lies in the z-plane, has magnitude, and makes an angle of 5 with the positive -ais. 58. v lies in the z-plane, has magnitude 0, and makes an angle of 60 with the positive z-ais. 8 Vectors in Space 60 cm L 5 cm A FIGURE FOR 8 in. 59 FIGURE FOR TENSION The lights in an auditorium are -pound disks of radius 8 inches. Each disk is supported b three equall spaced cables that are L inches long (see figure). (a) Write the tension T in each cable as a function of L. Determine the domain of the function. (b) Use the function from part (a) to complete the table. L T (c) Use a graphing utilit to graph the function in part (a). What are the asmptotes of the graph? Interpret their meaning in the contet of the problem. (d) Determine the minimum length of each cable if a cable can carr a maimum load of 0 pounds. EXPLORATION TRUE OR FALSE? In Eercises 6 and 6, determine whether the statement is true or false. Justif our answer. 6. If the dot product of two nonzero vectors is zero, then the angle between the vectors is a right angle. 6. If AB and AC are parallel vectors, then points A, B, and C are collinear. 6. What is known about the nonzero vectors u and v if u v < 0? Eplain. 6. CAPSTONE Consider the two nonzero vectors u and v. Describe the geometric figure generated b the terminal points of the vectors t v, u tv, and su tv where s and t represent real numbers.

100 8 Chapter Analtic Geometr in Three Dimensions. THE CROSS PRODUCT OF TWO VECTORS What ou should learn Find cross products of vectors in space. Use geometric properties of cross products of vectors in space. Use triple scalar products to find volumes of parallelepipeds. Wh ou should learn it The cross product of two vectors in space has man applications in phsics and engineering. For instance, in Eercise 6 on page 80, the cross product is used to find the torque on the crank of a biccle s brake. The Cross Product Man applications in phsics, engineering, and geometr involve finding a vector in space that is orthogonal to two given vectors. In this section, ou will stud a product that will ield such a vector. It is called the cross product, and it is convenientl defined and calculated using the standard unit vector form. Definition of Cross Product of Two Vectors in Space Let u u i u j u k and v v i v j v k be vectors in space. The cross product of u and v is the vector u v u v u v i u v u v j u v u v k. David L. Moore/Äisport/Alam It is important to note that this definition applies onl to three-dimensional vectors. The cross product is not defined for two-dimensional vectors. A convenient wa to calculate u v is to use the following determinant form with cofactor epansion. (This determinant form is used simpl to help remember the formula for the cross product it is technicall not a determinant because the entries of the corresponding matri are not all real numbers.) i j k u v u u u Put u in Row. v v v Put v in Row. i j k u u u v v v i i j k u u u v v v j i j k u u u k v v v u v u v i u v u v j u v u v k u v u v i u v u v j u v u v k Note the minus sign in front of the j-component. Recall from Section 8. that each of the three determinants can be evaluated b using the following pattern. a a b b a b a b TECHNOLOGY Some graphing utilities have the capabilit to perform vector operations, such as the cross product. Consult the user s guide for our graphing utilit for specific instructions.

101 Section. The Cross Product of Two Vectors 85 Eample Finding Cross Products Given u i j k and v i j k, find each cross product. a. u v b. v u c. v v a. u v b. c. v u i i j 6k i j 5k i j j i i i j 6 k i j 5k i j k v v 0 Now tr Eercise 5. The results obtained in Eample suggest some interesting algebraic properties of the cross product. For instance, u v v u k k and j j k k v v 0. These properties, and several others, are summarized in the following list. Algebraic Properties of the Cross Product Let u, v, and w be vectors in space and let c be a scalar.. u v v u. u v w u v u w. cu v cu v u cv. u 0 0 u 0 5. u u 0 6. u v w u v w For proofs of the Algebraic Properties of the Cross Product, see Proofs in Mathematics on page 85.

102 86 Chapter Analtic Geometr in Three Dimensions Geometric Properties of the Cross Product The first propert listed on the preceding page indicates that the cross product is not commutative. In particular, this propert indicates that the vectors u v and v u have equal lengths but opposite directions. The following list gives some other geometric properties of the cross product of two vectors. Geometric Properties of the Cross Product Let u and v be nonzero vectors in space, and let be the angle between u and v. z. u v is orthogonal to both u and v.. u v u v sin k = i j. u v 0 if and onl if u and v are scalar multiples of each other.. u v area of parallelogram having u and v as adjacent sides. i FIGURE. j -plane For proofs of the Geometric Properties of the Cross Product, see Proofs in Mathematics on page 86. Both u v and v u are perpendicular to the plane determined b u and v. One wa to remember the orientations of the vectors u, v, and u v is to compare them with the unit vectors i, j, and k i j, as shown in Figure.. The three vectors u, v, and u v form a right-handed sstem. Eample Using the Cross Product Find a unit vector that is orthogonal to both u i j k and v i 6j. z 8 6 ( 6,, 6) The cross product u v, as shown in Figure., is orthogonal to both u and v. 0 i j k u v 6 (,, ) u FIGURE. u v 6 (, 6, 0) 6 v Because 6i j 6k u v a unit vector orthogonal to both u and v is u v u v i j k. Now tr Eercise.

103 Section. The Cross Product of Two Vectors 87 In Eample, note that ou could have used the cross product v u to form a unit vector that is orthogonal to both u and v. With that choice, ou would have obtained the negative of the unit vector found in the eample. The fourth geometric propert of the cross product states that u v is the area of the parallelogram that has u and v as adjacent sides. A simple eample of this is given b the unit square with adjacent sides of i and j. Because i j k and k, it follows that the square has an area of. This geometric propert of the cross product is illustrated further in the net eample. Eample Geometric Application of the Cross Product Show that the quadrilateral with vertices at the following points is a parallelogram. Then find the area of the parallelogram. Is the parallelogram a rectangle? A5,, 0, B, 6,, C,, 7, D5, 0, 6 z 8 6 D (5, 0, 6) C (,, 7) B (, 6, ) 6 A (5,, 0) 8 8 FIGURE. From Figure. ou can see that the sides of the quadrilateral correspond to the following four vectors. CB \. Because CD \ AB \ and CB \ AD \, \ ou can conclude that AB is parallel to CD \ and AD \ is parallel to It follows that the quadrilateral is a parallelogram with AB \ and AD \ as adjacent sides. Moreover, because 6 i j k AB \ AD \ 6i 8j 6k 0 the area of the parallelogram is You can tell whether the parallelogram is a rectangle b finding the angle between the vectors AB \ and AD \. sin sin sin Because AB \ i j k CD \ i j k AB \ AD \ 0i j 6k CB \ 0i j 6k AD \ AB \ AD \ AB\ AD \ arcsin AB \ AD \ 90, the parallelogram is not a rectangle. Now tr Eercise.

104 88 Chapter Analtic Geometr in Three Dimensions The Triple Scalar Product For the vectors u, v, and w in space, the dot product of u and v w is called the triple scalar product of u, v, and w. proj v w u v w u w v v w Area of base Volume of parallelepiped FIGURE. u v w The Triple Scalar Product For u u i u j u and the triple scalar product is given b k, v v i v j v k, w w i w j w k, u u u u v w v v v w w w. If the vectors u, v, and w do not lie in the same plane, the triple scalar product u v w can be used to determine the volume of the parallelepiped (a polhedron, all of whose faces are parallelograms) with u, v, and w as adjacent edges, as shown in Figure.. Geometric Propert of the Triple Scalar Product The volume V of a parallelepiped with vectors u, v, and w as adjacent edges is given b V u v w. Eample Volume b the Triple Scalar Product (, 5, ) u FIGURE.5 6 (,, ) w z v (0,, ) Find the volume of the parallelepiped having u i 5j k, as adjacent edges, as shown in Figure.5. The value of the triple scalar product is 5 u v w 0 6. So, the volume of the parallelepiped is u v w 6 v j k, Now tr Eercise 57. and 0 w i j k

105 Section. The Cross Product of Two Vectors 89. EXERCISES See for worked-out solutions to odd-numbered eercises. VOCABULARY: Fill in the blanks.. To find a vector in space that is orthogonal to two given vectors, find the of the two vectors.. u u. u v. The dot product of u and v w is called the of u, v, and w. SKILLS AND APPLICATIONS In Eercises 5 0, find the cross product of the unit vectors and sketch the result. 5. j i 6. i j 7. i k 8. k i 9. j k 0. k j In Eercises 0, use the vectors u and v to find each epression. u i j k. u v. v u. v v. v u u 5. u v 6. u v 7. u v 8. u v 9. u u v 0. v v u In Eercises 0, find u v and show that it is orthogonal to both u and v.. u,,. u 6, 8, v 0,, v 5,, 5. u 0, 0, 6. u 7,, v 7, 0, 0 v,, 5. u 6i j k 6. u i j 5 k v i j k v i j k 7. u 6k 8. u i v i j k v j 9k 9. u i k 0. u i k v j k v j k v i j k In Eercises 6, find a unit vector orthogonal to u and v.. u i j. u i j v j k v i k. u i j 5k. u 7i j 5k v i j 0 k v i 8j 5k 5. u i j k 6. u i j k v i j k v i j k In Eercises 7, find the area of the parallelogram that has the vectors as adjacent sides. 7. u k 8. u i j k v i k v i k 9. u i j 7k v i j 6k 0. u i j k v i j k. u,, 6 v 0,, 6. u,, v 5, 0, In Eercises 6, (a) verif that the points are the vertices of a parallelogram, (b) find its area, and (c) determine whether the parallelogram is a rectangle.. A,,, B,,, C0, 5, 6, D,, 8. A,,, B,,, C6, 5,, D7, 7, 5 5. A,,, B,,, C, 5,, D, 5, 6. A,,, B,,, C,,, D, 6, In Eercises 7 50, find the area of the triangle with the given vertices. (The area A of the triangle having u and v as adjacent sides is given b A u v. ) 7. 0, 0, 0,,,,, 0, 0 8.,,,, 0,,,, 0 9.,, 5,,, 0,, 0, 6 50.,, 0,,, 0, 0, 0, In Eercises 5 5, find the triple scalar product. 5. u,,, v,, 0, w 0, 0, 6 5. u, 0,, v 0, 5, 0, w 0, 0, 5. u i j k, v i j, w i j k 5. u i j 7k, v i k, w j 6k

106 80 Chapter Analtic Geometr in Three Dimensions In Eercises 55 58, use the triple scalar product to find the volume of the parallelepiped having adjacent edges u, v, and w. 55. u i j v j k w i k 56. u i j k v j k w i k p (0,, ) (, 0, ) v w u w (, 0, ) u (,, 0) v (0,, ) u 0,, v 0, 0, w, 0, z 5 (, 0, ) 60 EXPLORATION TRUE OR FALSE? In Eercises 6 and 6, determine whether the statement is true or false. Justif our answer. w v (0, 0, ) u (,, ) In Eercises 59 and 60, find the volume of the parallelepiped with the given vertices. B, 0, 0, F 0, 5,, B,, 0, F,,, C,,, D 0,,, G 0,, 6, H,, 6 C, 0,, D 0,,, G,,, H,, 6. TORQUE The brakes on a biccle are applied b using a downward force of p pounds on the pedal when the si-inch crank makes a 0 angle with the horizontal (see figure). Vectors representing the position of the crank and the force are V cos 0 j sin 0 k and F pk, respectivel. 6 in. V F = p lb 0 F = 000 lb V (, 0, ) t 6f 0. v u 59. A 0, 0, 0, E, 5,, 60. A 0, 0, 0, E,,, 0 (,, ) u,, v,, w, 0, (0,, ) w 0 z 5 6. TORQUE Both the magnitude and direction of the force on a crankshaft change as the crankshaft rotates. Use the technique given in Eercise 6 to find the magnitude of the torque on the crankshaft using the position and data shown in the figure. (,, ) 5 T z z (a) The magnitude of the torque on the crank is given b V F. Using the given information, write the torque T on the crank as a function of p. (b) Use the function from part (a) to complete the table. 6. The cross product is not defined for vectors in the plane. 6. If u and v are vectors in space that are nonzero and nonparallel, then u v v u. 65. THINK ABOUT IT Calculate u v and v u for several values of u and v. What do our results impl? Interpret our results geometricall. 66. THINK ABOUT IT If the magnitudes of two vectors are doubled, how will the magnitude of the cross product of the vectors change? 67. THINK ABOUT IT If ou connect the terminal points of two vectors u and v that have the same initial points, a triangle is formed. Is it possible to use the cross product u v to determine the area of the triangle? Eplain. Verif our conclusion using two vectors from Eample. 68. CAPSTONE Define the cross product of two vectors in space, u and v, where u ui u j uk and v vi v j vk. Eplain, in our own words, what the cross product u v represents. What does it mean when u v 0? 69. PROOF Consider the vectors u cos, sin, 0 and v cos, sin, 0, where >. Find the cross product of the vectors and use the result to prove the identit sin sin cos cos sin.

107 Section. Lines and Planes in Space 8. LINES AND PLANES IN SPACE What ou should learn Find parametric and smmetric equations of lines in space. Find equations of planes in space. Sketch planes in space. Find distances between points and planes in space. Wh ou should learn it Equations in three variables can be used to model real-life data. For instance, in Eercise 6 on page 89, ou will determine how changes in the consumption of two tpes of beverages affect the consumption of a third tpe of beverage. Lines in Space In the plane, slope is used to determine an equation of a line. In space, it is more convenient to use vectors to determine the equation of a line. In Figure.6, consider the line L through the point P,, z and parallel to the vector v a, b, c. P(,, z) Direction vector for L z Q(,, z) v = a, b, c L PQ = tv FIGURE.6 JG Photograph/Alam The vector v is the direction vector for the line L, and a, b, and c are the direction numbers. One wa of describing the line L is to sa that it consists of all points Q,, z for which the vector PQ \ is parallel to v. This means that PQ \ is a scalar multiple of v, and ou can write PQ \ tv, where t is a scalar. PQ \,, z z at, bt, ct tv B equating corresponding components, ou can obtain the parametric equations of a line in space. Parametric Equations of a Line in Space A line L parallel to the vector v a, b, c and passing through the point P,, z is represented b the parametric equations at, bt, and z z ct. If the direction numbers a, b, and c are all nonzero, ou can einate the parameter t to obtain the smmetric equations of a line. a b z z c Smmetric equations

108 8 Chapter Analtic Geometr in Three Dimensions (,, ) 6 z Eample Finding Parametric and Smmetric Equations Find parametric and smmetric equations of the line L that passes through the point,, and is parallel to v,,. 6 L FIGURE.7 6 v =,, To find a set of parametric equations of the line, use the coordinates,, and z and direction numbers a, b, and c (see Figure.7). t, Because a, b, and c are all nonzero, a set of smmetric equations is Now tr Eercise 5. Parametric equations Smmetric equations Neither the parametric equations nor the smmetric equations of a given line are unique. For instance, in Eample, b letting t in the parametric equations ou would obtain the point,, 0. Using this point with the direction numbers a, b, and c produces the parametric equations t, t, z. t, and z t z t. Eample Parametric and Smmetric Equations of a Line Through Two Points Find a set of parametric and smmetric equations of the line that passes through the points,, 0 and,, 5. Begin b letting P,, 0 and Q,, 5. Then a direction vector for the line passing through P and Q is v PQ \,, 5 0,, 5 a, b, c. Using the direction numbers a, b, and c 5 with the initial point P,, 0, ou can obtain the parametric equations t, and Because a, b, and c are all nonzero, a set of smmetric equations is z 5. t, Now tr Eercise. z 5t. Parametric equations Smmetric equations

109 Section. Lines and Planes in Space 8 To check the answer to Eample, verif that the two original points lie on the line. For the point,, 0, ou can substitute in the parametric equations. t t z 5t t t 0 5t 0 t 0 t 0 t Tr checking the point,, 5 on our own. Note that ou can also check the answer using the smmetric equations. Planes in Space You have seen how an equation of a line in space can be obtained from a point on the line and a vector parallel to it. You will now see that an equation of a plane in space can be obtained from a point in the plane and a vector normal (perpendicular) to the plane. z P n Q n PQ = 0 FIGURE.8 Consider the plane containing the point P,, z having a nonzero normal vector n a, b, c, as shown in Figure.8. This plane consists of all points Q,, z for which the vector PQ \ is orthogonal to n. Using the dot product, ou can write n PQ \ 0 PQ \ is orthogonal to n. a, b, c,, z z 0 a b cz z 0. The third equation of the plane is said to be in standard form. Standard Equation of a Plane in Space The plane containing the point,, z and having normal vector n a, b, c can be represented b the standard form of the equation of a plane a b cz z 0. Regrouping terms ields the general form of the equation of a plane in space a b cz d 0. General form of equation of plane Given the general form of the equation of a plane, it is eas to find a normal vector to the plane. Use the coefficients of,, and z to write n a, b, c.

110 8 Chapter Analtic Geometr in Three Dimensions Eample Finding an Equation of a Plane in Three-Space Find the general form of the equation of the plane passing through the points,,, 0,,, and,,. z 5 (,, ) v (,, ) u (0,, ) FIGURE.9 To find the equation of the plane, ou need a point in the plane and a vector that is normal to the plane. There are three choices for the point, but no normal vector is given. To obtain a normal vector, use the cross product of vectors u and v etending from the point,, to the points 0,, and,,, as shown in Figure.9. The component forms of u and v are u 0,,,, 0 v,,, 0, and it follows that i j k n u v 0 0 9i 6j k a, b, c is normal to the given plane. Using the direction numbers for n and the initial point,, z,,, ou can determine an equation of the plane to be a b cz z z 0 Standard form 9 6 z 6 0 z 0. General form Check that each of the three points satisfies the equation z 0. Now tr Eercise 9. n θ n θ Two distinct planes in three-space either are parallel or intersect in a line. If the intersect, ou can determine the angle 90 between them from the angle between their normal vectors, as shown in Figure.0. Specificall, if vectors n and n are normal to two intersecting planes, the angle between the normal vectors is equal to the angle between the two planes and is given b cos n n n n. 0 Angle between two planes FIGURE.0 Consequentl, two planes with normal vectors and are. perpendicular if n n 0.. parallel if n is a scalar multiple of n. n n

111 Section. Lines and Planes in Space 85 Eample Finding the Line of Intersection of Two Planes Plane FIGURE. z θ 5.55 Line of Intersection Plane Find the angle between the two planes given b z 0 Equation for plane z 0 Equation for plane and find parametric equations of their line of intersection (see Figure.). The normal vectors for the planes are n,, and n,,. Consequentl, the angle between the two planes is determined as follows. cos n n 67 0 n n This implies that the angle between the two planes is You can find the line of intersection of the two planes b simultaneousl solving the two linear equations representing the planes. One wa to do this is to multipl the first equation b and add the result to the second equation. z 0 z 0 7 z 0 z 7 Substituting z7 back into one of the original equations, ou can determine that z7. Finall, b letting t z7, ou obtain the parametric equations t at t bt z 7t z ct. Parametric equation for Parametric equation for Parametric equation for z Because,, z 0, 0, 0 lies in both planes, ou can substitute for,, and z in these parametric equations, which indicates that a, b, and c 7 are direction numbers for the line of intersection. Now tr Eercise 7. Note that the direction numbers in Eample can also be obtained from the cross product of the two normal vectors as follows. i j k n n i i j 7k z 0 z 0 j k This means that the line of intersection of the two planes is parallel to the cross product of their normal vectors.

112 86 Chapter Analtic Geometr in Three Dimensions TECHNOLOGY Most three-dimensional graphing utilities and computer algebra sstems can graph a plane in space. Consult the user s guide for our utilit for specific instructions. Sketching Planes in Space As discussed in Section., if a plane in space intersects one of the coordinate planes, the line of intersection is called the trace of the given plane in the coordinate plane. To sketch a plane in space, it is helpful to find its points of intersection with the coordinate aes and its traces in the coordinate planes. For eample, consider the plane z. Equation of plane You can find the -trace b letting z 0 and sketching the line -trace in the -plane. This line intersects the -ais at, 0, 0 and the -ais at 0, 6, 0. In Figure., this process is continued b finding the z-trace and the z-trace and then shading the triangular region ling in the first octant. z z z (, 0, 0) (0, 0, ) (0, 6, 0) (, 0, 0) (, 0, 0) (0, 6, 0) (0, 0, ) (0, 6, 0) (a) -trace z 0: (b) z-trace 0: (c) z-trace 0: z z FIGURE. z (0, 0, ) If the equation of a plane has a missing variable, such as z, the plane must be parallel to the ais represented b the missing variable, as shown in Figure.. If two variables are missing from the equation of a plane, then it is parallel to the coordinate plane represented b the missing variables, as shown in Figure.. (, 0, 0) z z z ( 0, 0, d ) c Plane: + z = Plane is parallel to -ais FIGURE. ( d d ) ( 0,, 0 a, 0, 0 ) b (a) Plane a d 0 (b) Plane b d 0 (c) Plane cz d 0 is parallel to z-plane. is parallel to z-plane. is parallel to -plane. FIGURE.

113 Section. Lines and Planes in Space 87 n proj n PQ D proj n PQ \ FIGURE.5 P Q D Distance Between a Point and a Plane The distance D between a point Q and a plane is the length of the shortest line segment connecting Q to the plane, as shown in Figure.5. If P is an point in the plane, ou can find this distance b projecting the vector PQ \ onto the normal vector n. The length of this projection is the desired distance. Distance Between a Point and a Plane The distance between a plane and a point Q (not in the plane) is D proj n PQ \ PQ\ n n where P is a point in the plane and n is normal to the plane. To find a point in the plane given b a b cz d 0, where a 0, let 0 and z 0. Then, from the equation a d 0, ou can conclude that the point da, 0, 0 lies in the plane. Eample 5 Finding the Distance Between a Point and a Plane Find the distance between the point Q, 5, and the plane z 6. You know that n,, is normal to the given plane. To find a point in the plane, let 0 and z 0, and obtain the point P, 0, 0. The vector from P to Q is PQ \, 5 0, 0, 5,. The formula for the distance between a point and a plane produces D PQ\ n, 5,,, n Now tr Eercise 59. The choice of the point P in Eample 5 is arbitrar. Tr choosing a different point to verif that ou obtain the same distance.

114 88 Chapter Analtic Geometr in Three Dimensions. EXERCISES See for worked-out solutions to odd-numbered eercises. VOCABULARY: Fill in the blanks.. The vector for a line L is given b v.. The of a line in space are given b at, bt, and z z ct.. If the direction numbers a, b, and c of the vector v a, b, c are all nonzero, ou can einate the parameter to obtain the of a line.. A vector that is perpendicular to a plane is called. SKILLS AND APPLICATIONS In Eercises 5 0, find a set of (a) parametric equations and (b) smmetric equations for the line through the point and parallel to the specified vector or line. (For each line, write the direction numbers as integers.) Point 5. 0, 0, 0 6., 5, v 7.,, 0 8., 0, 9.,, 5 0., 0, Parallel to v,,, 7, 0 v i j k v i j k 5 t, 7 t, z t t, 5 t, z 7 t In Eercises 8, find (a) a set of parametric equations and (b) if possible, a set of smmetric equations of the line that passes through the given points. (For each line, write the direction numbers as integers.)., 0,,,,.,, 0, 0, 8,., 8, 5,,, 6.,,,, 5, 5.,,,,, 5 6.,, 5,,, 7. 8.,,,, 5,,,,,, 0 In Eercises 9 and 0, sketch a graph of the line. 9. t, t, 0. 5 t, t, z z 5 t t In Eercises 6, find the general form of the equation of the plane passing through the point and perpendicular to the specified vector or line. Point.,,., 0,. 5, 6,. 0, 0, 0 5., 0, , 0, 6 Perpendicular to n i n k n i j k n j 5k t, t, z t t, t, z t In Eercises 7 0, find the general form of the equation of the plane passing through the three points , 0, 0,,,,,,,,,, 5,,,,,,,,,,,, 0 5,,,,,,,, In Eercises 6, find the general form of the equation of the plane with the given characteristics.. Passes through, 5, and is parallel to the z-plane. Passes through,, and is parallel to the z-plane. Passes through 0,, and,, 0 and is perpendicular to the z-plane. Passes through,, and, 0, and is perpendicular to the z-plane 5. Passes through,, and,, and is perpendicular to z 6. Passes through,, 0 and,, and is perpendicular to z 6 In Eercises 7 0, determine whether the planes are parallel, orthogonal, or neither. If the are neither parallel nor orthogonal, find the angle of intersection z 8. z 7z 9 z 9. 8z z 8z z In Eercises 6, find a set of parametric equations of the line. (There are man correct answers.). Passes through,, and is parallel to the z-plane and the z-plane. Passes through, 5, and is parallel to the -plane and the z-plane. Passes through,, and is perpendicular to z 6

115 Section.. Passes through, 5, and is perpendicular to z 5 5. Passes through 5,, and is parallel to v,, 6. Passes through,, and is parallel to v 5i j In Eercises 7 50, (a) find the angle between the two planes and (b) find parametric equations of their line of intersection z 6 z z 0 5 z T (,, 8) In Eercises 57 60, find the distance between the point and the plane , 0, 0 8 z 8 59.,, z 58.,, z 60.,, 5 z 6. DATA ANALYSIS: BEVERAGE CONSUMPTION The table shows the per capita consumption (in gallons) of different tpes of beverages sold b a compan from 006 through 00. Consumption of energ drinks, soft drinks, and bottled water are represented b the variables,, and z, respectivel. Year z A model for the data is given b.5 0. z.5. (a) Complete a fifth column in the table using the model to approimate z for the given values of and. (b) Compare the approimations from part (a) with the actual values of z. z R (7, 7, 8) S (0, 0, 0) P (6, 0, 0) In Eercises 5 56, plot the intercepts and sketch a graph of the plane. 5. z 5. z z 6 89 (c) According to this model, an increases or decreases in consumption of two tpes of beverages will have what effect on the consumption of the third tpe of beverage? 6. MECHANICAL DESIGN A chute at the top of a grain elevator of a combine funnels the grain into a bin, as shown in the figure. Find the angle between two adjacent sides. 8. z 5z z 6 z 0 5. z z 6 Lines and Planes in Space Q (6, 6, 0) EXPLORATION TRUE OR FALSE? In Eercises 6 and 6, determine whether the statement is true or false. Justif our answer. 6. Ever two lines in space are either intersecting or parallel. 6. Two nonparallel planes in space will alwas intersect. 65. The direction numbers of two distinct lines in space are 0, 8, 0, and 5, 7, 0. What is the relationship between the lines? Eplain. 66. Consider the following four planes z z z z 5 What are the normal vectors for each plane? What can ou sa about the relative positions of these planes in space? 67. (a) Describe and find an equation for the surface generated b all points,, z that are two units from the point,,. (b) Describe and find an equation for the surface generated b all points,, z that are two units from the plane z CAPSTONE Give the parametric equations and the smmetric equations of a line in space. Describe what is required to find these equations.

116 80 Chapter Analtic Geometr in Three Dimensions CHAPTER SUMMARY What Did You Learn? Eplanation/Eamples Review Eercises Plot points in the three-dimensional coordinate sstem (p. 80). (,, ) (,, ) z 6 (, 5, 0) (,, ) Section. Section. Find distances between points in space and find midpoints of line segments joining points in space (p. 8). Write equations of spheres in standard form and find traces of surfaces in space (p. 8). Find the component forms of the unit vectors in the same direction of, the magnitudes of, the dot products of, and the angles between vectors in space (p. 87). The distance between the points,, z and,, z given b the Distance Formula in Space is d z z. The midpoint of the line segment joining the points,, z and,, z given b the Midpoint Formula in Space is,, z z. Standard Equation of a Sphere The standard equation of a sphere with center h, k, j and radius r is given b h k z j r. Vectors in Space. Two vectors are equal if and onl if their corresponding components are equal.. Magnitude of u u u u u u, u, u :. A unit vector u in the direction of v is u v v 0. v,. The sum of u u, u, u and v v, v, v is u v u v, u v, u v. 5. The scalar multiple of the real number c and u u, u, u is cu cu, cu, cu. 6. The dot product of u u, u, u and v v, v, v is u v u v u v u v. Angle Between Two Vectors If is the angle between two nonzero vectors u and v, then cos u v u v. Determine whether vectors in space are parallel or orthogonal (p. 89). Two nonzero vectors u and v are parallel if there is some scalar c such that u cv. 7 Use vectors in space to solve real-life problems (p. 8). Vectors can be used to solve equilibrium problems in space. (See Eample 7.) 5, 6

117 Chapter Summar 8 Section. Section. What Did You Learn? Eplanation/Eamples Review Eercises Find cross products of vectors in space (p. 8). Use geometric properties of cross products of vectors in space (p. 86). Use triple scalar products to find volumes of parallelepipeds (p. 88). Find parametric and smmetric equations of lines in space (p. 8). Find equations of planes in space (p. 8). Sketch planes in space (p. 86). Find distances between points and planes in space (p. 87). Definition of Cross Product of Two Vectors in Space Let u u i u j u k and v v i v j v k be vectors in space. The cross product of u and v, u v, is the vector u v u v i u v u v j u v u v k. Geometric Properties of the Cross Product Let u and v be nonzero vectors in space, and let be the angle between u and v.. u v is orthogonal to both u and v.. u v u v sin. u v 0 if and onl if u and v are scalar multiples of each other.. u v area of parallelogram having u and v as adjacent sides The Triple Scalar Product For u u i u and the triple scalar product is given b j u k, v v i v j v k, w w i w j w k, u u u u v w v v v w w w. Geometric Propert of the Triple Scalar Product The volume V of a parallelepiped with vectors u, v, and w as adjacent edges is given b V u v w. Parametric Equations of a Line in Space A line L parallel to the vector v a, b, c and passing through the point P,, z is represented b the parametric equations at, bt, and z z ct. Standard Equation of a Plane in Space The plane containing the point,, z and having normal vector n a, b, c can be represented b the standard form of the equation of a plane a b cz z 0. See Figure., which shows how to sketch the plane z. Distance Between a Point and a Plane The distance between a plane and a point Q (not in the plane) is D proj n PQ \ PQ\ n n where P is a point in the plane and n is normal to the plane ,

118 8 Chapter Analtic Geometr in Three Dimensions REVIEW EXERCISES See for worked-out solutions to odd-numbered eercises.. In Eercises and, plot each point in the same threedimensional coordinate sstem.. (a) 5,,. (a),, (b),, 0 (b) 0, 0, 5. Find the coordinates of the point in the -plane four units to the right of the z-plane and five units behind the z-plane.. Find the coordinates of the point located on the -ais and seven units to the left of the z-plane. In Eercises 5 8, find the distance between the points. 5., 0, 7, 5,, , 5, 6,,, 6 8. In Eercises 9 and 0, find the lengths of the sides of the right triangle. Show that these lengths satisf the Pthagorean Theorem. 9. z 0. (,, 0) (0, 5, ) (0,, ),,,,, 0 0, 0, 0,,, (,, ) (0, 0, ) (, 5, 5) In Eercises, find the midpoint of the line segment joining the points.. 8,,, 5, 6, 7. 7,,,,,. 0, 6,, 8,, 6. 5,,, 7, 9, 5 In Eercises 5 0, find the standard form of the equation of the sphere with the given characteristics. 5. Center:,, 5; radius: 6. Center:,, ; radius: 7. Center:, 5, ; diameter: 8. Center: 0,, ; diameter: 5 9. Endpoints of a diameter:,,,,, 0. Endpoints of a diameter:,,,, 5, z In Eercises, find the center and radius of the sphere.. z 8z 0. z 6 0. z 0 6 z 0. z z 0 In Eercises 5 and 6, sketch the graph of the equation and sketch the specified trace. 5. z 6 (a) z-trace (b) z-trace 6. z 9 (a) -trace (b) z-trace. In Eercises 7 0, (a) write the component form of the vector v, (b) find the magnitude of v, and (c) find a unit vector in the direction of v. 7. Initial point:,, Terminal point:,, 0 8. Initial point:,, Terminal point:,, 9. Initial point: 7,, Terminal point:,, 0 0. Initial point: 0,, Terminal point: 5, 8, 6 In Eercises, find the dot product of u and v.. u,,. u 8,, v 0, 6, 5 v, 5,. u i j k. u i j k v i k v i j k In Eercises 5 and 6, find the angle between the vectors. 5. u,, 0 6. u,, v,, v, 5, In Eercises 7 0, determine whether u and v are orthogonal, parallel, or neither. 7. u 9,, 8. u 8, 5, 8 v 6, 8, v,, 9. u 6i 5j 9k 0. u j k v 5i j 5k v i 8k

119 Review Eercises 8 In Eercises, use vectors to determine whether the points are collinear.. 6,,, 5, 8,, 7,, 5. 5,, 0,, 6,,,, 7. 5,, 7, 8, 5, 5,, 6,.,,,, 6, 9, 5,, 6 5. TENSION A load of 00 pounds is supported b three cables, as shown in the figure. Find the tension in each of the supporting cables. 6. TENSION Determine the tension in each of the supporting cables in Eercise 5 if the load is 00 pounds.. In Eercises 7 50, find u v. 7. u, 8, 8. u 0, 5, 5 v,, v 5,, 0 9. u i j k 50. u i j k v i j k v i In Eercises 5 5, find a unit vector orthogonal to u and v. 5. u i j k 5. u j k v i j k v i j 5. u i j 5k 5. u k v 0i 5j k v i k In Eercises 55 and 56, verif that the points are the vertices of a parallelogram and find its area. 55.,,, 5,,, 0,,,,, 56. 0,, 0,,,, 0, 6, 0,, 6, In Eercises 57 and 58, find the volume of the parallelepiped with the given vertices z (, 6, 0) B (, 6, 0) C A0, 0, 0, B, 0, 0, C0, 5,, D, 5,, E, 0, 5, F5, 0, 5, G, 5, 6, H5, 5, 6 A0, 0, 0, B, 0, 0, C,, 0, D0,, 0, E0, 0, 6, F, 0, 6, G,, 6, H0,, 6 O 00 lb (0, 0, 0) A. In Eercises 59 6, find a set of (a) parametric equations and (b) smmetric equations for the specified line. 59. Passes through,, 5 and, 6, 60. Passes through 0, 0, and 5, 0, 0 6. Passes through 0, 0, 0 and is parallel to v, 5, 6. Passes through,, and is parallel to the line given b z In Eercises 6 66, find the general form of the equation of the specified plane. 6. Passes through 0, 0, 0, 5, 0,, and,, 8 6. Passes through,,,,,, and, 8, Passes through 5,, and is parallel to the -plane 66. Passes through 0, 0, 6 and is perpendicular to the line given b t, t, and z t In Eercises 67 70, plot the intercepts and sketch a graph of the plane. 67. z z z z In Eercises 7 7, find the distance between the point and the plane. 7.,, 7.,, 0 z 0 z 7. 0, 0, , 0, 0 z 0 z EXPLORATION TRUE OR FALSE? In Eercises 75 and 76, determine whether the statement is true or false. Justif our answer. 75. The cross product is commutative. 76. The triple scalar product of three vectors in space is a scalar. In Eercises 77 and 78, let u v and w < w, w > < < v, w., v > u, u >, u,, v, u v w u v u w. 77. Show that 78. Show that u v w u v u w.

120 8 Chapter Analtic Geometr in Three Dimensions CHAPTER TEST See for worked-out solutions to odd-numbered eercises. Take this test as ou would take a test in class. When ou are finished, check our work against the answers given in the back of the book.. Plot each point in the same three-dimensional coordinate sstem. (a), 7, (b),, In Eercises, use the points A8,, 5, B6,,, and C,, 0, to solve the problem.. Consider the triangle with vertices A, B, and C. Is it a right triangle? Eplain.. Find the coordinates of the midpoint of the line segment joining points A and B.. Find the standard form of the equation of the sphere for which A and B are the endpoints of a diameter. Sketch the sphere and its z-trace. z 6 A B D C E F H G FIGURE FOR z T (,, ) R (8, 8, ) S (0, 0, 0) P (0, 0, 0) Q (0, 0, 0) FIGURE FOR 9 In Eercises 5 9, let u and v be the vectors from A8,, 5 to B6,, and from A to C,, 0, respectivel. 5. Write u and v in component form. 6. Find (a) u v and (b) u v. 7. Find (a) a unit vector in the direction of u and (b) a unit vector in the direction of v. 8. Find the angle between u and v. 9. Find a set of (a) parametric equations and (b) smmetric equations for the line through points A and B. In Eercises 0, determine whether u and v are orthogonal, parallel, or neither. 0. u i j k. u i j k. u,, 6 v j 6k v i j k v,,. Verif that the points A,,, B6, 5,, C, 6,, and D7,, are the vertices of a parallelogram, and find its area.. Find the volume of the parallelepiped at the left with the given vertices. A0, 0, 5, B0, 0, 5, C, 0, 5, D, 0, 5, E0,, 0, F0,, 0, G,, 0, H,, 0 In Eercises 5 and 6, plot the intercepts and sketch a graph of the plane z z 0 7. Find the general form of the equation of the plane passing through the points,,,,,, and,,. 8. Find the distance between the point,, 6 and the plane z A tractor fuel tank has the shape and dimensions shown in the figure. In fabricating the tank, it is necessar to know the angle between two adjacent sides. Find this angle.

121 PROOFS IN MATHEMATICS Notation for Dot and Cross Products The notation for the dot product and the cross product of vectors was first introduced b the American phsicist Josiah Willard Gibbs (89 90). In the earl 880s, Gibbs built a sstem to represent phsical quantities called vector analsis. The sstem was a departure from William Hamilton s theor of quaternions. Proof Let u u i u j u k, v v i v j v k, w w i w j w k, 0 0i 0j 0k, and let c be a scalar..... Algebraic Properties of the Cross Product (p. 85) Let u, v, and w be vectors in space and let c be a scalar.. u v v u. u v w u v u w. cu v cu v u cv. u 0 0 u 0 5. u u 0 6. u v w u v w u v u v u v i u v u v j u v u v k v u v u v u i v u v u j v u v u k So, this implies u v v u. u v w u v w u v w i [u v w u v u v i u v u v j u v u v k u w u w i u w u w j u w u w k u v u w cu v cu v cu v i cu v cu v j cu v cu v k cu v u v i u v u v j u v u v k cu v u 0 u 0 u 0i u 0 u 0j u 0 u 0k 0i 0j 0k 0 0 u 0 u 0 u i 0 u 0 u j 0 u 0 u k 0i 0j 0k 0 So, this implies u 0 0 u 0. u v w j u v w u v w k 5. u u u u u u i u u u u 6. u v w v v v u u u j u u u u k 0 and w w w w w w u v w w u v u u u v v v u v w u v w w v u v w w v u v w w v u v w u w v u v w u w v u v w u w v u w v u w v u w v u v w u v w u v w w u v u v w u v u v w u v u v u v w 85

122 Geometric Properties of the Cross Product (p. 86) Let u and v be nonzero vectors in space, and let be the angle between u and v.. u v is orthogonal to both u and v.. u v u v sin. u v 0 if and onl if u and v are scalar multiples of each other.. u v area of parallelogram having u and v as adjacent sides. v θ v sinθ u Proof Let u u i u j u k, v v i v j v k, and 0 0i 0j 0k.. u v u v u v i u v u v j u v u v k u v u u v u v u u v u v u u v u v u u u v u u v u u v u u v u u v u u v 0 u v v u v u v v u v u v v u v u v v u v v u v v u v v u v v u v v u v v 0 Because two vectors are orthogonal if their dot product is zero, it follows that u v is orthogonal to both u and v.. Note that cos u v So, uv.. If u and v are scalar multiples of each other, then u cv for some scalar c. If u v 0, then uv sin 0. (Assume u 0 and v 0. ) So, sin 0, and or In either case, because is the angle between the vectors, u and v are parallel. So, u cv for some scalar c. 0 uv sin uv cos u v cv v cv v c0 0.. The figure at the left is a parallelogram having v and u as adjacent sides. Because the height of the parallelogram is v sin, the area is Area baseheight uv sin u v. uv u v u v u v u v u u u v v v u v u v u v u v u v u v u v u v u v u v. 86

123 PROBLEM SOLVING This collection of thought-provoking and challenging eercises further eplores and epands upon concepts learned in this chapter.. Let u i j, v j k, and w au bv. (a) Sketch u and v. (b) If w 0, show that a and b must both be zero. (c) Find a and b such that w i j k. (d) Show that no choice of a and b ields w i j k.. The initial and terminal points of v are,, z and,, z, respectivel. Describe the set of all points,, z such that v.. You are given the component forms of the vectors u and v. Write a program for a graphing utilit in which the output is (a) the component form of u v, (b) u v, (c) u, and (d) v.. Run the program ou wrote in Eercise for the vectors u,, and v 5,.5, The vertices of a triangle are given. Determine whether the triangle is an acute triangle, an obtuse triangle, or a right triangle. Eplain our reasoning. (a),, 0, 0, 0, 0,,, 0 (b), 0, 0, 0, 0, 0,,, (c),,, 0,,,,, 0 (d), 7,,, 5, 8,, 6, 6. A television camera weighing 0 pounds is supported b a tripod (see figure). Represent the force eerted on each leg of the tripod as a vector. P(0, 0, ) Q (0,, 0) Q,, 0 ) z Q 7. A precast concrete wall is temporaril kept in its vertical position b ropes (see figure). Find the total force eerted on the pin at position A. The tensions in AB and AC are 0 pounds and 650 pounds, respectivel. ) ) ),, 0 D FIGURE FOR 7 8. Prove u v uv if u and v are orthogonal. 9. Prove u v w u wv u vw. 0. Prove that the triple scalar product of u, v, and w is given b u u u u v w v v v w w w.. Prove that the volume V of a parallelepiped with vectors u, v, and w as adjacent edges is given b V u v w. 6 ft. In phsics, the cross product can be used to measure torque, or the moment M of a force F about a point P. If the point of application of the force is Q, the moment of F about P is given b M PQ \ F. A force of 60 pounds acts on the pipe wrench shown in the figure. O 8 ft B 8 in. 0 (a) Find the magnitude of the moment about O. Use a graphing utilit to graph the resulting function of. (b) Use the result of part (a) to determine the magnitude of the moment when (c) Use the result of part (a) to determine the angle when the magnitude of the moment is maimum. Is the answer what ou epected? Wh or wh not? C 0 ft 5. A 8 ft A F 87

124 . A force of 00 pounds acts on the bracket shown in the figure. (a) Determine the vector AB \ and the vector F representing the force. F will be in terms of. (b) Find the magnitude of the moment (torque) about A b evaluating AB \ F. Use a graphing utilit to graph the resulting function of for (c) Use the result of part (b) to determine the magnitude of the moment when (d) Use the result of part (b) to determine the angle when the magnitude of the moment is maimum. (e) Use the graph in part (b) to approimate the zero of the function. Interpret the meaning of the zero in the contet of the problem.. Using vectors, prove the Law of Sines: If a, b, and c are three sides of the triangle shown in the figure, then sin A a sin B b sin C c. 5. Two insects are crawling along different lines in threespace. At time t (in minutes), the first insect is at the point,, z on the line given b 6 t, 8 t, z t. Also, at time t, the second insect is at the point,, z on the line given b t, t, z t. c B in. A B F 0. b 00 lb 5 in. Assume distances are given in inches. (a) Find the distance between the two insects at time t 0. (b) Use a graphing utilit to graph the distance between the insects from t 0 to t 0. a C A (c) Using the graph from part (b), what can ou conclude about the distance between the insects? (d) Using the graph from part (b), determine how close the insects get to each other. 6. The distance between a point Q and a line in space is given b D PQ\ u u where u is a direction vector for the line and P is a point on the line. Find the distance between the point and the line given b each set of parametric equations. (a), 5, t,, z t (b),, t, t, z t 7. Use the formula given in Eercise 6. (a) Find the shortest distance between the point Q, 0, 0 and the line determined b the points P 0, 0, and P 0,,. (b) Find the shortest distance between the point Q, 0, 0 and the line segment joining the points P 0, 0, and P 0,,. 8. Consider the line given b the parametric equations t, t, z t and the point,, s for an real number s. (a) Write the distance between the point and the line as a function of s. (Hint: Use the formula given in Eercise 6.) (b) Use a graphing utilit to graph the function from part (a). Use the graph to find the value of s such that the distance between the point and the line is a minimum. (c) Use the zoom feature of the graphing utilit to zoom out several times on the graph in part (b). Does it appear that the graph has slant asmptotes? Eplain. If it appears to have slant asmptotes, find them. 88

125 Limits and an Introduction to Calculus. Introduction to Limits. Techniques for Evaluating Limits. The Tangent Line Problem. Limits at Infinit and Limits of Sequences.5 The Area Problem In Mathematics If a function becomes arbitraril close to a unique number L as approaches c from either side, the it of the function as approaches c is L. In Real Life The fundamental concept of integral calculus is the calculation of the area of a plane region bounded b the graph of a function. For instance, in surveing, a civil engineer uses integration to estimate the areas of irregular plots of real estate. (See Eercises 9 and 50, page 897.) David Frazier/PhotoEdit IN CAREERS There are man careers that use it concepts. Several are listed below. Market Researcher Eercise 7, page 880 Aquatic Biologist Eercise 5, page 888 Business Economist Eercises 55 and 56, page 888 Data Analst Eercises 57 and 58, pages 888 and

126 850 Chapter Limits and an Introduction to Calculus. INTRODUCTION TO LIMITS What ou should learn Use the definition of it to estimate its. Determine whether its of functions eist. Use properties of its and direct substitution to evaluate its. Wh ou should learn it The concept of a it is useful in applications involving maimization. For instance, in Eercise 5 on page 858, the concept of a it is used to verif the maimum volume of an open bo. The Limit Concept The notion of a it is a fundamental concept of calculus. In this chapter, ou will learn how to evaluate its and how the are used in the two basic problems of calculus: the tangent line problem and the area problem. Eample Finding a Rectangle of Maimum Area You are given inches of wire and are asked to form a rectangle whose area is as large as possible. Determine the dimensions of the rectangle that will produce a maimum area. Let w represent the width of the rectangle and let l represent the length of the rectangle. Because w l Perimeter is. it follows that l w, as shown in Figure.. So, the area of the rectangle is A lw ww w w. Formula for area Substitute w for l. Simplif. w Dick Lurial/FPG/Gett Images l = w FIGURE. Using this model for area, ou can eperiment with different values of w to see how to obtain the maimum area. After tring several values, it appears that the maimum area occurs when w 6, as shown in the table. Width, w Area, A In it terminolog, ou can sa that the it of A as w approaches 6 is 6. This is written as A w w 6 w 6 w 6. Now tr Eercise 5.

127 Section. Introduction to Limits 85 Definition of Limit An alternative notation for f L is c f L as c which is read as f approaches L as approaches c. Definition of Limit If f becomes arbitraril close to a unique number L as approaches c from either side, the it of f as approaches c is L. This is written as f L. c Eample Estimating a Limit Numericall 5 (, ) Use a table to estimate numericall the it:. Let f. Then construct a table that shows values of f for two sets of -values one set that approaches from the left and one that approaches from the right. 5 FIGURE. f() = f ? From the table, it appears that the closer gets to, the closer f gets to. So, ou can estimate the it to be. Figure. adds further support for this conclusion. Now tr Eercise 7. In Figure., note that the graph of f is continuous. For graphs that are not continuous, finding a it can be more difficult. Eample Estimating a Limit Numericall f ( ) = 0 (0, ) 5 f() = + Use a table to estimate numericall the it: 0. Let f. Then construct a table that shows values of f for two sets of -values one set that approaches 0 from the left and one that approaches 0 from the right. f is undefined at = 0. FIGURE f ? From the table, it appears that the it is. The graph shown in Figure. verifies that the it is. Now tr Eercise 9.

128 85 Chapter Limits and an Introduction to Calculus In Eample, note that f has a it when 0 even though the function is not defined when 0. This often happens, and it is important to realize that the eistence or noneistence of f at c has no bearing on the eistence of the it of f as approaches c. Eample Estimating a Limit Estimate the it:. Numerical Let f. Then construct a table that shows values of f for two sets of -values one set that approaches from the left and one that approaches from the right. Graphical Let f. Then sketch a graph of the function, as shown in Figure.. From the graph, it appears that as approaches from either side, f approaches. So, ou can estimate the it to be f ? f? f() = + 5 f ( ) = (, ) From the tables, it appears that the it is. Now tr Eercise. FIGURE. f is undefined at =. Eample 5 Using a Graph to Find a Limit FIGURE.5 f () =, 0, = Find the it of f as approaches, where f is defined as f, 0, Because f for all other than and because the value of f is immaterial, it follows that the it is (see Figure.5). So, ou can write f. The fact that f 0 has no bearing on the eistence or value of the it as approaches. For instance, if the function were defined as f,,. the it as approaches would be the same. Now tr Eercise 7.

129 Section. Introduction to Limits 85 Limits That Fail to Eist Net, ou will eamine some functions for which its do not eist. Eample 6 Comparing Left and Right Behavior f() = FIGURE.6 f() = f() = Show that the it does not eist. 0 Consider the graph of the function given b see that for positive -values, and for negative -values, > 0 < 0. From Figure.6, ou can This means that no matter how close gets to 0, there will be both positive and negative -values that ield f and f. This implies that the it does not eist. Now tr Eercise. f. Eample 7 Unbounded Behavior f() = Discuss the eistence of the it. 0 Let f. In Figure.7, note that as approaches 0 from either the right or the left, f increases without bound. This means that b choosing close enough to 0, ou can force f to be as large as ou want. For instance, f will be larger than 00 if ou choose that is within of 0. That is, 0 < < 0 0 f > 00. FIGURE.7 Similarl, ou can force f to be larger than,000,000, as follows. 0 < < 000 Because f is not approaching a unique real number L as approaches 0, ou can conclude that the it does not eist. Now tr Eercise. f >,000,000

130 85 Chapter Limits and an Introduction to Calculus Eample 8 Oscillating Behavior Discuss the eistence of the it. 0 sin f() = sin Let f sin. In Figure.8, ou can see that as approaches 0, f oscillates between and. Therefore, the it does not eist because no matter how close ou are to 0, it is possible to choose values of and such that sin and sin, as indicated in the table sin? FIGURE.8 Now tr Eercise 5. Eamples 6, 7, and 8 show three of the most common tpes of behavior associated with the noneistence of a it. Conditions Under Which Limits Do Not Eist The it of f as c does not eist if an of the following conditions are true.. f approaches a different number from the right side of c than it approaches from the left side of c.. f increases or decreases without bound as approaches c.. f oscillates between two fied values as approaches c. Eample 6 Eample 7 Eample 8 f() = sin FIGURE.9 TECHNOLOGY A graphing utilit can help ou discover the behavior of a function near the -value at which ou are tring to evaluate a it. When ou do this, however, ou should realize that ou can t alwas trust the graphs that graphing utilities displa. For instance, if ou use a graphing utilit to graph the function in Eample 8 over an interval containing 0, ou will most likel obtain an incorrect graph, as shown in Figure.9. The reason that a graphing utilit can t show the correct graph is that the graph has infinitel man oscillations over an interval that contains 0.

131 Section. Introduction to Limits 855 Properties of Limits and Direct Substitution You have seen that sometimes the it of f as c is simpl f c, as shown in Eample. In such cases, it is said that the it can be evaluated b direct substitution. That is, f f c). c Substitute c for. There are man well-behaved functions, such as polnomial functions and rational functions with nonzero denominators, that have this propert. Some of the basic ones are included in the following list. Basic Limits Let b and c be real numbers and let n be a positive integer.. c Limit of a constant function. c Limit of the identit function. c n c n Limit of a power function. n c, n c for n even and c > 0 Limit of a radical function For a proof of the it of a power function, see Proofs in Mathematics on page 906. Trigonometric functions can also be included in this list. For instance, and sin sin 0 cos cos 0. 0 B combining the basic its with the following operations, ou can find its for a wide variet of functions. Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following its. f L c and g K c. Scalar multiple:. Sum or difference:. Product: bf bl c f ± g L ± K c fg LK c. Quotient: f c g L K, provided K 0 5. Power: c f n L n

132 856 Chapter Limits and an Introduction to Calculus Eample 9 Direct Substitution and Properties of Limits FIGURE.0 (, 6) = Find each it. a. b. c. d. e. cos f. 9 You can use the properties of its and direct substitution to evaluate each it. a. 6 b. 5 5 Propert tan tan c. Propert d. 9 9 e. cos cos Propert cos f. Properties and Now tr Eercise 7. When evaluating its, remember that there are several was to solve most problems. Often, a problem can be solved numericall, graphicall, or algebraicall. The its in Eample 9 were found algebraicall. You can verif the solutions numericall and/or graphicall. For instance, to verif the it in Eample 9(a) numericall, create a table that shows values of for two sets of -values one set that approaches from the left and one that approaches from the right, as shown below. From the table, ou can see that the it as approaches is 6. Now, to verif the it graphicall, sketch the graph of. From the graph shown in Figure.0, ou can determine that the it as approaches is 6. tan ?

133 Section. Introduction to Limits 857 The results of using direct substitution to evaluate its of polnomial and rational functions are summarized as follows. Limits of Polnomial and Rational Functions. If p is a polnomial function and c is a real number, then p pc. c. If r is a rational function given b r pq, and c is a real number such that qc 0, then pc r rc c qc. For a proof of the it of a polnomial function, see Proofs in Mathematics on page 906. Eample 0 Evaluating Limits b Direct Substitution Find each it. a. b. 6 The first function is a polnomial function and the second is a rational function with a nonzero denominator at. So, ou can evaluate the its b direct substitution. a b Now tr Eercise 5. CLASSROOM DISCUSSION Graphs with Holes Sketch the graph of each function. Then find the its of each function as approaches and as approaches. What conclusions can ou make? a. b. c. h g f Use a graphing utilit to graph each function above. Does the graphing utilit distinguish among the three graphs? Write a short eplanation of our findings.

134 858 Chapter Limits and an Introduction to Calculus. EXERCISES See for worked-out solutions to odd-numbered eercises. VOCABULARY: Fill in the blanks.. If f becomes arbitraril close to a unique number L as approaches c from either side, the of f as approaches c is L.. An alternative notation for f L is f L as c, which is read as f L as c. c. The it of f as c does not eist if f between two fied values.. To evaluate the it of a polnomial function, use. SKILLS AND APPLICATIONS 5. GEOMETRY You create an open bo from a square piece of material centimeters on a side. You cut equal squares from the corners and turn up the sides. (a) Draw and label a diagram that represents the bo. (b) Verif that the volume V of the bo is given b V. (c) The bo has a maimum volume when. Use a graphing utilit to complete the table and observe the behavior of the function as approaches. Use the table to find V V (d) Use a graphing utilit to graph the volume function. Verif that the volume is maimum when. 6. GEOMETRY You are given wire and are asked to form a right triangle with a hpotenuse of 8 inches whose area is as large as possible. (a) Draw and label a diagram that shows the base and height of the triangle. (b) Verif that the area A of the triangle is given b A 8. (c) The triangle has a maimum area when inches. Use a graphing utilit to complete the table and observe the behavior of the function as approaches. Use the table to find A A (d) Use a graphing utilit to graph the area function. Verif that the area is maimum when inches. In Eercises 7, complete the table and use the result to estimate the it numericall. Determine whether or not the it can be reached f? f? f?. 0 sin f? f f? f

135 Section. Introduction to Limits 859. tan f? f.. In Eercises 6, create a table of values for the function and use the result to estimate the it numericall. Use a graphing utilit to graph the corresponding function to confirm our result graphicall sin cos sin tan e e ln ln In Eercises 7 and 8, graph the function and find the it (if it eists) as approaches f,, f,, In Eercises 9 6, use the graph to find the it (if it eists). If the it does not eist, eplain wh < > cos 6. 0 In Eercises 7, use a graphing utilit to graph the function and use the graph to determine whether the it eists. If the it does not eist, eplain wh f 5 e, f ln7, f cos, 0 f sin, f, 5 f, f, f 0. f 7 f, f f f sin

136 860 Chapter Limits and an Introduction to Calculus In Eercises 5 and 6, use the given information to evaluate each it. 70. The it of the product of two functions is equal to the product of the its of the two functions. 5. c f, 7. THINK ABOUT IT From Eercises 7, select a it that can be reached and one that cannot be reached. (a) Use a graphing utilit to graph the corresponding functions using a standard viewing window. Do the graphs reveal whether or not the it can be reached? Eplain. (b) Use a graphing utilit to graph the corresponding functions using a decimal setting. Do the graphs reveal whether or not the it can be reached? Eplain. 7. THINK ABOUT IT Use the results of Eercise 7 to draw a conclusion as to whether or not ou can use the graph generated b a graphing utilit to determine reliabl if a it can be reached. 7. THINK ABOUT IT (a) If f, can ou conclude anthing about f? Eplain our reasoning. g 6 c (a) c g (b) c f g (c) c f g (d) c 6. c f 5, f g c (a) c f g (b) 6f g c 5g f (c) c f (d) c In Eercises 7 and 8, find (a) f, (b) g, (c) [ f g ], and (d) [ g ⴚ f ]. 7. f, 8. f g 5, g sin (b) If f, can ou conclude anthing In Eercises 9 68, find the it b direct substitution e ln 6. e sin tan 67. arcsin 68. arccos EXPLORATION TRUE OR FALSE? In Eercises 69 and 70, determine whether the statement is true or false. Justif our answer. 69. The it of a function as approaches c does not eist if the function approaches from the left of c and from the right of c. about f? Eplain our reasoning. 7. WRITING Write a brief description of the meaning of the notation f THINK ABOUT IT Use a graphing utilit to graph the tangent function. What are tan and tan? 0 What can ou sa about the eistence of the it tan? 76. CAPSTONE Use the graph of the function f to decide whether the value of the given quantit eists. If it does, find it. If not, eplain wh. (a) f 0 (b) f 5 (c) f 0 (d) f 77. WRITING Use a graphing utilit to graph the function 0 given b f. Use the trace feature 5 to approimate f. What do ou think f 5 equals? Is f defined at 5? Does this affect the eistence of the it as approaches 5?

137 Section. Techniques for Evaluating Limits 86. TECHNIQUES FOR EVALUATING LIMITS What ou should learn Use the dividing out technique to evaluate its of functions. Use the rationalizing technique to evaluate its of functions. Approimate its of functions graphicall and numericall. Evaluate one-sided its of functions. Evaluate its of difference quotients from calculus. Wh ou should learn it Limits can be applied in real-life situations. For instance, in Eercise 8 on page 870, ou will determine its involving the costs of making photocopies. Dividing Out Technique In Section., ou studied several tpes of functions whose its can be evaluated b direct substitution. In this section, ou will stud several techniques for evaluating its of functions for which direct substitution fails. Suppose ou were asked to find the following it. 6 Direct substitution produces 0 in both the numerator and denominator. 6 0 Numerator is 0 when. 0 Denominator is 0 when. 0 The resulting fraction, 0, has no meaning as a real number. It is called an indeterminate form because ou cannot, from the form alone, determine the it. B using a table, however, it appears that the it of the function as is ? Michael Krasowitz/TAXI/Gett Images When ou tr to evaluate a it of a rational function b direct substitution 0 and encounter the indeterminate form 0, ou can conclude that the numerator and denominator must have a common factor. After factoring and dividing out, ou should tr direct substitution again. Eample shows how ou can use the dividing out technique to evaluate its of these tpes of functions. Eample Find the it: Dividing Out Technique 6. From the discussion above, ou know that direct substitution fails. So, begin b factoring the numerator and dividing out an common factors. 6 Factor numerator. Divide out common factor. Simplif. 5 Direct substitution and simplif. Now tr Eercise.

138 86 Chapter Limits and an Introduction to Calculus The validit of the dividing out technique stems from the fact that if two functions agree at all but a single number c, the must have identical it behavior at c. In Eample, the functions given b f 6 and g agree at all values of other than. So, ou can use g to find the it of f. Eample Dividing Out Technique FIGURE. f () = (, ) + f is undefined when =. Find the it. Begin b substituting into the numerator and denominator. 0 Numerator is 0 when. 0 Denominator is 0 when. Because both the numerator and denominator are zero when, direct substitution will not ield the it. To find the it, ou should factor the numerator and denominator, divide out an common factors, and then tr direct substitution again. This result is shown graphicall in Figure.. Now tr Eercise 5. Factor denominator. Divide out common factor. Simplif. Direct substitution Simplif. In Eample, the factorization of the denominator can be obtained b dividing b or b grouping as follows.

139 Section. Techniques for Evaluating Limits 86 You can review the techniques for rationalizing numerators and denominators in Appendi A.. Rationalizing Technique Another wa to find the its of some functions is first to rationalize the numerator of the function. This is called the rationalizing technique. Recall that rationalizing the numerator means multipling the numerator and denominator b the conjugate of the numerator. For instance, the conjugate of is. Eample Rationalizing Technique Find the it: 0. ( ) FIGURE. f () = 0, + f is undefined when = 0. B direct substitution, ou obtain the indeterminate form Indeterminate form In this case, ou can rewrite the fraction b rationalizing the numerator. 0, Now ou can evaluate the it b direct substitution. Multipl. Simplif. Divide out common factor. Simplif You can reinforce our conclusion that the it is b constructing a table, as shown below, or b sketching a graph, as shown in Figure f ? Now tr Eercise The rationalizing technique for evaluating its is based on multiplication b a convenient form of. In Eample, the convenient form is.

140 86 Chapter Limits and an Introduction to Calculus Using Technolog The dividing out and rationalizing techniques ma not work well for finding its of nonalgebraic functions. You often need to use more sophisticated analtic techniques to find its of these tpes of functions. Eample Approimating a Limit Approimate the it: 0. Numerical Let f. Because ou are finding the it when 0, use the table feature of a graphing utilit to create a table that shows the values of f for starting at 0.0 and has a step of 0.00, as shown in Figure.. Because 0 is halfwa between 0.00 and 0.00, use the average of the values of f at these two -coordinates to estimate the it, as follows The actual it can be found algebraicall to be e.788. Graphical To approimate the it graphicall, graph the function f, as shown in Figure.. Using the zoom and trace features of the graphing utilit, choose two points on the graph of f, such as ,.785 and as shown in Figure.5. Because the -coordinates of these two points are equidistant from 0, ou can approimate the it to be the average of the -coordinates. That is, The actual it can be found algebraicall to be e f() = ( + )/ , FIGURE. Now tr Eercise 7. 0 FIGURE. FIGURE Eample 5 Approimating a Limit Graphicall Approimate the it: sin 0. FIGURE.6 f() = sin 0 Direct substitution produces the indeterminate form 0. To approimate the it, begin b using a graphing utilit to graph f sin, as shown in Figure.6. Then use the zoom and trace features of the graphing utilit to choose a point on each side of 0, such as , and , Finall, approimate the it as the average of the -coordinates of these two points, sin It can be shown algebraicall that this it is eactl. 0 Now tr Eercise.

141 Section. Techniques for Evaluating Limits 865 TECHNOLOGY The graphs shown in Figures. and.6 appear to be continuous at 0. However, when ou tr to use the trace or the value feature of a graphing utilit to determine the value of when 0, no value is given. Some graphing utilities can show breaks or holes in a graph when an appropriate viewing window is used. Because the holes in the graphs in Figures. and.6 occur on the -ais, the holes are not visible. One-Sided Limits In Section., ou saw that one wa in which a it can fail to eist is when a function approaches a different value from the left side of c than it approaches from the right side of c. This tpe of behavior can be described more concisel with the concept of a one-sided it. c f L c f L or f L as c Limit from the left or f L as c Limit from the right Eample 6 Evaluating One-Sided Limits f() = FIGURE.7 f() = f () = Find the it as 0 from the left and the it as 0 from the right for f. From the graph of f, shown in Figure.7, ou can see that f for all < 0. Therefore, the it from the left is. 0 Limit from the left: f as 0 Because f for all > 0, the it from the right is. 0 Now tr Eercise 55. Limit from the right: f as 0 In Eample 6, note that the function approaches different its from the left and from the right. In such cases, the it of f as c does not eist. For the it of a function to eist as c, it must be true that both one-sided its eist and are equal. Eistence of a Limit If f is a function and c and L are real numbers, then f L c if and onl if both the left and right its eist and are equal to L.

142 866 Chapter Limits and an Introduction to Calculus Eample 7 Finding One-Sided Limits Find the it of f as approaches. f,, < > FIGURE.8 f() =, < f() =, > Remember that ou are concerned about the value of f near rather than at. So, for <, f is given b, and ou can use direct substitution to obtain For >, f is given b, and ou can use direct substitution to obtain f. f. Because the one-sided its both eist and are equal to, it follows that f. The graph in Figure.8 confirms this conclusion. Now tr Eercise 59. Eample 8 Comparing Limits from the Left and Right Shipping cost (in dollars) For 0 <, f() = 8 7 Weight (in pounds) FIGURE.9 Overnight Deliver For <, f() = For <, f() = 0 To ship a package overnight, a deliver service charges $8 for the first pound and $ for each additional pound or portion of a pound. Let represent the weight of a package and let f represent the shipping cost. Show that the it of f as does not eist. f $8, $0, $, 0 < < < The graph of f is shown in Figure.9. The it of f as approaches from the left is f 0 whereas the it of f as approaches from the right is f. Because these one-sided its are not equal, the it of f as does not eist. Now tr Eercise 8.

143 Section. Techniques for Evaluating Limits 867 A Limit from Calculus In the net section, ou will stud an important tpe of it from calculus the it of a difference quotient. Eample 9 Evaluating a Limit from Calculus For the function given b f, find h 0 Direct substitution produces an indeterminate form. h 0 f h f. h f h f h h 0 h h h 0 9 6h h 9 h h 0 6h h h B factoring and dividing out, ou obtain the following. h 0 f h f h 0 0 6h h h6 h h 0 h h 0 h h 0 6 h So, the it is 6. Now tr Eercise For a review of evaluating difference quotients, refer to Section.. Note that for an -value, the it of a difference quotient is an epression of the form h 0 f h f. h Direct substitution into the difference quotient alwas produces the indeterminate form For instance, f h f f 0 f h 0 h f f

144 868 Chapter Limits and an Introduction to Calculus. EXERCISES See for worked-out solutions to odd-numbered eercises. VOCABULARY: Fill in the blanks.. To evaluate the it of a rational function that has common factors in its numerator and denominator, use the. 0. The fraction 0 has no meaning as a real number and therefore is called an.. The it f L is an eample of a. c f h f. The it of a is an epression of the form. h 0 h SKILLS AND APPLICATIONS In Eercises 5 8, use the graph to determine each it visuall (if it eists). Then identif another function that agrees with the given function at all but one point h g (a) (b) (c) 6 6 g g g 0 (a) g (a) h 0 (b) g (b) h 0 (c) g (c) h f g 6 (a) f (b) f (c) f In Eercises 9 6, find the it (if it eists). Use a graphing utilit to verif our result graphicall t a t t a a z 7. 0 z 0 z sec csc.. 0 tan cot sin cos cos sin

145 Section. Techniques for Evaluating Limits 869 In Eercises 7 8, use a graphing utilit to graph the function and approimate the it accurate to three decimal places. e e ln 0. 0 ln 0. sin 0.. tan GRAPHICAL, NUMERICAL, AND ALGEBRAIC ANALYSIS In Eercises 9 5, (a) graphicall approimate the it (if it eists) b using a graphing utilit to graph the function, (b) numericall approimate the it (if it eists) b using the table feature of a graphing utilit to create a table, and (c) algebraicall evaluate the it (if it eists) b the appropriate technique(s) In Eercises 55 6, graph the function. Determine the it (if it eists) b evaluating the corresponding one-sided its f where 60. f where 6. f where 6. f where 0 f,, f,, f,, f,, sin cos > < > 0 > 0 In Eercises 6 68, use a graphing utilit to graph the function and the equations and in the same viewing window. Use the graph to find f f cos 6. f sin 65. f sin 66. f cos In Eercises 69 and 70, state which it can be evaluated b using direct substitution. Then evaluate or approimate each it. 69. (a) sin 0 sin (b) (a) (b) In Eercises 7 78, find f sin f cos 0 cos cos 0 f f f f f f f f FREE-FALLING OBJECT position function st 6t 56 h 0 f h f. h In Eercises 79 and 80, use the which gives the height (in feet) of a free-falling object. The velocit at time t a seconds is given b /a t. t a 79. Find the velocit when t second. 80. Find the velocit when t seconds.

146 870 Chapter Limits and an Introduction to Calculus 8. SALARY CONTRACT A union contract guarantees an 8% salar increase earl for ears. For a current salar of $0,000, the salaries f t (in thousands of dollars) for the net ears are given b 0.000, 0 < t f t.00, < t.99, < t 5 where represents the weight of the package (in pounds). Show that the it of f as does not eist. 8. CONSUMER AWARENESS The cost of hooking up and towing a car is $85 for the first mile and $5 for each additional mile or portion of a mile. A model for the cost C (in dollars) is C 85 5, where is the distance in miles. (Recall from Section.6 that f the greatest integer less than or equal to.) (a) Use a graphing utilit to graph C for 0 < 0. (b) Complete the table and observe the behavior of C as approaches 5.5. Use the graph from part (a) and the table to find C ? C C ? EXPLORATION TRUE OR FALSE? In Eercises 85 and 86, determine whether the statement is true or false. Justif our answer. 85. When our attempt to find the it of a rational function ields the indeterminate form 00, the rational function s numerator and denominator have a common factor. 86. If f c L, then f L. c 87. THINK ABOUT IT (a) Sketch the graph of a function for which f is defined but for which the it of f as approaches does not eist. (b) Sketch the graph of a function for which the it of f as approaches is but for which f. 88. CAPSTONE f Given, 0,, > 0 find each of the following its. If the it does not eist, eplain wh. (a) f (b) f (c) f 0 0.5, 0 < 5 0.0, 5 < 00 C. 0.07, 00 < , > WRITING Consider the it of the rational function given b p q. What conclusion can ou make if direct substitution produces each epression? Write a short paragraph eplaining our reasoning. p 0 c q (a) 8. CONSUMER AWARENESS The cost C (in dollars) of making photocopies at a cop shop is given b the function 500 (d) Eplain how ou can use the graph in part (a) to verif that the its in part (c) do not eist. 0 (c) Complete the table and observe the behavior of C as approaches 5. Does the it of C as approaches 5 eist? Eplain. 05 (c) Create a table of values to show numericall that each it does not eist. (i) C (ii) C (iii) C where t represents the time in ears. Show that the it of f as t does not eist. 8. CONSUMER AWARENESS The cost of sending a package overnight is $5 for the first pound and $.0 for each additional pound or portion of a pound. A plastic mailing bag can hold up to pounds. The cost f of sending a package in a plastic mailing bag is given b 5.00, 0 < f 6.0, < 7.60, < (a) Sketch a graph of the function. (b) Find each it and interpret our result in the contet of the situation. (i) C (ii) C (iii) C (b) p q (c) p q 0 (d) p 0 q 0 c c c

147 Section. The Tangent Line Problem 87. THE TANGENT LINE PROBLEM What ou should learn Use a tangent line to approimate the slope of a graph at a point. Use the it definition of slope to find eact slopes of graphs. Find derivatives of functions and use derivatives to find slopes of graphs. Wh ou should learn it The slope of the graph of a function can be used to analze rates of change at particular points on the graph. For instance, in Eercise 7 on page 880, the slope of the graph is used to analze the rate of change in book sales for particular selling prices. Tangent Line to a Graph Calculus is a branch of mathematics that studies rates of change of functions. If ou go on to take a course in calculus, ou will learn that rates of change have man applications in real life. Earlier in the tet, ou learned how the slope of a line indicates the rate at which a line rises or falls. For a line, this rate (or slope) is the same at ever point on the line. For graphs other than lines, the rate at which the graph rises or falls changes from point to point. For instance, in Figure.0, the parabola is rising more quickl at the point, than it is at the point,. At the verte,, the graph levels off, and at the point,, the graph is falling. (, ) (, ) (, ) (, ) FIGURE.0 Bob Rowan, Progressive Image/Corbis To determine the rate at which a graph rises or falls at a single point, ou can find the slope of the tangent line at that point. In simple terms, the tangent line to the graph of a function f at a point P, is the line that best approimates the slope of the graph at the point. Figure. shows other eamples of tangent lines. P P P FIGURE. From geometr, ou know that a line is tangent to a circle if the line intersects the circle at onl one point. Tangent lines to noncircular graphs, however, can intersect the graph at more than one point. For instance, in the first graph in Figure., if the tangent line were etended, it would intersect the graph at a point other than the point of tangenc.

148 87 Chapter Limits and an Introduction to Calculus Slope of a Graph Because a tangent line approimates the slope of the graph at a point, the problem of finding the slope of a graph at a point is the same as finding the slope of the tangent line at the point. Eample Visuall Approimating the Slope of a Graph f() = 5 FIGURE. Use the graph in Figure. to approimate the slope of the graph of f at the point,. From the graph of f, ou can see that the tangent line at, rises approimatel two units for each unit change in. So, ou can estimate the slope of the tangent line at, to be Slope. change in change in Because the tangent line at the point, has a slope of about, ou can conclude that the graph of f has a slope of about at the point,. Now tr Eercise 5. When ou are visuall approimating the slope of a graph, remember that the scales on the horizontal and vertical aes ma differ. When this happens (as it frequentl does in applications), the slope of the tangent line is distorted, and ou must be careful to account for the difference in scales. Eample Approimating the Slope of a Graph Temperature ( F) FIGURE. Monthl Normal Temperatures (0, 69) Month Figure. graphicall depicts the monthl normal temperatures (in degrees Fahrenheit) for Dallas, Teas. Approimate the slope of this graph at the indicated point and give a phsical interpretation of the result. (Source: National Catic Data Center) From the graph, ou can see that the tangent line at the given point falls approimatel 6 units for each two-unit change in. So, ou can estimate the slope at the given point to be Slope change in change in 6 8 degrees per month. This means that ou can epect the monthl normal temperature in November to be about 8 degrees lower than the normal temperature in October. Now tr Eercise 7.

149 Section. The Tangent Line Problem 87 ( + h, f( + h)) (, f()) FIGURE. h f ( + h) f ( ) Slope and the Limit Process In Eamples and, ou approimated the slope of a graph at a point b creating a graph and then eeballing the tangent line at the point of tangenc. A more precise method of approimating tangent lines makes use of a secant line through the point of tangenc and a second point on the graph, as shown in Figure.. If, f is the point of tangenc and h, f h is a second point on the graph of f, the slope of the secant line through the two points is given b m sec change in change in f h f. h Slope of secant line The right side of this equation is called the difference quotient. The denominator h is the change in, and the numerator is the change in. The beaut of this procedure is that ou obtain more and more accurate approimations of the slope of the tangent line b choosing points closer and closer to the point of tangenc, as shown in Figure.5. ( + h, f( + h)) ( + h, f( + h )) ( + h, f( + h)) (, f()) f ( + h) f ( ) (, f()) f ( + h) f ( ) (, f()) f ( + h) f ( ) Tangent line (, f()) h h h As h approaches 0, the secant line approaches the tangent line. FIGURE.5 Using the it process, ou can find the eact slope of the tangent line at, f. Definition of the Slope of a Graph The slope m of the graph of f at the point, f is equal to the slope of its tangent line at, f, and is given b m h 0 m sec h 0 f h f h provided this it eists. From the definition above and from Section., ou can see that the difference quotient is used frequentl in calculus. Using the difference quotient to find the slope of a tangent line to a graph is a major concept of calculus.

150 87 Chapter Limits and an Introduction to Calculus Eample Finding the Slope of a Graph Tangent line at (, ) m = FIGURE.6 5 f() = Find the slope of the graph of f at the point,. Find an epression that represents the slope of a secant line at,. m sec f h f h h h h h h h h h h h h h, h 0 Net, take the it of as h approaches 0. m sec Set up difference quotient. Substitute in f. Epand terms. Simplif. Factor and divide out. Simplif. m m h h 0 sec h 0 The graph has a slope of at the point,, as shown in Figure.6. Now tr Eercise 9. Eample Finding the Slope of a Graph FIGURE.7 f() = + m = Find the slope of f. m h 0 f h f h h 0 h h h 0 h h h h 0 h Set up difference quotient. Substitute in f. Epand terms. Divide out. Simplif. You know from our stud of linear functions that the line given b f has a slope of, as shown in Figure.7. This conclusion is consistent with that obtained b the it definition of slope, as shown above. Now tr Eercise.

151 Section. The Tangent Line Problem 875 TECHNOLOGY Tr verifing the result in Eample 5 b using a graphing utilit to graph the function and the tangent lines at, and, 5 as in the same viewing window. Some graphing utilities even have a tangent feature that automaticall graphs the tangent line to a curve at a given point. If ou have such a graphing utilit, tr verifing Eample 5 using this feature. Tangent line at (, ) f() = + FIGURE Tangent line at (, 5) It is important that ou see the difference between the was the difference quotients were set up in Eamples and. In Eample, ou were finding the slope of a graph at a specific point c, f c. To find the slope in such a case, ou can use the following form of the difference quotient. m h 0 f c h f c h Slope at specific point In Eample, however, ou were finding a formula for the slope at an point on the graph. In such cases, ou should use, rather than c, in the difference quotient. m h 0 f h f h Formula for slope Ecept for linear functions, this form will alwas produce a function of, which can then be evaluated to find the slope at an desired point. Eample 5 Finding a Formula for the Slope of a Graph Find a formula for the slope of the graph of f. What are the slopes at the points, and, 5? m sec h h h h h h h h h h h h, Net, take the it of as h approaches 0. m h 0 m sec Set up difference quotient. Substitute in f. Epand terms. Simplif. Factor and divide out. Simplif. Formula for finding slope Using the formula m for the slope at, f, ou can find the slope at the specified points. At,, the slope is m and at, 5, the slope is m. f h f h h 0 m sec h 0 h The graph of f is shown in Figure.8. Now tr Eercise 7.

152 876 Chapter Limits and an Introduction to Calculus The Derivative of a Function In Eample 5, ou started with the function f and used the it process to derive another function, m, that represents the slope of the graph of f at the point, f. This derived function is called the derivative of f at. It is denoted b f, which is read as f prime of. In Section.5, ou studied the slope of a line, which represents the average rate of change over an interval. The derivative of a function is a formula which represents the instantaneous rate of change at a point. Definition of Derivative The derivative of f at is given b f h 0 f h f h provided this it eists. Remember that the derivative f is a formula for the slope of the tangent line to the graph of f at the point, f. Eample 6 Finding a Derivative Find the derivative of f. f h 0 f h f h So, the derivative of f is f 6. Now tr Eercise. Note that in addition to f, other notations can be used to denote the derivative of f. The most common are d d, h 0 h h h h 0 6h h h h h 0 6h h h h h 0 h6 h h h 0 6 h 6, d f, d and D.

153 Section. The Tangent Line Problem 877 Eample 7 Using the Derivative Find f for f. Then find the slopes of the graph of f at the points, and,. Remember that in order to rationalize the numerator of an epression, ou must multipl the numerator and denominator b the conjugate of the numerator. 5 f() = (, ) FIGURE.9 m = (, ) m = f h 0 f h f h h 0 h h Because direct substitution ields the indeterminate form 0, ou should use the rationalizing technique discussed in Section. to find the it. h h h 0 h h h 0 h h 0 h h h 0 h At the point,, the slope is f. At the point,, the slope is f. h h h The graph of f is shown in Figure.9. Now tr Eercise. 0 CLASSROOM DISCUSSION Using a Derivative to Find Slope In man applications, it is convenient to use a variable other than as the independent variable. Complete the following it process to find the derivative of ft /t. Then use the result to find the slope of the graph of ft /t at the point,. ft h ft ft h 0 h h 0 t h t h Write a short paragraph summarizing our findings....

154 878 Chapter Limits and an Introduction to Calculus. EXERCISES VOCABULARY: Fill in the blanks.. is the stud of the rates of change of functions. See for worked-out solutions to odd-numbered eercises.. The to the graph of a function at a point is the line that best approimates the slope of the graph at the point.. A is a line through the point of tangenc and a second point on the graph.. The slope of the tangent line to a graph at, f is given b. SKILLS AND APPLICATIONS In Eercises 5 8, use the figure to approimate the slope of the curve at the point, (, ) (, ) 9. f 0. (a) 0, (b),. f. (a) 5, (b) 0, f (a) 0, (b), f (a) 5, (b) 8, (, ) (, ) In Eercises 8, sketch a graph of the function and the tangent line at the point, f. Use the graph to approimate the slope of the tangent line.. f. f 5. f 6. f 7. f 8. f In Eercises 9 6, use the it process to find the slope of the graph of the function at the specified point. Use a graphing utilit to confirm our result g, f 0,,,.. g 5, h 5,,,. g,. g,,, 5. h, 9, 6. h 0,, In Eercises 7, find a formula for the slope of the graph of f at the point, f. Then use it to find the slope at the two given points. 7. f 8. f (a) 0, (a), (b), (b), 8 In Eercises 9, find the derivative of the function. 9. f 5 0. f. g 9. f 5. f. f 5. f 6. f 7. f 8. f 8 9. f 0. f 6 5. f. hs 9 s In Eercises 50, (a) find the slope of the graph of f at the given point, (b) use the result of part (a) to find an equation of the tangent line to the graph at the point, and (c) graph the function and the tangent line.. f,,. 5. f,, f,, 6. f,, 6

155 Section. The Tangent Line Problem f,, f,, f,, 5 f,, In Eercises 5 5, use a graphing utilit to graph f over the interval [, ] and complete the table. Compare the value of the first derivative with a visual approimation of the slope of the graph f f 5. f 5. f f f In Eercises 55 58, find an equation of the line that is tangent to the graph of f and parallel to the given line. 0 Function Line 55. f f f f In Eercises 59 6, find the derivative of f. Use the derivative to determine an points on the graph of f at which the tangent line is horizontal. Use a graphing utilit to verif our results. 59. f 60. f 6 6. f 9 6. f In Eercises 6 70, use the function and its derivative to determine an points on the graph of f at which the tangent line is horizontal. Use a graphing utilit to verif our results f, f, f f 65. f cos, f sin, over the interval 0, 66. f sin, f cos, over the interval 0, f e, f e, f e e f e e f ln, f ln, 7. PATH OF A BALL The path of a ball thrown b a child is modeled b 5 f ln ln f where is the height of the ball (in feet) and is the horizontal distance (in feet) from the point from which the ball was thrown. Using our knowledge of the slopes of tangent lines, show that the height of the ball is increasing on the interval 0, and decreasing on the interval, 5. Eplain our reasoning. 7. PROFIT The profit P (in hundreds of dollars) that a compan makes depends on the amount (in hundreds of dollars) the compan spends on advertising. The profit function is given b P Using our knowledge of the slopes of tangent lines, show that the profit is increasing on the interval 0, 0 and decreasing on the interval 0, The table shows the revenues (in millions of dollars) for eba, Inc. from 000 through 007. (Source: eba, Inc.) Year Revenue, (a) Use the regression feature of a graphing utilit to find a quadratic model for the data. Let represent the time in ears, with 0 corresponding to 000. (b) Use a graphing utilit to graph the model found in part (a). Estimate the slope of the graph when 5 and give an interpretation of the result. (c) Use a graphing utilit to graph the tangent line to the model when 5. Compare the slope given b the graphing utilit with the estimate in part (b).

156 880 Chapter Limits and an Introduction to Calculus 7. MARKET RESEARCH The data in the table show the number N (in thousands) of books sold when the price per book is p (in dollars). (c) (d) 5 Price, p Number of books, N $0 $5 $0 $5 $0 $ f 79. f (a) Use the regression feature of a graphing utilit to find a quadratic model for the data. (b) Use a graphing utilit to graph the model found in part (a). Estimate the slopes of the graph when p $5 and p $0. (c) Use a graphing utilit to graph the tangent lines to the model when p $5 and p $0. Compare the slopes given b the graphing utilit with our estimates in part (b). (d) The slopes of the tangent lines at p $5 and p $0 are not the same. Eplain what this means to the compan selling the books. EXPLORATION TRUE OR FALSE? In Eercises 75 and 76, determine whether the statement is true or false. Justif our answer. 75. The slope of the graph of is different at ever point on the graph of f. 76. A tangent line to a graph can intersect the graph onl at the point of tangenc. In Eercises 77 80, match the function with the graph of its derivative. It is not necessar to find the derivative of the function. [The graphs are labeled (a), (b), (c), and (d).] (a) (b) 80. f 78. f 8. THINK ABOUT IT Sketch the graph of a function whose derivative is alwas positive. 8. THINK ABOUT IT Sketch the graph of a function whose derivative is alwas negative. 8. THINK ABOUT IT Sketch the graph of a function for which f < 0 for <, f 0 for >, and f CONJECTURE Consider the functions f and g. (a) Sketch the graphs of f and f on the same set of coordinate aes. (b) Sketch the graphs of g and g on the same set of coordinate aes. (c) Identif an pattern between the functions f and g and their respective derivatives. Use the pattern to make a conjecture about h if h n, where n is an integer and n. 85. Consider the function f. (a) Use a graphing utilit to graph the function. (b) Use the trace feature to approimate the coordinates of the verte of this parabola. (c) Use the derivative of f to find the slope of the tangent line at the verte. (d) Make a conjecture about the slope of the tangent line at the verte of an arbitrar parabola. 86. CAPSTONE Eplain how the slope of the secant line is used to derive the slope of the tangent line and the definition of the derivative of a function f at a point, f. Include diagrams or sketches as necessar. 5 5 PROJECT: ADVERTISING To work an etended application analzing the amount spent on advertising in the United States, visit this tet s website at academic.cengage.com. (Data Source: Universal McCann)

157 Section. Limits at Infinit and Limits of Sequences 88. LIMITS AT INFINITY AND LIMITS OF SEQUENCES What ou should learn Evaluate its of functions at infinit. Find its of sequences. Wh ou should learn it Finding its at infinit is useful in man tpes of real-life applications. For instance, in Eercise 58 on page 889, ou are asked to find a it at infinit to determine the number of militar reserve personnel in the future. Limits at Infinit and Horizontal Asmptotes As pointed out at the beginning of this chapter, there are two basic problems in calculus: finding tangent lines and finding the area of a region. In Section., ou saw how its can be used to solve the tangent line problem. In this section and the net, ou will see how a different tpe of it, a it at infinit, can be used to solve the area problem. To get an idea of what is meant b a it at infinit, consider the function given b f. The graph of f is shown in Figure.0. From earlier work, ou know that is a horizontal asmptote of the graph of this function. Using it notation, this can be written as follows. f Horizontal asmptote to the left Karen Kasmauski/Corbis f Horizontal asmptote to the right These its mean that the value of f gets arbitraril close to as decreases or increases without bound. = f() = + The function f is a rational function. You can review rational functions in Section.6. FIGURE.0 Definition of Limits at Infinit If f is a function and and are real numbers, the statements and L Limit as approaches Limit as approaches f L denote the its at infinit. The first statement is read the it of f as approaches is L, and the second is read the it of f as approaches is L. f L L

158 88 Chapter Limits and an Introduction to Calculus To help evaluate its at infinit, ou can use the following definition. Limits at Infinit If r is a positive real number, then Limit toward the right Furthermore, if r is defined when < 0, then 0. r 0. r Limit toward the left Limits at infinit share man of the properties of its listed in Section.. Some of these properties are demonstrated in the net eample. Eample Evaluating a Limit at Infinit Find the it. Algebraic Use the properties of its listed in Section.. 0 So, the it of f as approaches is. Now tr Eercise 9. Graphical Use a graphing utilit to graph. Then use the trace feature to determine that as gets larger and larger, gets closer and closer to, as shown in Figure.. Note that the line is a horizontal asmptote to the right. 5 = = 0 0 FIGURE. In Figure., it appears that the line is also a horizontal asmptote to the left. You can verif this b showing that. The graph of a rational function need not have a horizontal asmptote. If it does, however, its left and right horizontal asmptotes must be the same. When evaluating its at infinit for more complicated rational functions, divide the numerator and denominator b the highest-powered term in the denominator. This enables ou to evaluate each it using the its at infinit at the top of this page.

159 Section. Limits at Infinit and Limits of Sequences 88 Eample Comparing Limits at Infinit Find the it as approaches for each function. a. b. f f c. f In each case, begin b dividing both the numerator and denominator b highest-powered term in the denominator., the a b. 0 0 c. In this case, ou can conclude that the it does not eist because the numerator decreases without bound as the denominator approaches. Now tr Eercise 9. In Eample, observe that when the degree of the numerator is less than the degree of the denominator, as in part (a), the it is 0. When the degrees of the numerator and denominator are equal, as in part (b), the it is the ratio of the coefficients of the highest-powered terms. When the degree of the numerator is greater than the degree of the denominator, as in part (c), the it does not eist. This result seems reasonable when ou realize that for large values of, the highest-powered term of a polnomial is the most influential term. That is, a polnomial tends to behave as its highest-powered term behaves as approaches positive or negative infinit.

160 88 Chapter Limits and an Introduction to Calculus Limits at Infinit for Rational Functions Consider the rational function f ND, where N a and D b m m... n n... a 0 b 0. The it of f as approaches positive or negative infinit is as follows. ± 0, n < m f a n, n m b m If n > m, the it does not eist. Eample Finding the Average Cost You are manufacturing greeting cards that cost $0.50 per card to produce. Your initial investment is $5000, which implies that the total cost C of producing cards is given b C The average cost C per card is given b C C Find the average cost per card when (a) 000, (b) 0,000, and (c) 00,000. (d) What is the it of C as approaches infinit? Average cost per card (in dollars) 6 5 = 0.5 C C C = = Average Cost ,000 60,000 00,000 Number of cards As, the average cost per card approaches $0.50. FIGURE. a. When 000, the average cost per card is C b. When 0,000, the average cost per card is C c. When 00,000, the average cost per card is C $ , ,000 $ , ,000 $0.55. d. As approaches infinit, the it of C is $0.50. The graph of C is shown in Figure.. Now tr Eercise ,000 00,000

161 Section. Limits at Infinit and Limits of Sequences 885 You can review sequences in Sections TECHNOLOGY There are a number of was to use a graphing utilit to generate the terms of a sequence. For instance, ou can displa the first 0 terms of the sequence a n n using the sequence feature or the table feature. Limits of Sequences Limits of sequences have man of the same properties as its of functions. For instance, consider the sequence whose nth term is a n n.,, 8, 6,,... As n increases without bound, the terms of this sequence get closer and closer to 0, and the sequence is said to converge to 0. Using it notation, ou can write n 0. n The following relationship shows how its of functions of can be used to evaluate the it of a sequence. Limit of a Sequence Let f be a function of a real variable such that f L. If a n is a sequence such that f n a n for ever positive integer n, then n a n L. A sequence that does not converge is said to diverge. For instance, the terms of the sequence,,,,,... oscillate between and. Therefore, the sequence diverges because it does not approach a unique number. Eample Finding the Limit of a Sequence You can use the definition of its at infinit for rational functions on page 88 to verif the its of the sequences in Eample. Find the it of each sequence. (Assume n begins with.) a. b. c. a. b. c. a n b n n n n n c n n n n n n n n n 0 n n n Now tr Eercise 9. 5, 5 6, 7 7, 9 8, 9, 0,... 5, 5 8, 7, 9 0, 9, 0,... 0, 9 6, 9 6, 6, 5 00, 7,...

162 886 Chapter Limits and an Introduction to Calculus In the net section, ou will encounter its of sequences such as that shown in Eample 5. A strateg for evaluating such its is to begin b writing the nth term in standard rational function form. Then ou can determine the it b comparing the degrees of the numerator and denominator, as shown on page 88. Eample 5 Finding the Limit of a Sequence Find the it of the sequence whose nth term is a n 8 n nnn 6. Algebraic Begin b writing the nth term in standard rational function form as the ratio of two polnomials. a n 8 n nnn 6 8nn n 6n 8n n n n Write original nth term. Multipl fractions. Write in standard rational form. From this form, ou can see that the degree of the numerator is equal to the degree of the denominator. So, the it of the sequence is the ratio of the coefficients of the highest-powered terms. 8n n n 8 n n Now tr Eercise 9. Numerical Construct a table that shows the value of becomes larger and larger, as shown below. as n From the table, ou can estimate that as n approaches, gets closer and closer to a n n a n , a n CLASSROOM DISCUSSION Comparing Rates of Convergence In the table in Eample 5 above, the value of 8 a n approaches its it of rather slowl. (The first term to be accurate to three decimal places is a ) Each of the following sequences converges to 0. Which converges the quickest? Which converges the slowest? Wh? Write a short paragraph discussing our conclusions. a. a b. b n n c. n n c n n d. d e. h n n n n! n!

163 Section. Limits at Infinit and Limits of Sequences 887. EXERCISES VOCABULARY: Fill in the blanks.. A at can be used to solve the area problem in calculus. See for worked-out solutions to odd-numbered eercises.. When evaluating its at infinit for complicated rational functions, ou can divide the numerator and denominator b the term in the denominator.. A sequence that has a it is said to.. A sequence that does not have a it is said to. SKILLS AND APPLICATIONS In Eercises 5 8, match the function with its graph, using horizontal asmptotes as aids. [The graphs are labeled (a), (b), (c), and (d).] (a) (b) (c) (d) 5. f 6. f 7. f 8. f 6 6 In Eercises 9 8, find the it (if it eists). If the it does not eist, eplain wh. Use a graphing utilit to verif our result graphicall t t 8. t 9. t t t t t t t 5t t In Eercises 9, use a graphing utilit to graph the function and verif that the horizontal asmptote corresponds to the it at infinit NUMERICAL AND GRAPHICAL ANALYSIS In Eercises 5 8, (a) complete the table and numericall estimate the it as approaches infinit, and (b) use a graphing utilit to graph the function and estimate the it graphicall. 5. f f 9 f 8. f f

164 888 Chapter Limits and an Introduction to Calculus In Eercises 9 8, write the first five terms of the sequence and find the it of the sequence (if it eists). If the it does not eist, eplain wh. Assume n begins with. 9. a 0. a n n n n n n. n a n n. a n n n.. a n n a n n n n 5. n! n! a n 6. a n n! n! 7. a 8. a n n n n n n In Eercises 9 5, find the it of the sequence. Then verif the it numericall b using a graphing utilit to complete the table n a n a n n n nn n a n n n n nn a n 6 n nnn 6 nn a n n nnn 5. OXYGEN LEVEL Suppose that f t measures the level of ogen in a pond, where f t is the normal (unpolluted) level and the time t is measured in weeks. When t 0, organic waste is dumped into the pond, and as the waste material oidizes, the level of ogen in the pond is given b f t t t t. (a) What is the it of f as t approaches infinit? (b) Use a graphing utilit to graph the function and verif the result of part (a). (c) Eplain the meaning of the it in the contet of the problem. 5. TYPING SPEED The average tping speed S (in words per minute) for a student after t weeks of lessons is given b (a) What is the it of S as t approaches infinit? (b) Use a graphing utilit to graph the function and verif the result of part (a). (c) Eplain the meaning of the it in the contet of the problem. 55. AVERAGE COST The cost function for a certain model of personal digital assistant (PDA) is given b C.50 5,750, where C is measured in dollars and is the number of PDAs produced. (a) Write a model for the average cost per unit produced. (b) Find the average costs per unit when 00 and 000. (c) Determine the it of the average cost function as approaches infinit. Eplain the meaning of the it in the contet of the problem. 56. AVERAGE COST The cost function for a compan to reccle tons of material is given b C.5 0,500, where C is measured in dollars. (a) Write a model for the average cost per ton of material reccled. (b) Find the average costs of reccling 00 tons of material and 000 tons of material. (c) Determine the it of the average cost function as approaches infinit. Eplain the meaning of the it in the contet of the problem. 57. DATA ANALYSIS: SOCIAL SECURITY The table shows the average monthl Social Securit benefits B (in dollars) for retired workers aged 6 or over from 00 through 007. (Source: U.S. Social Securit Administration) A model for the data is given b B Year Benefit, B t.0 0.8t 0.00t, t 7 where t represents the ear, with t corresponding to 00. S 00t 65 t, t > 0.

165 Section. (a) Use a graphing utilit to create a scatter plot of the data and graph the model in the same viewing window. How well does the model fit the data? (b) Use the model to predict the average monthl benefit in 0. (c) Discuss wh this model should not be used for long-term predictions of average monthl Social Securit benefits. 58. DATA ANALYSIS: MILITARY The table shows the numbers N (in thousands) of U.S. militar reserve personnel for the ears 00 through 007. (Source: U.S. Department of Defense) Year Number, N A model for the data is given b N t, t t 6. THINK ABOUT IT Find the functions f and g such that both f and g increase without bound as approaches c, but f g eists. c 6. THINK ABOUT IT function given b EXPLORATION TRUE OR FALSE? In Eercises 59 6, determine whether the statement is true or false. Justif our answer. 59. Ever rational function has a horizontal asmptote. 60. If f increases without bound as approaches c, then the it of f eists. 6. If a sequence converges, then it has a it. 6. When the degrees of the numerator and denominator of a rational function are equal, the it does not eist. Use a graphing utilit to graph the. f How man horizontal asmptotes does the function appear to have? What are the horizontal asmptotes? In Eercises 65 68, create a scatter plot of the terms of the sequence. Determine whether the sequence converges or diverges. If it converges, estimate its it. 65. an.5 n 67. an an 0.5 n 68. an 0.5 n n 69. Use a graphing utilit to graph the two functions given b and in the same viewing window. Wh does not appear to the left of the -ais? How does this relate to the statement at the top of page 88 about the infinite it 7 where t represents the ear, with t corresponding to 00. (a) Use a graphing utilit to create a scatter plot of the data and graph the model in the same viewing window. How well does the model fit the data? (b) Use the model to predict the number of militar reserve personnel in 0. (c) What is the it of the function as t approaches infinit? Eplain the meaning of the it in the contet of the problem. Do ou think the it is realistic? Eplain. 889 Limits at Infinit and Limits of Sequences? r 70. CAPSTONE Use the graph to estimate (a) f, (b) f, and (c) the horizontal asmptote of the graph of f. (i) (ii) 6 f 6 f 7. Use a graphing utilit to complete the table below to verif that Make a conjecture about

166 890 Chapter Limits and an Introduction to Calculus.5 THE AREA PROBLEM Adam Woolfitt/Corbis What ou should learn Find its of summations. Use rectangles to approimate areas of plane regions. Use its of summations to find areas of plane regions. Wh ou should learn it The its of summations are useful in determining areas of plane regions. For instance, in Eercise 50 on page 897, ou are asked to find the it of a summation to determine the area of a parcel of land bounded b a stream and two roads. Recall from Section 9. that the sum of a finite geometric sequence is given b n i a r i a rn Furthermore, if then r n 0 as n. r. 0 < r <, Limits of Summations Earlier in the tet, ou used the concept of a it to obtain a formula for the sum S of an infinite geometric series S a a r a r... a r i a r, Using it notation, this sum can be written as S n n a r i i for n rn 0 The following summation formulas and properties are used to evaluate finite and infinite summations. Eample Evaluate the summation. Evaluating a Summation 00 i i Using the second summation formula with n 00, ou can write n nn i i i i 0,00 0,00. a r n n r Now tr Eercise 5. i Summation Formulas and Properties n i. c cn, c is a constant... n nn (n i. 6 i n i a r. n i r < ka i k n a i ± b i n n a i ± a i, k is a constant. i b i i n i i nn n i n n i i r <