1 x 3 9x x. Chapter 4 Curve Part Sketch a graph of the function with the following. properties: 1. What is the second derivative test?

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1 Chapter 4 Curve Part 1 1. What is the second derivative test?. Restate the following using calculus: Our prices are rising slower than any place in town. For numbers 3-19, find all critical points of the given function. Determine where the graph is rising or falling, and find where the graph is concave up or down. 3. y = ( + 0) 8( + 0) y = y = y = u 4 + 6u 3 4u y = (t 3 + t) 8. y = t 3 3t 4 9. y = 5t 6 6t y = y = t e -3t 1. y = e e 13. y =ln( ) e e 14. y = 4/3 ( - 7) 15. y = y = θ + cos θ for 0 θ π 17. y = 1 sin t + cos t for -π t π y = sin -1 for y = tan + 3 on (, ) 0. Sketch a graph of the function with the following properties: f () > 0 when < -1 f () > 0 when > 3 f () < 0 when < 3 f () < 0 when < f () > 0 when > 1. Sketch a graph of the function with the following properties: f () > 0 when < and when < < 5 f () < 0 when > 5 f () = 0 f () < 0 when < and when 4 < < 7 f () > 0 when < < 4 and when > 7. Sketch a graph of the function with the following properties: f () > 0 when < 1 f () < 0 when > 1 f () > 0 when < 1 f () > 0 when > 1 3. Sketch the graph of a function with the following properties: There are relative etrema at (-1,7) and (3,). There is an inflection point at (1,4). The graph is concave down only when < 1. The -intercept is (-4,0), and the y-intercept is (0,5). 4. Eamine the graphs : a. f() = ( + 1) 1/3 b. f() = ( + 1) /3 c. f() = ( + 1) 4/3 d. f() = ( + 1) 5/3 Generalize to make a statement about the effect of a positive integer n on the graph of f() = ( + 1) n/3. 5. Find constants A, B, and C that guarantee that the function f() = A 3 + B + C will have a relative etremum at (,11) and an inflection point at (1,5). Sketch the graph of f. 6. Let S() =.5(e e - ) and C() =.5(e + e - ). These functions are known as the hyperbolic sine and hyperbolic cosine, respectively. These functions are eamined in Section 7.8. a. Show that S () = C() and C () = S(). b. Sketch the graphs of S and C.

2 7. Set up an appropriate model to answer the given question. Be sure to state your assumptions. At noon on a certain day, Frank sets out to assemble five stereo sets. His rate of assembly increases steadily throughout the afternoon until 4 PM at which time he has completed three sets. After that he assembles sets at a slower and slower rate until he finally completes the fifth set at 8 PM. Sketch a rough graph of a function that represents the number of sets Frank has completed after t hours of work. 37. Given a sphere of radius R, find the radius r and altitude h of the right circular cylinder with the largest lateral surface area that can be inscribed in the sphere. Hint: the lateral surface area is S = πrh. 38. Find the dimensions of the right circular cylinder of largest volume that can be inscribed in a right circular cone of radius R and altitude H. 8. Describe an optimization procedure. 9. A woman plans to fence off a rectangular garden whose area is 64 ft. What should be the dimensions of the garden if she wants to minimize the amount of fencing used? 30. Pull out a sheet of 8.5-in. by 11-in. binder paper. Cut squares from the corners and fold the sides up to form a container. Show that the maimum volume of such a container is about 1 liter. 31. Why is it important to check endpoints when finding an optimum value? 3. The highway department is planning to build a rectangular picnic area for motorists along a major highway. It is to have an area of 5,000 yd and is to be fenced off on the three sides not adjacent to the highway. What is the least amount of fencing that will be needed to complete the job? 33. Farmer Jones has to build a fence to enclose a 1,00 m rectangular area ABCD. Fencing costs $3 per meter, but Farmer Smith has agreed to pay half the cost of fencing CD, which borders the property. Given is the length of sidecd, what is the minimum amount (to the nearest cent) Jones has to pay? 34. Find the rectangle of largest area that can be inscribed in a semicircle of radius R, assuming that one side of the rectangle lied on the diameter of the semicircle. 39. A truck is 50 mi due east of a sports car and is traveling west at a constant speed of 60 mi/h. Meanwhile, the sports car is going north at 80 mi/h. When will the truck and the car be closest to each other? What is the minimum distance between them? Hint: Minimize the square of the distance. 40. Show that of all rectangles with a given perimeter, the square has the largest area. 41. Show that of all rectangles with a given area, the square has the smallest perimeter. 4. A closed bo with a square base is to be built to house an ant colony. The bottom of the bo and all four sides are to be made of material costing $1/ft, and the top is to be constructed of glass costing $5/ft. What are the dimensions of the bo of greatest volume that can be constructed for $7? 43. According to postal regulations, the girth plus the length of a parcel sent by fourth-class mail may not eceed 108 in. What is the largest possible volume of a rectangular parcel with two square sides that can be sent by fourth-class mail? 35. A tinsmith wants to make an open-topped bo out of a rectangular sheet of tin 4 in. wide and 45 in. long. The tinsmith plans to cut congruent squares out of each corner of the sheet and then bend the edges of the sheet upward to form the sides of the bo. What are the dimensions of the largest bo that can be made in this fashion? 36. Find the dimensions of the right circular cylinder of largest volume that can be inscribed in a sphere of radius R.

3 44. A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground. When he is 10 feet from the base of the light, a) at what rate is the tip of his shadow moving? b) at what rate is the length of his shadow changing? 48. A trough is 1 feet long and 3 feet across the top. Its ends are isosceles triangles with altitudes of 3 feet. If water is being pumped into the trough at cubic feet per minute, how fast is the water level rising when the water is 1 foot deep? 49. An airplane is flying at an altitude of 6 miles and passes directly over a radar antenna. When the plane is 10 miles away (s = 10), the radar detects the distance s is changing at a rate of 40 miles per hour. What is the speed of the plane? 45. As a spherical raindrop falls, it reaches a layer of dry air and begins to evaporate at a rate that is proportional to its surface area (S = 4 πr ). Show that the radius of the raindrop decreases at a constant rate. 46. When a certain polyatomic gas undergoes adiabatic epansion, its pressure p and volume v satisfy the equation pv 1.3 = k where k is a constant. Find the relationship between the related rates dp/ and dv/. 47. A fish is reeled in at a rate of 1 foot per second from a point 15 feet above the water. At what rate is the angle between the line and the water changing when there are 5 feet of line out? 50. Missy Smith is at a point A on the north bank of a long, straight river 6 mi. wide. Directly across from her on the south bank is a point B, and she wishes to reach a cabin C located s miles down the river from B. Given that Missy can row at 6 mi/h (including the effect of the current) and run at 10 mi/h, what is the minimum time (to the nearest minute) required for her to travel from A to C in each case? a) s = 4 b) s = 6

4 51. Two towns A and B are 1 mi apart and are located 5 and 3 mi, respectively, from a long, straight, highway. A construction company has a contract to build a road from A to the highway and then to B. Analyze a model to determine the length (to the nearest mile) of the shortest road that meets these requirements. 5. A poster is to contain 108 cm of printed matter, with margins of 6 cm each at top and bottom and cm on the sides. What is the minimum cost of the poster if it is to be made of material costing 0 /cm? 55. A cylindrical container with no top is to be constructed to hold a fied volume of liquid. The cost of the material used for the bottom is 50 /in, and the cost of the material used for the curved face is 30 /in. Use calculus to find the radius of the least epensive container. 56. Use the fact that 1 oz 355 ml = 55 cm 3 to find the dimensions of the 1-oz Coke can that can be constructed using the least amount of metal. Compare these dimensions with a Coke from your refrigerator. What do you think accounts for the difference? 57. A stained glass window in the form of an equilateral triangle is built on top of a rectangular window. The rectangular part of the window is one of clear glass and transmits twice as much light per square foot as the triangular part, which is made of stained glass. If the entire window has a perimeter of 0 ft, find the dimensions (to the nearest ft) of the window that will admit the most light. 53. An isosceles trapezoid has a base of 14 cm and slant sides of 6 cm. What is the largest area of such a trapezoid? 54. It is noon. The spy has returned from space and is driving a jeep through the sandy desert in the tiny principality of Alta Loma. He is 3 km from the nearest point on a straight, paved road. Down the road 16 km is a power plant in which a band of international terrorists has placed a time bomb set to eplode at 1:50 PM. The jeep can travel at 48 km/h in the sand and at 80 km/h on the paved road. If he arrives at the power plant in the shortest possible time, how long will our hero have to defuse the bomb? 58. y =. a) Given d = 3, find dy when = 4 dy d b) Given =, find when = A point is moving along the graph of y = 1 1. d dy = cm/sec. Find for the given values of. a) = - b) = c) = 0 d) = 10

5 60. Using the graph below, determine whether: dy a) increases or decreases for increasing and d constant d b) increases or decreases for increasing y and dy constant. 63. At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approimately three times the altitude. At what rate is the height of the pile changing when the pile is 15 feet high? Part 1. Numerical, Graphical, and Analytic Analysis Find two positive numbers whose sum is 110 and whose product is a maimum. (a) Analytically complete si rows of a table such as the one below. (The first two rows are shown.) First Number Second Number Product P (110 10) = (110 0) = 1800 (b) Use a graphing utility to generate additional rows of the table. Use the table to estimate the solution. (Hint: Use the table feature of the graphing utility.) 61. Find the rate of change of the distance between the origin and a moving point on the graph of y = sin d if = centimeters per second. 65. The included angle of the two sides of constant equal length s of an isosceles triangle is θ. a) Show that the area of the triangle is given by A =.5s sin θ. b) If θ is increasing at the rate of.5 radians per minute, find the rate of change of the area when θ = and θ =. 6 3 c) Eplain why the rate of change of the area of the area of the triangle is not constant even though d is constant. 6. All edges of a cube are epanding at a rate of 3 centimeters per second. How fast is the volume changing when each edge is: (c) Write the Product P as a function of. (d) Use a graphing utility to graph the function in part (c) and estimate the solution from the graph. (e) Use the calculus to find the critical number of the function in part (c). Then find the two numbers. In Eercises -6, find two positive numbers that satisfy the given requirements.. The sum is S and the product is a maimum. 3. The product is 19 and the sum is a minimum. 4. The product is 19 and the sum of the first plus three times the second is a minimum. 5. The second number is the reciprocal of the first and the sum is a minimum. 6. The sum of the first and twice the second is 100 and the product is a maimum. a) 1 centimeter? b) 10 centimeters?

6 In Eercise 7 and 8, find the length and wih of a rectangle that has the given perimeter and a maimum area. 7. Perimeter: 100 meters 8. Perimeter: P units In Eercise 9 and 10, find the length and wih of a rectangle that has the given area and a minimum perimeter. 9. Area: 64 square feet 10. Area: A square centimeters In Eercises 11 and 1, find the point on the graph of the function that is closest to the given point. (b) (c) Find the volume of each. Determine the dimensions of a rectangular solid (with a square base) of maimum volume if its surface area is 150 square inches. Function Point 11. f() = (4, 0) 1. f() = (, ½) 13. Chemical Reaction In an autocatalytic chemical reaction the product formed is a catalyst for the reaction. If Q 0 is the amount of the original substance and is the amount of catalyst formed, the rate of chemical reaction is dq d k( Q0 ). For what value of will the rate of chemical reaction be greatest? 14. Traffic Control On a given day, the flow rate F (cars per hour) on a congested roadway is v F 0.0v where v is the speed of the traffic in miles per hour. What speed will maimize the flow rate on the road? 18. Numerical, Graphical, and Analytic Analysis An open bo of maimum volume is to be made from a square piece of material, 4 inches on a side, by cutting equal squares from the corners and turning up the sides (see figure). (a) Analytically complete si rows of a table such as the one below. (The first two rows are shown.) Use the table to guess the maimum volume. Height Length and Wih Volume 1 4 (1) 1[4 (1)] = () [4 ()] = 800 (b) Write the volume V as a function of. (c) Use calculus to find the critical number of the function in part (b) and find the maimum value. (d) Use a graphing utility to graph the function in part (b) and verify the maimum volume from the graph. 15. Area A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river? 17. Volume (a) Verify that each of the rectangular solids shown in the figure has a surface area of 150 square inches. 19. (a) Solve Eercise 18 given that the square piece of material is s meters on a side.

7 (b) If the dimensions of the square piece of material are doubled, how does the volume change? What length and wih should the rectangle have so that its area is a maimum? 0. Numerical, Graphical, and Analytic Analysis A physical fitness room consists of a rectangle with a semicircle on each end. A 00-meter running track runs around the outside of the room. (a) Draw a figure to represent the problem. Let and y represent the length and wih of the rectangle. Length 10 0 (b) Analytically complete si rows of a table such as the one below. (The first two rows are shown.) Use the table to guess the maimum area of the rectangular region. Wih y (100 10) (100 0) Area ( 10) (100 10) 573 ( 0) (100 0) 1019 (c) Write the area A as a function of. (d) Use calculus to find the critical number of the function in part (c) and find the maimum value. 3. Length A right triangle is formed in the first quadrant by the - and y-aes and a line through the point (1, ) (see figure). (a) (b) (c) Write the length L of the hypotenuse as a function of. Use a graphing utility to graphically approimate such that the length of the hypotenuse is a minimum. Find the vertices of the triangle such that its area is a minimum. 4. Area Find the dimensions of the largest isosceles triangle that can be inscribed in a circle of radius 4 (see figure). (e) Use a graphing utility to graph the function in part (c) and verify the maimum area from the graph. 1. Area A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). Find the dimensions of a Norman window of maimum area if the total perimeter is 16 feet. 5. Area A rectangle is bounded by the -ais y 5 and the semicircle (see figure). What length and wih should the rectangle have so that its area is a maimum?. Area A rectangle is bounded by the - and y- aes and the graph of y = (6-)/ (see figure).

8 Use an appropriate local linear approimation to estimate the value of the given quantity ( 3.0) 3 ( 1.97) Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs (with a graphing device) that display the major features of the curve. Use these graphs to estimate the maimum and minimum values. ( 4)( - 3) 13. f() = 4 ( - 1) 14. f() = 4 10( - 1) 3 ( - ) ( 1) Part Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f' and f" to estimate the intervals of increase and decrease, etreme values, intervals of concavity, and inflection points. 1. f() = f() = f() = f() = - 5. f() = f() = tan + 5 cos 7. f() = sin, -7 7 e 8. f() = Produce graphs of f that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease, etreme values, intervals of concavity, and inflection points, and use calculus to find these quantities eactly. 9. f() = f() = Produce a graph of f that shows all the important aspects of the curve. Estimate the local maimum and minimum values and then use calculus to find these values eactly. Use a graph of f" to estimate the inflection points. 11. f() = e ^3-1. f() =e cos