Section 1.2 The Slope of a Tangent

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1 Section 1.2 Te Slope of a Tangent You are familiar wit te concept of a tangent to a curve. Wat geometric interpretation can be given to a tangent to te grap of a function at a point? A tangent is te straigt line tat most resembles te grap near a point. Its slope tells ow steep te grap is at te point of tangency. In te figure below, four tangents ave been drawn. T 2 T 1 T 3 y = f() T 4 Te goal of tis section is to develop a metod for determining te slope of a tangent at a given point on a curve. We begin wit a brief review of lines and slopes. Lines and Slopes y 2 ( 2, y 2 ) l 1 ( 1, y 1 ) 0 D Dy Te slope m of te line joining points 1 1 1, y 1 2 and 2 1 2, y 2 2 is defined as m y y 2 y y Te equation of te line l in point-slope form is y 1 m or y y 1 m Te equation in slope y-intercept form is y m b, were b is te y-intercept of te line. To determine te equation of a tangent to a curve at a given point, we first need to know te slope of te tangent. Wat can we do wen we only ave one point? We proceed as follows: y Q Q Q tangent at 0 y = f() secant THE SLOE OF A TANGENT NEL

2 Consider a curve y f 12 and a point tat lies on te curve. Now consider anoter point Q on te curve. Te line joining and Q is called a secant. Tink of Q as a moving point tat slides along te curve toward, so tat te slope of te secant Q becomes a progressively better estimate of te slope of te tangent at. Tis suggests te following definition of te slope of te tangent: Slope of a Tangent Te slope of te tangent to a curve at a point is te limiting slope of te secant Q as te point Q slides along te curve toward. In oter words, te slope of te tangent is said to be te limit of te slope of te secant as Q approaces along te curve. We will illustrate tis idea by finding te slope of te tangent to te parabola y 2 at 13, 92. INVESTIGATION 1 A. Determine te y-coordinates of te following points tat lie on te grap of te parabola y 2 : i) Q , y2 ii) Q , y2 iii) Q , y2 iv) Q , y2 B. Calculate te slopes of te secants troug 13, 92 and eac of te points Q 1, Q 2, Q 3, and Q 4. C. Determine te y-coordinates of eac point on te parabola, and ten repeat part B using te following points. i) Q , y2 ii) Q , y2 iii) Q , y2 iv) Q , y2 D. Use your results from parts B and C to estimate te slope of te tangent at point 13, 92. E. Grap y 2 and te tangent to te grap at 13, 92. In tis investigation, you found te slope of te tangent by finding te limiting value of te slopes of a sequence of secants. Since we are interested in points Q tat are close to 13, 92 on te parabola y 2 it is convenient to write Q as 13, , were is a very small nonzero number. Te variable determines te position of Q on te parabola. As Q slides along te parabola toward, will take on values successively smaller and closer to zero. We say tat approaces zero and use te notation S 0. NEL CHATER 1 11

3 INVESTIGATION 2 A. Using tecnology or grap paper, draw te parabola f B. Let be te point 11, 12. C. Determine te slope of te secant troug Q 1 and 11, 12, Q 2 and 11, 12 and so on, for points Q , f , Q , f , Q , f , Q , f , and Q , f D. Draw tese secants on te same grap you created in part A. E. Use your results to estimate te slope of te tangent to te grap of f at point. F. Draw te tangent at point 11, 12. INVESTIGATION 3 A. Determine an epression for te slope of te secant Q troug points 13, 92 and Q13, B. Eplain ow you could use te epression in a part A to predict te slope of te tangent to te parabola f 12 2 at point 13, 92. Te slope of te tangent to te parabola at point is te limiting slope of te secant line Q as point Q slides along te parabola; tat is, as S 0, we write lim as te abbreviation for limiting value as approaces 0. Terefore, from te investigation, te slope of te tangent at a point is lim 1slope of te secant Q2. EXAMLE 1 Reasoning about te slope of a tangent as a limiting value Determine te slope of te tangent to te grap of te parabola f 12 2 at 13, 92. Solution Using points 13, 92 and Q13, , 0, te slope of te secant Q is y y 2 y (Substitute) (Epand) (Simplify and factor) (Divide by te common factor of ) THE SLOE OF A TANGENT NEL

4 As S 0, te value of 16 2 approaces 6, and tus lim We conclude tat te slope of te tangent at 13, 92 to te parabola y 2 is 6. EXAMLE 2 Tec Support For elp graping functions using a graping calculator, see Tecnology Appendi p Selecting a strategy involving a series of secants to estimate te slope of a tangent a. Use your calculator to grap te parabola y lot te 8 points on te parabola from 1to 6, were is an integer. b. Determine te slope of te secants using eac point from part a and point 15, c. Use te result of part b to estimate te slope of te tangent at 15, Solution a. Using te -intercepts of 1 and 7, te equation of te ais of symmetry is 1 7 3, so te -coordinate of te verte is 3. 2 Substitute 3 into y y Terefore, te verte is 13, Te y-intercept of te parabola is 8. Te points on te parabola are 1 1, 02, 10, , 11, 1.52, 12, , 13, 22, 14, , 15, 1.52, and 16, Te parabola and te secants troug eac point and point 15, 1.52 in red. Te tangent troug 15, 1.52 is sown in green. are sown y (5, 1.5) b. Using points 1 1, 02 and 15, 1.52, te slope is m Using te oter points and 15, 1.52, te slopes are 0.125, 0, 0.125, 0.25, 0.375, and 0.625, respectively. c. Te slope of te tangent at 15, 1.52 is between and It can be determined to be 0.5 using points closer and closer to 15, NEL CHATER 1 13

5 0 y tangent at (a, f(a)) Q(a +, f(a + )) y = f() Te Slope of a Tangent at an Arbitrary oint We can now generalize te metod used above to derive a formula for te slope of te tangent to te grap of any function y f 12. Let 1a, f 1a22 be a fied point on te grap of y f 12, and let Q1, y2 Q1, f 122 represent any oter point on te grap. If Q is a orizontal distance of units from, ten a and y f 1a 2. oint Q ten as coordinates Q1a, f 1a 22. Te slope of te secant Q is y f 1a 2 f 1a2 a a Tis quotient is fundamental to calculus and is referred to as te difference quotient. Terefore, te slope m of te tangent at 1a, f 1a22 is f 1a 2 f 1a2 lim 1slope of te secant Q2, wic may be written as m. f 1a 2 f 1a2. Slope of a Tangent as a Limit Te slope of te tangent to te grap y f 12 at point 1a, f 1a22 is y f 1a 2 f 1a2 m, if tis limit eists. S0 EXAMLE 3 Connecting limits and te difference quotient to te slope of a tangent a. Using te definition of te slope of a tangent, determine te slope of te tangent to te curve y at te point determined by 3. b. Determine te equation of te tangent. c. Sketc te grap of y and te tangent at 3. Solution a. Te slope of te tangent can be determined using te epression above. In tis eample, f and a 3. Ten f and f THE SLOE OF A TANGENT NEL

6 Te slope of te tangent at 13, 42 f 13 2 f 132 m is (Substitute) (Simplify and factor) (Divide by te common factor) (Evaluate) Te slope of te tangent at 3 is 2. b. Te equation of te tangent at 13, 42 y 4 is or y , c. Using graping software, we obtain y 10 y = (3, 4) y = EXAMLE 4 Selecting a limit strategy to determine te slope of a tangent 3 6 Determine te slope of te tangent to te rational function f 12 at point 12, 62. Solution Using te definition, te slope of te tangent at 12, 62 is f 12 2 f 122 m (Substitute) (Determine a common denominator) (Simplify) NEL CHATER 1 15

7 (Multiply by te reciprocal) 3 6 Terefore, te slope of te tangent to f 12 at 12, 62 is 1.5. (Evaluate) EXAMLE 5 Determining te slope of a line tangent to a root function Find te slope of te tangent to f 12 at 9. Solution f 192 V9 3 f 19 2 V9 Using te limit of te difference quotient, te slope of te tangent at 9 is m f 19 2 f (Substitute) (Rationalize te numerator) (Simplify) (Divide by te common factor of ) (Evalute) 1 6 Terefore, te slope of te tangent to f 12 V at 9 is THE SLOE OF A TANGENT NEL

8 INVESTIGATION 4 Tec Support For elp graping functions, tracing, and using te table feature on a graping calculator, see Tecnology Appendices p. 597 and p A graping calculator can elp us estimate te slope of a tangent at a point. Te eact value can ten be found using te definition of te slope of te tangent using te difference quotient. For eample, suppose tat we wis to find te slope of te tangent to y f 12 3 at 1. A. Grap Y B. Eplain wy te values for te WINDOW were cosen. f 1a 2 f 1a2 Observe tat te function entered in Y 1 is te difference quotient for f 12 3 and Remember tat tis approimates te slope of te tangent and not te grap of f C. Use te TRACE function to find X , Y Tis means tat te slope of te secant passing troug te points were 1 and is about 3.2. Te value 3.2 could be used as an approimation for te slope of te tangent at 1. D. Can you improve tis approimation? Eplain ow you could improve your estimate. Also, if you use different WINDOW values, you can see a different-sized, or differently centred, grap. E. Try once again by setting X min 9, X ma 10, and note te different appearance of te grap. Use te TRACE function to find X , Y , and ten X , Y Wat is your guess for te slope of te tangent at 1 now? Eplain wy only estimation is possible. F. Anoter way of using a graping calculator to approimate te slope of te tangent is to consider as te variable in te difference quotient. For tis f 1a 2 f 1a2 eample, at look at f , 3. G. Trace values of as S 0. You can use te table or grap function of your 11 2 calculator. Grapically, we say tat we are looking at 3 1 in te neigbourood of To do tis, grap y and eamine te value of te function as S 0. NEL CHATER 1 17

9 IN SUMMARY Key Ideas Te slope of te tangent to a curve at a point is te limit of te slopes of te secants Q as Q moves closer to. m tangent 1slope of secant Q2 QS Te slope of te tangent to te grap of y f12 at 1a, f1a22 is given by y f1a 2 f1a2 m tangent. S0 Need to Know To find te slope of te tangent at a point 1a, f 1a22, find te value of f 1a2 find te value of f 1a 2 f1a 2 f1a2 evaluate lim Eercise 1.2 ART A 1. Calculate te slope of te line troug eac pair of points. a. 12, 72, 1 3, 82 b. a 1 2, 3 2 b, a 7 2, 7 2 b c. 16.3, 2.62, 11.5, Determine te slope of a line perpendicular to eac of te following: a. y 3 5 b. 13 7y State te equation and sketc te grap of eac line described below. a. passing troug 1 4, 42 and Q 5 3, 5 3 R b. aving slope 8 and y-intercept 6 c. aving -intercept 5 and y-intercept 3 d. passing troug 15, 62 and 15, THE SLOE OF A TANGENT NEL

10 4. Simplify eac of te following difference quotients: a. d b. e c. f. 5. Rationalize te numerator of eac epression to obtain an equivalent epression. V16 4 V V5 V5 a. b. c. K ART B 6. Determine an epression, in simplified form, for te slope of te secant Q. a. 11, 32, Q11, f 11 22, were f b. 11, 32, Q11, c. 19, 32, Q19, Consider te function f a. Copy and complete te following table of values. and Q are points on te grap of f 12. Q 12, 2 13, 2 12, , 2 12, , 2 12, , 2 12, 2 11, 2 12, , 2 12, , 2 12, , 2 Slope of Line Q b. Use your results for part a to approimate te slope of te tangent to te grap of f 12 at point. c. Calculate te slope of te secant Q, were te -coordinate of Q is 2. d. Use your result for part c to calculate te slope of te tangent to te grap of f 12 at point. NEL CHATER 1 19

11 e. Compare your answers for parts b and d. f. Sketc te grap of f 12 and te tangent to te grap at point. 8. Determine te slope of te tangent to eac curve at te given value of. a. y 3 2, 2 b. y 2, 3 c. y 3, 2 9. Determine te slope of te tangent to eac curve at te given value of. a. y 2, 3 b. y 5, 9 c. y 5 1, Determine te slope of te tangent to eac curve at te given value of. a. y 8 b. y 8 c. y 1 3, 2 3, 1 2, 11. Determine te slope of te tangent to eac curve at te given point. a. y 2 3, 12, 22 d. y 7, 116, 32 C b. f , 22 e. y 25 2, 13, 42, c. y f. y 4 33, 11, 32, 18, Sketc te grap of te function in question 11, part e. Sow tat te slope of te tangent can be found using te properties of circles. 13. Eplain ow you would approimate te slope of te tangent at a point witout using te definition of te slope of te tangent. 14. Using tecnology, sketc te grap of y 3 Eplain ow te 4 V16 2. slope of te tangent at 10, 32 can be found witout using te difference quotient. 15. Determine te equation of te tangent to y at (3, 1). 16. Determine te equation of te tangent to y were For f , find a. te coordinates of point A, were 3, b. te coordinates of point B, were 5 c. te equation of te secant AB d. te equation of te tangent at A e. te equation of te tangent at B THE SLOE OF A TANGENT NEL

12 18. Copy te following figures. Draw an approimate tangent for eac curve at point and estimate its slope. a. d. b. e. c. f. A T 19. Find te slope of te demand curve D1p2 20, p 7 1, at point 15, 102. Vp It is projected tat, t years from now, te circulation of a local newspaper will be C1t2 100t 2 400t Find ow fast te circulation is increasing after 6 monts. Hint: Find te slope of te tangent wen t Find te coordinates of te point on te curve f were te tangent is parallel to te line y Find te points on te grap of y 1 at wic te tangent is orizontal. ART C 23. Sow tat, at te points of intersection of te quadratic functions y 2 and y 1 te tangents to te functions are perpendicular. 2 2, 24. Determine te equation of te line tat passes troug (2, 2) and is parallel to te line tangent to y at 1 1, a. Determine te slope of te tangent to te parabola y at te point wose -coordinate is a. b. At wat point on te parabola is te tangent line parallel to te line 10 2y 18 0? c. At wat point on te parabola is te tangent line perpendicular to te line 35y 7 0? NEL CHATER 1 21