(For help, go to Sections 1.2 and 1.6.) 34. f u 1 1 u, u g x f u cot p u. 36. f u u, cos 2 u g x px, x f u u. 38.

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1 Section.6 Chain Rule 5 Quick Review.6 (For help, go to Sections. an.6.) In Eercises 5, let f sin, g, an h 7. Write a simplifie epression for the composite function.. f g sin ( ). f gh sin (49 ). g h h g f ( g h ) sin 7 In Eercises 6 0, let f cos, g, an h. Write the given function as a composite of two or more of f, g, an h. For eample, cos is f h. 6. cos g(f()) 7. cos g(h(f())) 8. cos 6 h(g(f())) 9. cos 7 4 f(h(h())) 0. cos f(g(h())). sin () 4. ( ) sec ( ) Section.6 Eercises 6. 0 csc 9. sin t. ( ) 6. 8 ( 5) ( 5) 4 ( 5) (4 5) In Eercises 8, use the given substitution an the Chain Rule to fin y. cos ( ) 5 cos (7 5). y sin ( ), u. y sin (7 5), u 7 5. y cos (), u 4. y tan ( ), u ) 5. y ( sin cos sin sin, u cos ( cos ) 6. y 5 cot ( ), u 7. y cos (sin ), u sin sin (sin ) cos 8. y sec (tan ), u tan sec (tan ) tan (tan ) sec In Eercises 9, an object moves along the -ais so that its position at any time t 0 is given by (t) s(t). Fin the velocity of the object as a function of t. 4t sin ( 4t) cos ( 4t) 9. s cos ( p t ) 0. s t cos p 4t 4. s p sin t 4 cos 5t 5p. s sin ( p t ) ( cos 7 p t 4 ) 4 4 cos t sin 5t t cos 7 7t sin 4 4 In Eercises 4, fin y. If you are unsure of your answer, use csc NDER to support your computation. csc cot. y 4. y csc cot 5. y sin 5 cos 6. y y sin tan 4 8. y 4sec tan 9. y ( ) / 0. y ( ) /. y sin. y cos In Eercises 8, fin the value of f g at the given value of.. f u u 5, u g, 5/ 4. f u u, u g, 5. f u cot p u, u g 5, f u u, cos u g p, 5 u 4 u 7. f u u, u g 0, 0 0 ) 8. f u ( u u, u g, 8 What happens if you can write a function as a composite in ifferent ways? Do you get the same erivative each time? The Chain Rule says you shoul. Try it with the functions in Eercises 9 an Fin y if y cos 6 by writing y as a composite with (a) y cos u an u 6. 6 sin (6 ) (b) y cos u an u. 6 sin (6 ) 40. Fin y if y sin by writing y as a composite with (a) y sin u an u. cos ( ) (b) y sin u an u. cos ( ) In Eercises 4 48, fin the equation of the line tangent to the curve at the point efine by the given value of t. 4. cos t, y sin t, t p4 y. y cos 7 4. y tan 5 5 (tan 5)/ sec 5 4. sin pt, y cos pt, t 6 y In Eercises 5 8 fin ru. 4. sec t, y tan t, t p4 y 5. r tan u sec ( ) 6. r sec u tan u 44. sec t, y tan t, t p6 y 7. r u sin u c os sin 8. r usecu sec ( tan ) sin 45. t, y t, t 4 y 4 In Eercises 9, fin y. 46. t, y t 4, t y 4 9. y tan sec tan 0. y cot csc cot 47. t sin t, y cos t, t p y. y cot. y 9 tan sec tan 48. cos t, y sin t, t p y 5. 5 sin 6 cos cos sin 8. sec sec tan 7. 4 sin sec 4 sin cos tan 4. 6 sin ( ) cos ( ) sin (6 4). 4 ( cos ) sin 6. sec sec tan. 4( cos 7) cos 7 sin 7. 8 csc ( ) cot ( )

2 54 Chapter Derivatives 49. Let t t, an let y sin t. cos t (a) Fin y as a function of t. t (b) Fin ( t y ) as a function of t. y )(sin t) cos t t (c) Fin ( y ) (t (t ) as a function of t. y )(sin t) cos t (t (t ) Use the Chain Rule an your answer from part (b). () Which of the epressions in parts (b) an (c) is y? part (c) 50. A circle of raius an center 0, 0 can be parametrize by the equations cos t an y sin t. Show that for any value of t, the line tangent to the circle at cos t, sin t is perpenicular to the raius. See page 56. ( cos t, sin t) 5. Let s cos u. Evaluate st when u p an ut Let y 7 5. Evaluate yt when an t. 5. What is the largest value possible for the slope of the curve y sin? 54. Write an equation for the tangent to the curve y sin m at the origin. y m 55. Fin the lines that are tangent an normal to the curve y tan p4 at. Support your answer graphically. 56. Working with Numerical Values Suppose that functions f an g an their erivatives have the following values at an. f g f g 8 4 p 5 Evaluate the erivatives with respect to of the following combinations at the given value of. (a) f at / (b) f g at 5 (c) f g at () f g at 7/6 5 8 (e) f g at (f) f at /() 58. Working with Numerical Values Suppose that the functions f an g an their erivatives with respect to have the following values at 0 an. f g f g Evaluate the erivatives with respect to of the following combinations at the given value of. (a) 5f g, (b) f g, 0 6 f (c), () f g, 0 /9 g (e) g f, 0 40/ (f) g f, 6 (g) f g, 0 4/9 59. Orthogonal Curves Two curves are sai to cross at right angles if their tangents are perpenicular at the crossing point. The technical wor for crossing at right angles is orthogonal. Show that the curves y sin an y sin are orthogonal at the origin. Draw both graphs an both tangents in a square viewing winow. 60. Writing to Learn Eplain why the Chain Rule formula y y u u is not simply the well-known rule for multiplying fractions. 6. Running Machinery Too Fast Suppose that a piston is moving straight up an own an that its position at time t secons is s A cos pbt, with A an b positive. The value of A is the amplitue of the motion, an b is the frequency (number of times the piston moves up an own each secon). What effect oes oubling the frequency have on the piston s velocity, acceleration, an jerk? (Once you fin out, you will know why machinery breaks when you run it too fast.) (g) g at 5/ (h) f g at 5/(7) 57. Etension of Eample 8 Show that cos ( ) is Figure.45 The internal forces in the engine get so large that they p sin ( ). tear the engine apart when the velocity is too great Because the symbols y y,, an u are not fractions. The iniviual u 55. Tangent: y ; Normal: y symbols y, u, an o not have numerical values. 6. The amplitue of the velocity is ouble. The amplitue of the acceleration is quaruple. The amplitue of the jerk is multiplie by 8.

3 Section.6 Chain Rule Group Activity Temperatures in Fairbanks, Alaska. The graph in Figure.46 shows the average Fahrenheit temperature in Fairbanks, Alaska, uring a typical 65-ay year. The equation that approimates the temperature on ay is y 7 sin [ p 65 0] 5. On the 0 st ay (April th ) (a) On what ay is the temperature increasing the fastest? (b) About how many egrees per ay is the temperature increasing when it is increasing at its fastest? About 0.67 egrees per ay Temperature ( F ) y Jan Jun Jul Feb Mar Apr May Aug Sep Oct Nov Jan Dec... Feb Mar Figure.46 Normal mean air temperatures at Fairbanks, Alaska, plotte as ata points, an the approimating sine function (Eercise 6). 6. Particle Motion The position of a particle moving along a coorinate line is s 4t, with s in meters an t in secons. Fin the particle s velocity an acceleration at t 6 sec. See page Constant Acceleration Suppose the velocity of a falling boy is v ks msec (k a constant) at the instant the boy has fallen s meters from its starting point. Show that the boy s acceleration is constant. See page Falling Meteorite The velocity of a heavy meteorite entering the earth s atmosphere is inversely proportional to s when it is s kilometers from the earth s center. Show that the meteorite s acceleration is inversely proportional to s. See page Particle Acceleration A particle moves along the -ais with velocity t f. Show that the particle s acceleration is f f. See page Temperature an the Perio of a Penulum For oscillations of small amplitue (short swings), we may safely moel the relationship between the perio T an the length L of a simple penulum with the equation T p L g, where g is the constant acceleration of gravity at the penulum s location. If we measure g in centimeters per secon square, we measure L in centimeters an T in secons. If the penulum is mae of metal, its length will vary with temperature, either increasing or ecreasing at a rate that is roughly proportional to L. In symbols, with u being temperature an k the proportionality constant, L kl. u Assuming this to be the case, show that the rate at which the perio changes with respect to temperature is kt. See page Writing to Learn Chain Rule Suppose that f an g. Then the composites f g an g f are both ifferentiable at 0 even though g itself is not ifferentiable at 0. Does this contraict the Chain Rule? Eplain. 69. Tangents Suppose that u g is ifferentiable at an that y f u is ifferentiable at u g. If the graph of y f g has a horizontal tangent at, can we conclue anything about the tangent to the graph of g at or the tangent to the graph of f at u g? Give reasons for your answer. Stanarize Test Questions You shoul solve the following problems without using a graphing calculator. 70. True or False (sin ) cos, if is measure in egrees or raians. Justify your answer. False. See eample True or False The slope of the normal line to the curve cos t, y sin t at t p4 is. Justify your answer. 7. Multiple Choice Which of the following is y if y tan (4)? E (A) 4 sec (4) tan (4) (B) sec (4) tan (4) (C) 4 cot (4) (D) sec (4) (E) 4 sec (4) 7. Multiple Choice Which of the following is y if y cos ( )? C (A) ( ) (B) ( ) cos ( ) sin ( ) (C) ( ) cos ( ) sin ( ) (D) ( ) cos ( ) sin ( ) (E) ( ) In Eercises 74 an 75, use the curve efine by the parametric equations t cos t, y sin t. 74. Multiple Choice Which of the following is an equation of the tangent line to the curve at t 0? A (A) y (B) y (C) y (D) y (E) y 75. Multiple Choice At which of the following values of t is y 0? B (A) t p4 (B) t p (C) t p4 (D) t p (E) t p 7. False. It is.

4 56 Chapter Derivatives Eplorations 76. The Derivative of sin Graph the function y cos for.5. Then, on the same screen, graph sin h sin y h for h.0, 0.5, an 0.. Eperiment with other values of h, incluing negative values. What o you see happening as h 0? Eplain this behavior. 77. The Derivative of cos ( ) Graph y sin for. Then, on screen, graph cos h cos y h for h.0, 0.7, an 0.. Eperiment with other values of h. What o you see happening as h 0? Eplain this behavior. Etening the Ieas 78. Absolute Value Functions Let u be a ifferentiable function of. u (a) Show that u u. u (b) Use part (a) to fin the erivatives of f 9 an g sin. 79. Geometric an Arithmetic Mean The geometric mean of u an v is G uv an the arithmetic mean is A u v. Show that if u, v c, c a real number, then G A. G Quick Quiz for AP* Preparation: Sections.4.6 You shoul solve the following problems without using a graphing calculator.. Multiple Choice Which of the following gives y/ for y sin 4 ()? B (A) 4 sin () cos () (B) sin () cos () (C) sin () cos () (D) sin () (E) sin () cos (). Multiple Choice Which of the following gives y for y cos tan? A (A) cos sec tan (C) sin sec (E) cos sec tan (B) cos sec tan (D) cos sec tan. Multiple Choice Which of the following gives y/ for the parametric curve sin t, y cos t? C (A) cot t (B) cot t (C) tan t (D) tan t (E) tan t 4. Free Response A particle moves along a line so that its position at any time t 0 is given by s(t) t t, where s is measure in meters an t is measure in secons. (a) What is the initial position of the particle? s(0) m (b) Fin the velocity of the particle at any time t. v(t) s(t) t m/s (c) When is the particle moving to the right? () Fin the acceleration of the particle at any time t. (e) Fin the spee of the particle at the moment when s(t) 0. (c) The particle moves to the right when v(t) 0; that is, when 0 t /. () a(t) v(t) m/s (e) s(t) (t )(t ), so s(t) 0 when t. The spee at that moment is v() m/s. Answers to Section.6 Eercises 50. Since the raius goes through (0, 0) an ( cos t, sin t), it has slope given by tan t. But y y/ t c os t cot t, which is the / t sin t negative reciprocal of tan t. This means that the raius an the tangent are perpenicular. 6. Velocity m/sec 5 4 acceleration m/sec Acceleration v v s v v t s t s k (ks) k s k 65. Given: v s acceleration: v v s v v t s t s k k k s / s s 66. Acceleration v f ( ) t t f( ) t f()f() 67. T u T L L u T gl kl k L g k

5 6 Chapter Derivatives. y 9, y 9 EXAMPLE 6 Using the Rational Power Rule (a) Notice that is efine at 0, but is not. (b) The original function is efine for all real numbers, but the erivative is unefine at 0. Recall Figure., which showe that this function s graph has a cusp at 0. (c) cos 5 5 cos 65 cos 5 cos 65 sin 5. y, y Quick Review.7 (For help, go to Section. an Appeni A.5.) 5 sin cos 65 Now try Eercise. y 8. y y In Eercises 5, sketch the curve efine by the equation an fin two functions y an y whose graphs will combine to give the curve.. y 0 y, y. 4 9y 6. 4y 0 y, y 4. y 9 5. y y 9, y 9 In Eercises 6 8, solve for y in terms of y an. 6. yy 4 y y 4 y y Section.7 Eercises 4. y In Eercises 8, fin y. ( y) or y. y y 6 y y. y 8y 6 y y y 6. y 4. y y( ) y 5. tan y cos y 6. sin y sec y y 7. tan y 0 See page sin y y cos y In Eercises 9, fin y an fin the slope of the curve at the inicate point. y, 9. y y, (, ) y 0. y 9, (0, ),0 y y. ( ) (y ), (, 4), y. ( ) (y ) 5, (, 7) See page 64. In Eercises 6, fin where the slope of the curve is efine.. y y 4 See page cos y See page y y See page y 4y 6 See page 64. In Eercises 7 6, fin the lines that are (a) tangent an (b) normal to the curve at the given point. 7. y y,, (a) y 7 4 (b) y y 5,, 4 (a) y 4 5 (b) y y 9,, See page ysin cos yy 8. y y y y y y cos sin In Eercises 9 an 0, fin an epression for the function using rational powers rather than raicals (a) y (b) y 0. y 4y 0,, (a) y (b) y. 6 y y 7y 6 0,, 0 See page 64.. y y 5,, (a) y = (b) y. y p sin y p,, p See page sin y y cos, p4, p (a) y (b) y y sin p y,, 0 6. cos y sin y 0, 0, p (a) y (b) 0 In Eercises 7 0, use implicit ifferentiation to fin y an then y. 7. y See page y See page y See page y y See page 64. In Eercises 4, fin y.. y 94 (9/4) 5/4. y 5 (/5) 8/5. y (/) / 4. y 4 (/4) /4 5. y 5 ( 5) 6. y 6 4( 6) / 7. y 8. y ( ) / ( ) / ( ) / 9. y See page y See page y csc See page y sin 5 54 See page 64.

6 Section.7 Implicit Differentiation 6 4. Which of the following coul be true if f? (b), (c), an () (a) f 9 (b) f (c) f 4 () f Which of the following coul be true if gt t 4? (a) an (c) (a) gt 4 4 t 4 (b) gt 4 4 t (c) gt t 7 65t 54 () gt 4t The Eight Curve (a) Fin the slopes of the figure-eightshape curve y 4 y at the two points shown on the graph that follows. (b) Use parametric moe an the two pairs of parametric equations t t t, 4 y t t, t t t, 4 y t t, to graph the curve. Specify a winow an a parameter interval. 46. The Cissoi of Diocles (ates from about 00 B.c.) (a) Fin equations for the tangent an normal to the cissoi of Diocles, y, at the point, as picture below. (b) Eplain how to reprouce the graph on a grapher. (a) Tangent: y normal: y y 0 y, 4, 4 y 4 y y ( ) (b) One way is to graph the equations y. 47. (a) Confirm that, is on the curve efine by y cos py. () () cos () is true since both sies equal:. (b) Use part (a) to fin the slope of the line tangent to the curve at,. The slope is /. 48. Grouping Activity (a) Show that the relation There are three values:, 5 y y cannot be a function of by showing that there is more than one possible y-value when. f(), (b) On a small enough square with center,, the part f () 4 of the graph of the relation within the square will efine a function y f. For this function, fin f an f. 49. Fin the two points where the curve y y 7 crosses the -ais, an show that the tangents to the curve at these points are parallel. What is the common slope of these tangents? 50. Fin points on the curve y y 7 (a) where the tangent is parallel to the -ais an (b) where the tangent is parallel to the y-ais. (In the latter case, y is not efine, but y is. What value oes y have at these points?) 5. Orthogonal Curves Two curves are orthogonal at a point of intersection if their tangents at that point cross at right angles. Show that the curves y 5 an y are orthogonal at, an,. Use parametric moe to raw the curves an to show the tangent lines. 5. The position of a boy moving along a coorinate line at time t is s 4 6t, with s in meters an t in secons. Fin the boy s velocity an acceleration when t sec. 5. The velocity of a falling boy is v 8s t feet per secon at the instant tsec the boy has fallen s feet from its starting point. Show that the boy s acceleration is ftsec. 54. The Devil s Curve (Gabriel Cramer [the Cramer of Cramer s Rule], 750) Fin the slopes of the evil s curve y 4 4y 4 9 at the four inicate points. (, ) (, ) y y 4 4y 4 9 (, ) (, ) At (, ): 7 ; 8 at (, ): 7 ; 8 at (, ): 7 ; 8 at (, ): (, ) 55. The Folium of Descartes (See Figure.47 on page 57) (a) Fin the slope of the folium of Descartes, y 9y 0 at the points 4, an, 4. 5 (a) At (4, ): 4 ; at (, 4): 4 5 (b) At what point other than the origin oes the folium have a horizontal tangent? At (, 4) (.780, 4.76) (c) Fin the coorinates of the point A in Figure.47, where the folium has a vertical tangent. At ( 4, ) (4.76,.780)

7 64 Chapter Derivatives 56. The line that is normal to the curve y y 0 at, intersects the curve at what other point? (, ) 57. Fin the normals to the curve y y 0 that are parallel to the line y 0. At (, ): y ;at (, ): y 58. Show that if it is possible to raw these three normals from the point a, 0 to the parabola y shown here, then a must be greater than. One of the normals is the -ais. For what value of a are the other two normals perpenicular? y y 6. Multiple Choice Which of the following is equal to y if y /4? E (A) (B) 4 4 (C) 4 4 (D) (E) 4 4 /4 4 /4 64. Multiple Choice Which of the following is equal to the slope of the tangent to y at (,)? C (A) (B) (C) (D) (E) 0 Etening the Ieas 65. Fining Tangents (a) Show that the tangent to the ellipse Stanarize Test Questions You shoul solve the following problems without using a graphing calculator. 59. True or False The slope of y at (, ) is. Justify your answer. False. It is equal to. 60. True or False The erivative of y is. Justify your / answer. True. By the power rule. In Eercises 6 an 6, use the curve y y. 6. Multiple Choice Which of the following is equal to y? A (A) y y (C) y 0 (a, 0) The normal at the point (b, b) is: y b b b. This line intersects the -ais at b, which must be greater than if b 0. The two normals are perpenicular when a /4. (B) y y (D) y y (E) y 6. Multiple Choice Which of the following is equal to y? 6 0y (A) (y ) (B) 0 0y (y ) (C) 8 4y 8y ( y) (D) 0 0y ( y) (E) 7. cos (y) y y.,4 y y y y., efine at every point ecept where 0 or y y y 4., efine at every point ecept where y k, k any integer sin y y y 5., efine at every point ecept where y y y 4y 6., efine at every point ecept where y 4 8y A a y b at the point, y has equation a y y b. (b) Fin an equation for the tangent to the hyperbola a y b at the point, y. 66. En Behavior Moel Consier the hyperbola Show that (a) y b a a. a y b. (b) g ba is an en behavior moel for f ba a. (c) g ba is an en behavior moel for. (a) y (b) y y y y ( y ) y y 8. y y y / 4 / y / y/ 4/ y / 9. 4 ( / ) / / 40. ( ) (csc ) cot [sin ( 5)]/4 cos ( 5) f ba a. 9. (a) y 6 (b) y 8. (a) y (b) y 9. y y y y ) (y y 0. y y y (y )

8 70 Chapter Derivatives Quick Review.8 (For help, go to Sections.,.5, an.6.) In Eercises 5, give the omain an range of the function, an evaluate the function at.. y sin Domain: [, ]; Range: [, ] At :. y cos Domain: [, ]; Range: [0, ] At : 0. y tan Domain: all reals; Range: (, ) At : 4 4. y sec Domain: (, ] [, ); Range: [0, ) (, ] At : 0 5. y tan tan Domain: all reals; Range: all reals At : In Eercises 6 0, fin the inverse of the given function. 6. y 8 f () 8 7. y 5 f () 5 8. y 8 f () 8 9. y f () 0. y arctan f () tan, Section.8 Eercises 9. (a) f() sin an f() 0. So f has a ifferentiable inverse by Theorem. In Eercises 8, fin the erivative of y with respect to the appropriate 7. (a) Fin an equation for the line tangent to the graph of variable. y tan at the point p4,. y. y cos 4. y cos (b) Fin an equation for the line tangent to the graph of. y sin t 4. y sin y tan t at the point, p4. y 4 t t t 8. Let f y sin 6 t 6. y s s cos s s (a) Fin f an f. f (), f() tt 4 9 s 7. y sin (b) Fin f an f. f (), (f )() 8. y sin (sin ) 4 sin 9. Let f cos. In Eercises 9, a particle moves along the -ais so that its position (a) Show that f has a ifferentiable inverse. at any time t 0 is given by (t). Fin the velocity at the ini- (b) Fin f 0 an f 0. f(0), f(0) cate value of t (c) Fin f an f. f () 0, (f )() 9. (t) sin t 4 sin 4t, t 4 0. Group Activity Graph the function f sin sin in the viewing winow p, p by 4, 4. Then answer the. (t) tan t, t 5. (t) tan (t ), t following questions: (a) What is the omain of f? In Eercises, fin the erivatives of y with respect to the (b) What is the range of f? appropriate variable. (c) At which points is f not ifferentiable?. y sec s 4. y sec 5s s 5s See page 7. () Sketch a graph of y f without using NDER or 5. y csc, 0 6. y csc computing the erivative. 7. y sec See page 7. 4 t, 0 t 8. y (e) Fin f algebraically. Can you reconcile your answer cot t See page 7. t(t ) with the graph in part ()? 9. y cot t 0. y s sec s. Group Activity A particle moves along the -ais so that its See page 7. position at any time t 0 is given by arctan t.. y tan csc s s, 0, > s s (a) Prove that the particle is always moving to the right. (b) Prove that the particle is always ecelerating.. y cot tan 0, 0 In Eercises 6, fin an equation for the tangent to the graph of y at the inicate point. y y y sec, 4. y tan 5, 5. y sin 4, 6. y tan ( ), y y 0.5 (c) What is the limiting position of the particle as t approaches infinity? In Eercises 4, use the inverse function inverse cofunction ientities to erive the formula for the erivative of the function.. arccosine See page 7.. arccotangent See page arccosecant See page 7.. (a) v(t) t t which is always positive. (b) a(t) v t ( tt ) which is always negative.

9 Section.8 Derivatives of Inverse Trigonometric Functions 7 Stanarize Test Questions You may use a graphing calculator to solve the following problems. 5. True or False The omain of y sin is. Justify your answer. True. By efinition of the function. 6. True or False The omain of y tan is. Justify your answer. False. The omain is all real numbers. 7. Multiple Choice Which of the following is sin? (A) 4 (B) 4 (C) 4 (D) 4 (E) 4 8. Multiple Choice Which of the following is tan ()? (A) 9 (B) 9 (C) 9 E D Etening the Ieas 47. Ientities Confirm the following ientities for 0. (a) cos sin p (b) tan cot p (c) sec csc p 48. Proof Without Wors The figure gives a proof without wors that tan tan tan p. Eplain what is going on. (D) 9 (E) 9 9. Multiple Choice Which of the following is sec ( )? (A) 4 (B) (C) 4 (D) (E) Multiple Choice Which of the following is the slope of the tangent line to y tan () at? C (A) 5 (B) 5 (C) 5 (D) 5 (E) 5 Eplorations In Eercises 4 46, fin (a) the right en behavior moel, (b) the left en behavior moel, an (c) any horizontal tangents for the function if they eist. 4. y tan 4. y cot 4. y sec 44. y csc 45. y sin 46. y cos A 49. (Continuation of Eercise 48) Here is a way to construct tan, tan, an tan by foling a square of paper. Try it an eplain what is going on. Fol Fol Fol tan tan tan.. s s s 5. ( ) 7. t 9. t t. cos () sin 0 sin. cot tan 0 tan 4. csc () 0 sec sec 4. (a) y (b) y (c) None 4. (a) y 0 (b) y (c) None 4. (a) y (b) y (c) None 44. (a) y 0 (b) y 0 (c) None 45. (a) None (b) None (c) None 46. (a) None (b) None (c) None

10 78 Chapter Derivatives [ 5, 0] by [ 0, 0] Figure.59 The graph of Pt, the rate of sprea of the flu in Eample 8. The graph of P is shown in Figure.58. (b) To fin the rate at which the flu spreas, we fin Pt. To fin Pt, we nee to invoke the Chain Rule twice: P 00 e t t t 00 e t e t t 00 e t 0 e t t t 00 e t e t 00e e t t At t, then, Pt The flu is spreaing to 5 stuents per ay. (c) We coul estimate when the flu is spreaing the fastest by seeing where the graph of y Pt has the steepest upwar slope, but we can answer both the when an the what parts of this question most easily by fining the maimum point on the graph of the erivative (Figure.59). We see by tracing on the curve that the maimum rate occurs at about ays, when (as we have just calculate) the flu is spreaing at a rate of 5 stuents per ay. Now try Eercise 5. Quick Review.9 (For help, go to Sections. an.5.). Write log 5 8 in terms of natural logarithms. l n 8 ln 5. Write 7 as a power of e. e ln 7 In Eercises 7, simplify the epression using properties of eponents an logarithms.. ln e tan tan 4. ln 4 ln ln ( ) 5. log log 4 5 log ln ln ln ln (4 4 ). csc (ln )(csc cot ) Section.9 Eercises 6., ln ln In Eercises 8, fin y. Remember that you can use NDER to support your computations.. y e e. y e e. y e e 4. y e 5 5e 5 5. y e e 6. y e 4 4 e/4 7. y e e e e 8. y e e 9. y e e/ 0. y e e ( ). y 8 8 ln 8. y 9 9 ln 9 e e e. y csc 4. y cot cot (ln )(csc ) 5. y ln 6. y ln l n 7. y ln See page y ln 0 See page y ln ln l 0. y ln ln n. y log 4 See page 80.. y log 5 See page 80.. y log, 0 4. y log l n (ln )( log ) 5. y ln log 6. y log ln, 0 In Eercises 8 0, solve the equation algebraically using logarithms. Give an eact answer, such as ln, an also an approimate answer to the nearest hunreth l n 9 ln t ln 5 8 ln 8 ln (ln 5).50 ln 5 0. ln.7 ln ln 7. y log 0 e 8. y ln 0 ln 0 ln 0 9. At what point on the graph of y is the tangent line parallel to the line y 5? (.79, 5.55) 0. At what point on the graph of y e is the tangent line perpenicular to the line y? (.79, 0.667). A line with slope m passes through the origin an is tangent to y ln (). What is the value of m? e. A line with slope m passes through the origin an is tangent to y ln (). What is the value of m? e In Eercises 6, fin y.. y p 4. y ( ) 5. y 6. y e ( e) e In Eercises 7 4, fin f() an state the omain of f. 7. f () ln ( ), 8. f () ln ( ),

11 40., all reals 9. f () ln ( cos ) 40. f () ln ( ) 4. f () log ( ) 4. f () log 0 4., ( )ln 0 sin, all reals cos (, ) ln Group Activity In Eercises 4 48, use the technique of logarithmic ifferentiation to fin y. 4. y sin, 0 p (sin ) [ cot ln (sin )] 44. y tan, 0 tan tan (ln )(sec ) 45. y y ( ) / ( ) 47. y ln ln (ln ) 48. y (/ln ) 0, Fin an equation for a line that is tangent to the graph of y e an goes through the origin. y e 50. Fin an equation for a line that is normal to the graph of y e an goes through the origin. y 5. Sprea of a Rumor The sprea of a rumor in a certain school is moele by the equation 00 P(t), 4t where P(t) is the total number of stuents who have hear the rumor t ays after the rumor first starte to sprea. (a) Estimate the initial number of stuents who first hear the rumor. 8 (b) How fast is the rumor spreaing after 4 ays? 5 stuents per ay (c) When will the rumor sprea at its maimum rate? What is that rate? After 4 ays; 5 stuents per ay 5. Sprea of Flu The sprea of flu in a certain school is moele by the equation 00 P(t), e5t where P(t) is the total number of stuents infecte t ays after the flu first starte to sprea. (a) Estimate the initial number of stuents infecte with this flu. (b) How fast is the flu spreaing after 4 ays? 9 stuents per ay (c) When will the flu sprea at its maimum rate? What is that rate? After 5 ays; 50 stuents per ay 5. Raioactive Decay The amount A (in grams) of raioactive plutonium remaining in a 0-gram sample after t ays is given by the formula A 0 t40. At what rate is the plutonium ecaying when t ays? Answer in appropriate units. rate grams/ay 54. For any positive constant k, the erivative of ln k is. Prove this fact (a) by using the Chain Rule. See page 80. (b) by using a property of logarithms an ifferentiating. 45. ) 4 ( ( ( ) 5) 4 6 /5 5( ) 5( ) 5( 5) Section.9 Derivatives of Eponential an Logarithmic Functions 79 See page Let f. (a) Fin f 0. ln (b) Use the efinition of the erivative to write f 0 as a limit. (c) Deuce the eact value of f (0) lim h h 0 h lim h 0 h. h ln () What is the eact value of lim h 0 7h? ln 7 h 56. Writing to Learn The graph of y ln looks as though it might be approaching a horizontal asymptote. Write an argument base on the graph of y e to eplain why it oes not. [, 6] by [, ] Stanarize Test Questions You shoul solve the following problems without using a graphing calculator. 57. True or False The erivative of y is. Justify your answer. False. It is (ln ). 58. True or False The erivative of y e is (ln ) e. Justify your answer. False. It is e. 59. Multiple Choice If a flu is spreaing at the rate of 50 P(t), e4t which of the following is the initial number of persons infecte? B (A) (B) (C) 7 (D) 8 (E) Multiple Choice Which of the following is the omain of f () if f () log ( )? D (A) (B) (C) (D) (E) 6. Multiple Choice Which of the following gives y if y log 0 ( )? A (A) (B) (C) ( )ln 0 ( )ln0 (D) (E) 6. Multiple Choice Which of the following gives the slope of the tangent line to the graph of y at? E (A) (B) (C) (D) (E) ln

12 80 Chapter Derivatives Eploration 7., 0 8., 0. l n4 6. Let y a, y NDER y, y y y, an y 4 e y. (a) Describe the graph of y 4 for a,, 4, 5. Generalize your escription to an arbitrary a. (b) Describe the graph of y for a,, 4, 5. Compare a table of values for y for a,, 4, 5 with ln a. Generalize your escription to an arbitrary a. (c) Eplain how parts (a) an (b) support the statement., 0 l n ln Which is Bigger, p e or e p? Calculators have taken some of the mystery out of this once-challenging question. (Go ahea an check; you will see that it is a surprisingly close call.) You can answer the question without a calculator, though, by using the result from Eample of this section. Recall from that eample that the line through the origin tangent to the graph of y ln has slope e. a a if an only if a e. () Show algebraically that y y if an only if a e. Etening the Ieas 64. Orthogonal Families of Curves Prove that all curves in the family y k (k any constant) are perpenicular to all curves in the family y ln c (c any constant) at their points of intersection. (See accompanying figure.) [, 6] by [, ] (a) Fin an equation for this tangent line. (b) Give an argument base on the graphs of y ln an the tangent line to eplain why ln e for all positive e. (c) Show that ln e for all positive e. () Conclue that e e for all positive e. (e) So which is bigger, p e or e p? [, 6] by [, ] 54. (a) ln (k) k k k k (b) ln (k) (ln k ln ) 0 ln Quick Quiz for AP* Preparation: Sections k an (ln c). Therefore, at any given value of, these two curves will have perpenicular tangent lines. 65. (a) y e (b) Because the graph of ln lies below the graph of the tangent line for all positive e. (c) Multiplying by e, e(ln ), or ln e. () Eponentiate both sies of the inequality in part (c). (e) Let to see that e e. You may use a graphing calculator to solve the following problems.. Multiple Choice Which of the following gives y at if y 9? E (A) (B) 5 (C) (D) 5 (E). Multiple Choice Which of the following gives y if y cos ( )? A (A) 9 cos ( ) sin ( ) (B) cos ( ) sin ( ) (C) 9 cos ( ) sin ( ) (D) 9 cos ( ) (E) cos ( ). Multiple Choice Which of the following gives y if y sin ()? C (A) 4 (B) 4 (C) 4 (D) 4 (E) 4 4. Free Response A curve in the y-plane is efine by y y 6. (a) Fin y/. (b) Fin an equation for the tangent line at each point on the curve with -coorinate. (c) Fin the -coorinate of each point on the curve where the tangent line is vertical.

13 Review Eercises 8 Calculus at Work Iwork at Ramsey County Hospital an other community hospitals in the Minneapolis area, both with patients an in a laboratory. I have wante to be a physician since I was about years ol, an I began attening meical school when I was 0 years ol. I am now working in the fiel of internal meicine. Cariac patients are common in my fiel, especially in the iagnostic stages. One of the machines that is sometimes use in the emergency room to iagnose problems is calle a Swan-Ganz catheter, name after its inventors Harol James Swan an William Ganz. The catheter is inserte into the pulmonary artery an then is hooke up to a cariac monitor. A program measures cariac output by looking at changes of slope in the curve. This information alerts me to left-sie heart failure. Lupe Boling, M.D. Ramsey County Hospital Minneapolis, MN Chapter Key Terms acceleration (p. 0) average velocity (p. 8) Chain Rule (p. 49) Constant Multiple Rule (p. 7) Derivative of a Constant Function (p. 6) erivative of f at a (p. 99) ifferentiable function (p. 99) ifferentiable on a close interval (p. 04) isplacement (p. 8) free-fall constants (p. 0) implicit ifferentiation (p. 57) instantaneous rate of change (p. 7) instantaneous velocity (p. 8) Intermeiate Value Theorem for Derivatives (p. ) 7. / Chapter Review Eercises inverse function inverse cofunction ientities (p. 68) jerk (p. 44) left-han erivative (p. 04) local linearity (p. 0) logarithmic ifferentiation (p. 77) marginal cost (p. 4) marginal revenue (p. 4) nth erivative (p. ) normal to the surface (p. 59) numerical erivative NDER (p. ) orthogonal curves (p. 54) orthogonal families (p. 80) Power Chain Rule (p. 5) Power Rule for Arbitrary Real Powers (p. 76) 7. cos Power Rule for Negative Integer Powers of (p. ) Power Rule for Positive Integer Powers of (p. 6) Power Rule for Rational Powers of (p. 6) Prouct Rule (p. 9) Quotient Rule (p. 0) right-han erivative (p. 04) sensitivity to change (p. ) simple harmonic motion (p. 4) spee (p. 9) Sum an Difference Rule (p. 7) symmetric ifference quotient (p. ) velocity (p. 8) 8. l n The collection of eercises marke in re coul be use as a chapter test. In Eercises 0, fin the erivative of the function. 5 csc 5 cot 5 csc 5. y csc 5. y ln, 0. y ln e e 4. y e e e e 5. y e ln e 6. y ln sin. y 5 7. r ln cos 8. r log u 8 4. y s log 5 t 7 0. s 8 t 8 t ln 8. y sin cos 4. y 4, t 7. y ln See page 84.. y ( ) (t 7)ln 5 cos sin cos 5. s cos t sin ( t) 6. s cot. y e tan e t an t t csc See page 84. t 4. y sin u 7. y 8. y 5. y t sec t ln t 6. y t cot t See page 84. See page r sec u 0. r tan u 7. y z cos z z 8. y csc sec ( ) tan ( ) 4 tan ( ) sec ( ) cos z c sc 6. cot, where is an interval of the form (k, (k ) ), k even

14 8 Chapter Derivatives 9. y csc sec, 0 p 0. r ( sin u sin sin cos u cos cos ( cos ) ) In Eercises 4, fin all values of for which the function is ifferentiable.. y ln For all 0. y sin cos For all real. y For all 4. y 7 5 For all 7 In Eercises 5 8, fin y. 5. y y y y y y (y)/5 or 8. y y( ) In Eercises 9 4, fin y by implicit ifferentiation. 9. y y5 40. y y 4 y 4. y y cos 4. y 4 In Eercises 4 an 44, fin all erivatives of the function. 4. y y 0 In Eercises 45 48, fin an equation for the (a) tangent an (b) normal to the curve at the inicate point. 45. y, (a) y (b) y y 4 cot csc, p (a) y (b) y 47. y 9,, 48. y 6, 4, In Eercises 49 5, fin an equation for the line tangent to the curve at the point efine by the given value of t. 49. sin t, y cos t, t p4 y 50. cos t, y 4 sin t, t p4 5. sec t, y 5 tan t, t p6 y cos t, y t sin t, t p4 y ( ) 4 5. Writing to Learn or y (a) Graph the function, 0 f {,. 4. (y ) cos y sin 4. (y ) 4/ y / 5/ y / 8 5/ y / 50. y 4 4 (a) [, 0) (0, 4], f { (b) At 0, 0 4 (c) Nowhere in its omain, 0 (a) [, 0) (0, ] 58. g { (b) Nowhere, 0 (c) Nowhere in its omain In Eercises 59 an 60, use the graph of f to sketch the graph of f. 59. Sketching f from f y 60. Sketching f from f 6. Recognizing Graphs The following graphs show the istance travele, velocity, an acceleration for each secon of a -minute automobile trip. Which graph shows (a) istance iii? (b) velocity? i (c) acceleration? ii (i) (iii) 0 t (ii) y 0 y = f() y = f() t (b) Is f continuous at? Eplain. (c) Is f ifferentiable at? Eplain. 54. Writing to Learn For what values of the constant m is sin, 0 f { m, 0 (a) continuous at 0? Eplain. (b) ifferentiable at 0? Eplain. In Eercises 55 58, etermine where the function is (a) ifferentiable, (b) continuous but not ifferentiable, an (c) neither continuous nor ifferentiable. 55. f 45 (a) For all g sin (b) At 0 (c) Nowhere (b) Nowhere (c) Nowhere (a) For all 4. y, y6, y, y (4), an the rest are all zero y, 4 y 6, y, y (4), y (5), an the rest are all zero Sketching f from f Sketch the graph of a continuous function f with f 0 5 an, f { 0.5,. 6. Sketching f from f Sketch the graph of a continuous function f with f an, f, 4 {, (a) y 4 9 (b) y (a) y (b) y t

15 Review Eercises Which of the following statements coul be true if f? Answer is D: i an iii only coul be true 9 i. f ii. f 8 7 iii. f iv. f A. i only B. iii only C. ii an iv only D. i an iii only 65. Derivative from Data The following ata give the coorinates of a moving boy for various values of t. t (sec) s (ft) (a) Make a scatter plot of the t, s ata an sketch a smooth curve through the points. (b) Compute the average velocity between consecutive points of the table. (c) Make a scatter plot of the ata in part (b) using the mipoints of the t values to represent the ata. Then sketch a smooth curve through the points. () Writing to Learn Why oes the curve in part (c) approimate the graph of st? 66. Working with Numerical Values Suppose that a function f an its first erivative have the following values at 0 an. f f Fin the first erivative of the following combinations at the given value of. (a) f, /0 (b) f, 0 / (c) f, /0 () f 5 tan, 0 f (e), 0 / (f) 0 sin cos ( p ) f, 67. Working with Numerical Values Suppose that functions f an g an their first erivatives have the following values at an 0. f g f g Fin the first erivative of the following combinations at the given value of. (a) f g, 5 (b) f g, 0 0 (c) g f, 8 () f g, f (e), 0 6 (f) g f, 0 g 68. Fin the value of ws at s 0 if w sin r an r 8 sin s p Fin the value of rt at t 0 if r u 7 an u t u. /6 70. Particle Motion The position at time t 0 of a particle moving along the s-ais is st 0 cos t p4. (a) Give parametric equations that can be use to simulate the motion of the particle. (b) What is the particle s initial position t 0? (c) What points reache by the particle are farthest to the left an right of the origin? () When oes the particle first reach the origin? What are its velocity, spee, an acceleration then? 7. Vertical Motion On Earth, if you shoot a paper clip 64 ft straight up into the air with a rubber ban, the paper clip will be st 64t 6t feet above your han at t sec after firing. (a) Fin st an st. s 64 t s t t (b) How long oes it take the paper clip to reach its maimum height? sec (c) With what velocity oes it leave your han? 64 ft/sec () On the moon, the same force will sen the paper clip to a height of st 64t.6t ft in t sec. About how long will it take the paper clip to reach its maimum height, an how high 64 will it go? 5.. sec; ft 7. Free Fall Suppose two balls are falling from rest at a certain height in centimeters above the groun. Use the equation s 490t to answer the following questions. (a) How long oes it take the balls to fall the first 60 cm? What is their average velocity for the perio? 4 sec; 80 cm/sec 7 (b) How fast are the balls falling when they reach the 60-cm mark? What is their acceleration then? 560 cm/sec; 980 cm/sec 7. Filling a Bowl If a hemispherical bowl of raius 0 in. is fille with water to a epth of in., the volume of water is given by V p0. Fin the rate of increase of the volume per inch increase of epth. p(0 ) 74. Marginal Revenue A bus will hol 60 people. The fare charge ( p ollars) is relate to the number of people who use the bus by the formula p 40. (a) Write a formula for the total revenue per trip receive by the bus company. r() (b) What number of people per trip will make the marginal revenue equal to zero? What is the corresponing fare? 40 people; $4.00 (c) Writing to Learn Do you think the bus company s fare policy is goo for its business? One possible answer: Probably not, since the company charges less overall for 60 passengers than it oes for 40 passengers.

16 84 Chapter Derivatives 75. Searchlight The figure shows a boat km offshore sweeping the shore with a searchlight. The light turns at a constant rate, ut 0.6 rasec. (a) How fast is the light moving along the shore when it reaches point A? 0.6 km/sec (b) How many revolutions per minute is 0.6 rasec? km A 8/ 5.7 revolutions/min (a) Estimate the initial number of stuents infecte with measles. P(0).9, so initially, one stuent was infecte (b) About how many stuents in all will get the measles? 00 (c) When will the rate of sprea of measles be greatest? What is this rate? After 5 ays, when the rate is 50 stuents/ay 79. Graph the function f tan tan in the winow p, p by 4, 4. Then answer the following questions. (a) What is the omain of f? (b) What is the range of f? k, where k is an o integer 4 (,) (c) At which points is f not ifferentiable? () Describe the graph of f. 80. If y, fin y at the point,. /() 76. Horizontal Tangents The graph of y sin sin appears to have horizontal tangents at the -ais. Does it? Yes 77. Funamental Frequency of a Vibrating Piano String We measure the frequencies at which wires vibrate in cycles (trips back an forth) per sec. The unit of measure is a hertz: cycle per sec. Mile A on a piano has a frequency 440 hertz. For any given wire, the funamental frequency y is a function of four variables: r: the raius of the wire; l: the length; : the ensity of the wire; T: the tension (force) holing the wire taut. With r an l in centimeters, in grams per cubic centimeter, an T in ynes (it takes about 00,000 ynes to lift an apple), the funamental frequency of the wire is y rl T p. If we keep all the variables fie ecept one, then y can be alternatively thought of as four ifferent functions of one variable, yr, yl, y, an yt. How woul changing each variable affect the string s funamental frequency? To fin out, calculate yr, yl, y, an yt. 78. Sprea of Measles The sprea of measles in a certain school is given by 00 Pt, e5t where t is the number of ays since the measles first appeare, an Pt is the total number of stuents who have caught the measles to ate. T 77. y(r) r l, so increasing r ecreases the frequency. T y(l) r l, so increasing l ecreases the frequency. T y() 4 rl, so increasing ecreases the frequency. y(t) 4rl T, so increasing T increases the frequency. 79. (c) Where it s not efine, at k, k an o integer 4 () It has perio an continues to repeat the pattern seen in this winow. AP* Eamination Preparation You may use a graphing calculator to solve the following problems. 8. A particle moves along the -ais so that at any time t 0 its position is given by (t) t t 5. (a) Fin the velocity of the particle at any time t. (b) Fin the acceleration of the particle at any time t. (c) Fin all values of t for which the particle is at rest. () Fin the spee of the particle when its acceleration is zero. (e) Is the particle moving towar the origin or away from the origin when t? Justify your answer. e 8. Let y e. (a) Fin y. (b) Fin y. (c) Fin an equation of the line tangent to the curve at. () Fin an equation of the line normal to the curve at. (e) Fin any points where the tangent line is horizontal. 8. Let f () ln ( ). (a) State the omain of f. (b) Fin f (). (c) State the omain of f. () Prove that f () 0 for all in the omain of f. Aitional Answers: ) ln. (ln ( )[ ln ln ]. or ( ) / () ln u 4. u u u 4 u u t 5. t t sec t t 6. t t cot t 4t