CHAPTER 2 Differentiation
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1 CHAPTER Differentiation Section. The Derivative and the Slope of a Graph Section. Some Rules for Differentiation Section. Rates of Change: Velocit and Marginals Section. The Product and Quotient Rules Section.5 The Chain Rule Section.6 Higher-Order Derivatives Section.7 Implicit Differentiation Section.8 Related Rates Review Eercises
2 CHAPTER Differentiation Section. Solutions to Even-Numbered Eercises The Derivative and the Slope of a Graph. The tangent line at, has a negative slope.. The tangent line at, has zero slope. The tangent line The tangent line at, has a positive slope. at, has a positive slope. 6. The slope is m. 8. The slope is m. 0. The slope is m.. For 998, t 8 and m 500. For 00, t and m 90. For 00, t and m (a) At t, ft > gt, so the runner given b f is running faster. (b) At t, gt > ft, so the runner given b g is running faster. The runner given b f has traveled farther. (c) At t, the runners are at the same location, but the runner given b g is running faster. (d) The runner given b g will finish first, because that runner finishes the distance at a lesser value of t. 6. f 8. f f 0 f f 0 0 f f lim 0 0 f 5 5 f f f f f f lim 0 0. f f f f f f f lim 0 6
3 6 Chapter Differentiation. f f ) f f f f f lim 0. ft t t t t t t t t tt t t tt t ft t ft t t tt t tt t ft t ft t ft t ft lim t t 0 t t t tt t t t 6. gs s s s gs s gs gs s gs s s s s gs s gs lim s 0 s s s s s s s s s s s s s s 8. f 0. f 6 f f f f f f lim 0 At, 6, the slope of the tangent line is m. f f f f 0 0 f f lim 0 0 At, 6, the slope of the tangent line is m 0.
4 Section. The Derivative and the Slope of a Graph 65. f f f f f At,, the slope of the tangent line is m. f f lim 0. f f f f f f f lim 0 f At,, the slope of the tangent line is m 5. The figure shows the graph of f and the tangent line., ) ) 5 6. f f f f f f f lim 0 At 9,, the slope of the tangent line is m 9.
5 66 Chapter Differentiation 8. f 0. f f f f f f f lim 0 At the point,, the slope of the tangent line is m. The equation of the tangent line is. f f f f f f lim 0 At the point 0,, the slope of the tangent line is m 0 0. The equation of the horizontal tangent line is.. From Eercise, At the point 7 6 f f lim 0 7,, the slope of the tangent line is m 6... From Eercise 6, At the point,, the slope is. f f lim (, ) 8 6 (7, ) f f f f f f f lim (Slope of tangent line) 0 Since the slope of the given line is, we have so and f. Therefore, at the point,, the tangent line parallel to 0 is.
6 Section. The Derivative and the Slope of a Graph f f f f f f f lim 0 Slope of line 6 0 is m. Equating slopes: and f Tangent line. 50. is differentiable everwhere ecept the two cusps at ±. 5. is not differentiable at 0. At 0, 0 the graph has a cusp. 5. is differentiable everwhere ecept ±, where f is 56. Since is a nonremovable discontinuit, is not defined. differentiable everwhere ecept at f f Analticall, the slope of f is f f m lim lim 0 0 lim lim f f f f 8
7 68 Chapter Differentiation 6. f f 6 f lim lim lim lim Graphs of f 6 and f 6 (see figure) The -intercept of the derivative indicates a point of horizontal tangenc for f f f f lim 0 lim lim lim The -intercepts of the derivative, 0,, indicate points of horizontal tangenc for f. 66. One possible answer is. 68. True 70. True (See page 89.)
8 Section. Some Rules for Differentiation 69 Section. Some Rules for Differentiation. (a) At,,. (b) At,,.. (a) At,,. (b) At,,. 6. f 08. g 0. t h 5 8. g 0. st t t g Function Rewrite Differentiate Simplif f t t t 0. ft t t f At, 8, f f At 5, 0, f5 0. f 5 f f 8. f f 6 f
9 70 Chapter Differentiation 0. f f 5 0 f f. f 6 f 6 6. f f At,, the slope is m. The equation of the tangent line is. f f 5 f Tangent line: when 5. when 0,. The function has horizontal tangent lines at the points 0, 0 and,.. The function has a horizontal tangent line at the point,. 56. (a) (c) (b) f g h f g h 58. R.7879t 8.6t 69.99t 80.t t 6 corresponds to 996; 6 t. (a) Rt.756t 5.09t t : R : R0. 00: R. (b) These results are close to the estimates. (c) The units of R are millions of dollars per ear.
10 Section. Rates of Change: Velocit and Marginals C C 7.75, which equals the variable cost. 6. (a) More men and women seem to suffer from migraines between 0 and 0 ears old. More males than females suffer from migraines. Fewer people whose income is greater than or equal to $0,000 suffer from migraines than people whose income is less than $0,000. (b) The derivatives are positive up to approimatel 7 ears old and negative after about 7 ears of age. The percent of adults suffering from migraines increases up to about 7 ears old, then decreases. The units of the derivative are percent of adults suffering from migraines per ear f has horizontal tangents at., 0 and 0.67, True. c is a constant. Section. Rates of Change: Velocit and Marginals (a) $9 billion per ear. 0 (b) (c) (d) (e) (f) Answers will var. $9 billion per ear $7 billion per ear $9 billion per ear $ billion per ear $ billion per ear h, 0,, h Average rate of change: Instantaneous rates of change: h h0 0 0 h0 h 6. f 6 Average rate of change: f f 0 6 Instantaneous rates of change: f 8, f 0 8. f f Average rate of change: f f 6 (, 6) Instantaneous rates of change: f f 6, (, 0) 5 5
11 7 Chapter Differentiation 0. g Average rate of change: g g 0 Instantaneous rates of change: g, g. (a) From t to t :, : The average rate of change is increasing at the greatest rate. From t 5 to t 6: 5, 6: the average rate of change is decreasing at the greatest rate. (b) At t, m 0. Over, 5, the average rate of change is also 0. [Other answers possible.]. (a) Rate of change of heat loss. (b) Hv 0 v 5 v H kilocaloriesmeters H kilocaloriesmeters 6. First leg: 0.75 km in 0 seconds Second leg: 0.75 km in 5 seconds (a) (b) kmsec 7.5 msec kmsec. msec dc dr 0. dc dr dp dp (a) (b) R5 R ,650,56 9 dollars R R R 6 dollars (c) The answers are nearl the same.
12 Section. Rates of Change: Velocit and Marginals 7. P t 5t 0,000 (a) P0 0,000 people P0,70 people P5 5,70 people P0 9,80 people The population is growing quadraticall. dp (b) t 5 (c) P0 5 people per ear P0 9 people per ear P5 7 people per ear P0 9 people per ear The rate of growth is increasing.. dp (a) When (b) When (c) When (d) When (e) When (f) When 50, 75, 00, 5, dp $8.6. dp $77.. dp $7.. dp $68.7. dp 50, $6.76. dp 75, $ (a) TR 0Q 60Q (b) (c) Q TR MR 0Q 60 Model Table ,000, 6,,000, 7 Slope (a) 7 6,000 6, p 6 6, p 8 (demand function) 000 P R C p C ,000 (b) 00, , ,000 (c) P8,000 > 0 positive slope P6,000 < 0 negative slope P P8, dollars per ticket P6, dollars per ticket 0. C v k. Marginal cost: C v 0 v Thus, the marginal cost is independent of the fied cost. dc dq,008,000 6 Q 0 C5 C50 $.9 dc $.9 per unit when Q 50. dq
13 7 Chapter Differentiation 5,000 milesear. C (.0 dollars/gallon) milesgallon 9,500 dollarsear C 9, C C The car that gets 5 miles per gallon will benefit more. 6. (a) (b) f, f 8 5 f f (c) 8, :, :, :, :, :, :, 5: average rate of change average rate of change average rate of change 8 average rate of change average rate of change 0.67 average rate of change 0. average rate of change f f f 0 when,,. Therefore, f has horizontal tangents at, 0,,, and, 0.
14 Section. The Product and Quotient Rules 75 Section. The Product and Quotient Rules. f 5. f 6 0 f 7 7 f 7 6. g g 0 h h f. f f f f 9 5. g 5 g 5 0 g0 Function Rewrite Differentiate Simplif , 0 8 5, 0, 8, 0,., 5. ht t 5 t 7t 6. ht t 5 8t 7 5t t 7t 8t 6 7t 5 8t 7 0t 6 5t 5 5t 8t 6 t 5 5t 8t 7 h p p h p p 6 p h p 6p 5 p 6 p p
15 76 Chapter Differentiation 8. f 0. f f f 6 6. f 5 f 5 f. ht t t 5t 6 6. f ht t 5t 6 t t 5 t 5t 6 f t t t t f t t t t, t Equivalentl, note that ht t t t t, t. 8. f 5 f 5 5 f h,, 9. h h f f f,, (, 5) 5 7 8
16 Section. The Product and Quotient Rules 77. g 5 0, 0, g 0 7 (0, 0) g f f 0 when 0, which implies that 0. Thus, the horizontal tangent line occurs at 0, f f 0 when 0 and ±. Thus, the horizontal tangent line occurs at the points 0,,,, and,. 50. f 5. f f f 5. p p 0 dp p When p p p p p p 9 6 p, 9 dp The initial temperature is (a) When t, (b) When t, (c) When t 5, (d) When t 0, T deg. 0 dt 0 t t 08t 6 t 6t 75t t t 0 dt deghr. 5 dt deghr. 96 dt deghr. 55 dt deghr t 0 0 t t 0 700t t t 0
17 78 Chapter Differentiation 58. dp 50t t t t t 500t (a) When t, (b) When t 0, dp percentda. 5 dp percentda t 87 5t 60. (a) (b) P a b c 6. When 0, P 50: 50 00a 0b c. When, P 60: 60 a b c. When, P 65: 65 96a b c. Solving this sstem, we have Thus, a 5 b 75 8, and c 75., P C , Marginal cost: Average cost: (a) 9 dc 0 87 C (b) Point of intersection: P 5 75 (c) Marginal profit: 0 5 This is the maimum point on the graph of P C dc When 6.68, Thus, the point of intersection is 6.68, At this point average cost is at a minimum. 6. Mt 00t t 8 (a) Mt t 00 00tt t (b) M 98 (c) M M 0.8 M t t 66. Answers will var.
18 Section.5 The Chain Rule 79 Section.5 The Chain Rule. fg u g u fu u 6. 9 u 9 u. fg u g u fu u 8. u u 0. f Quotient Rule (d). f Simple Power Rule (a). (a) First rewrite f 7, then use the Simple Power Rule. 6. (c) Rewrite as f 5 and use the General Power Rule ht t t 8t t. f 6. ft 9t 9 69t 6 9t 6. g 8. g f 5 f 5 5. f g 9,, 5,8 g g 9 96,5 5,8 96,5 96,5 77,56 00,
19 80 Chapter Differentiation 8. f g Point:, f, 6 f g 8 g 5 6 g When, the slope is f and the equation of the tangent line is f f f is never 0.. f f f has a horizontal tangent when 5 f 0. f f f f 6. st t t t t 8. st t t t t t t f f g g
20 Section.5 The Chain Rule 8 5. f f tt tt t t t tt t t t t t t t t gt t t t gt t t 6t t t t t t t t t t t 6t t t t t 6t t 6t t t t t t 9t 6t t t tt t t t 6. g 6. g s,, s 5 s
21 8 Chapter Differentiation Pn 0.5n 5n 5 n 5 P,000 P n 5n 5, ,000 5, k 7. (a) V t When t 0, V 0,000. (b) 0,000 dv 5000t 5000 t When t, (c) When k k 0,000 0 V 0,000 t V 0,000t dv $ per ear. dv t, per ear $ (a) Using the General Power Rule, rt 0.9t.99t 9.085t 6.07t t.87t 78.67t 6.07 (b) (c) rt rt is changing most rapidl near t. (d) is changing the least when rt 0, near t and t True.
22 Section.6 Higher-Order Derivatives 8 Section.6 Higher-Order Derivatives. f. f 0 f 6 f 6 6. f 8. f f 6 gt t gt t gt 9 t7 9t 7 0. f. gt t t f gt 8t f 9 9 gt t t. hs s (s s s 5 s s 6. f hs 5s 8s s f 6 hs 0s s 6s s0s s f f 8. f 0. f f f 0 f f f f 6 6. f 9. f f f5 ft t t ft t t ft t t ft t 5 f t 5 6. g g 0 0 g g g0 8 f 0 6 f
23 8 Chapter Differentiation 0. f. f f f f 6. f 9 6. f 9 9 f 8 0 f 0 when 0. f ( 9 6 f 6 f 6 f 0 6 ± 6 ± f 0. f f f 0 when 0. Note: ±6 are not in the domain of f. f f f 0 f 0 when 0, ±.. (a) st 6t 50 (b) vt st t at vt (c) st 0 when 6t 50, or t sec. (d) v ftsec. st 8.5t 66t vt st 6.50t 66 at st 6.50 t 0 5 st vt at
24 Section.7 Implicit Differentiation f 9 8. f 9 9 f 8 f ' f '' f 0 f f f The degrees of the successive derivatives decrease b. 0 The degrees of the successive derivatives decrease b. 50. (a) st 6t 8t 6 (b) vt st t 8 at vt (c) st 6t 8t 6 0 6t t 0 t t 0 t sec (d) vt t 8 0 when t 8 or t sec. (e) s feet high (f) The position function is a quadratic function, the velocit function is a linear function, and the acceleration function is a constant function. 5. True. The fifth derivative of a fourth degree polnomial is True. The n st derivative of an n th degree polnomial is 0. dc 56. True. where vt C is constant. 0 Section.7 Implicit Differentiation
25 86 Chapter Differentiation At, 0, is undefined. At,, At,, is undefined. At,,.. At,, At,, At 0,, 0. At,,.
26 Section.7 Implicit Differentiation 87. Implicitl: Eplicitl: At =,, d. 9 8 d 0 ± 9 d ± ± 9 9 6± d 9 6 ( ),. Implicitl: Eplicitl: At,, 8 0 ± 7 8. ± 7 ± 7 ± 7 ± 7 = At 0, : At, 5 : 0 m m (0, ) (, 5) At, : 6 5 At 5, :
27 88 Chapter Differentiation 0. 6 At, : m 0 At, : m 0 6 (, ) 0 (, ). p , dp dp dp p 500, p 500 p 500 p p dp dp p dp dp p dp p p dp p p 0 < t 6.8t 78.0t 756.6t.6t 6.8t 78.0t 756.6t 6 t t corresponds to 99. (a) 00 0 The number of cases decrease until around 000 t 0. (b) The graph seems to decrease most rapidl around t 6.7, or during 996. (c) t
28 Section.8 Related Rates 89 Section.8 Related Rates., (a) When and (b) When and d,, d 5, d, d d. 5, (a) When (b) When,, d 0,, and, and 8, d, d, d d dv 6. V dr r dr r, 8r,8. (a) When r 6, (b) When r, dv in min. dv in min. V r, dv r dr If dr is constant, dv is not constant since it is proportional to the square of r. 0. V r h r r r dv r dr 6r (a) When r 6, (b) When r, since h r since dr dv 66 6 in min. dv 6 56 in min.. C 75,000.05, (a) (b) dc 5,000 dollarsweek (c) P R C R dollarsweek dr dp dr dc 5, ,77.5 dollarsweek. A da 6,, 6 6. (a) When (b) When da, 6 6 cm sec. da 0, cm sec., (a) When, 8 5 cmmin. (b) When (c) When (d) When, d, 8 5 cmmin. 0, 0 0 cmmin , 0 0,0 cmmin.
29 90 Chapter Differentiation 8. ft ft sec 0. 6 s ds s d d d s ds ds When s 0, 8 and 0: The speed of the plane is 00 mi/hr mihr. 8 When, 5 and 0. ftsec. 5 6 mi s As 0, increases.. S ds ds $7,500 per week. dp P R C 6. (a) C 58p00 p and 5%r. p C dc 58p00 p dp 5800 pdp dp When and 5, dp $650week. (b) p dp 5800 pdp 00 p dc $.70r p0 70 dc $65r. p60 50, As p 00, C increases without bound.
30 Review Eercises for Chapter 9 Review Eercises for Chapter. Slope. Slope 6. t : slope 8000 thousand per ear per ear 8. (a) At t, ft > gt, so rafter f is traveling faster. t 8: slope,500 thousand per ear per ear t 8: slope 7,500 thousand per ear per ear (b) At t, gt > ft, so rafter g is faster. (c) At t, g is faster. (d) g finishes first, because g completes 0 miles at a lesser value of t. 0. f f f lim 0. f f f lim lim 0 lim lim 0 7 lim 0 f 7 lim 0 f. f f f lim 0 lim 0 lim 0 lim 0 f f f f lim 0 lim 0 8. f f f lim 0 lim 0 f
31 9 Chapter Differentiation 0. f. f. f f f f f f f 8 6. is not differentiable at is not differentiable at. 0. h 9, h 9 h 8 9, 8. f,, f f (, ) , 6. f, 9, 6 f f f 0. f,, f f 8 Average rate of change f f f f f
32 Review Eercises for Chapter 9 7,560 57,09. (a) 997 to 00: 8,050 thousand per ear per ear 7 (b) 997: S7,867 thousand per ear per ear 00: S7 7,56 thousand per ear per ear (c) Answers will var.. T.89t 6.06t 55.6t 509. (a) 50 0 (b) Tt.89t 6.06t 55.6t t.t : T.8 million tons per ear 998: T8 0.5 million tons per ear 00: T. million tons per ear (c) Yes, Tt > 0 for t. The graph is increasing on t. The amount of reccled paper products is increasing. 6. S t, 0 t 8 v t t v P R C p C C C 5 dc R R 50 R dr dp s t t t t t t s 8t t 8t t
33 9 Chapter Differentiation 6. f f 66. f f 68. g g gt gt t t t t 6 t t t t 6 t t t t t t t 7. f f 5 5 f 8 f fs s s fs s 5 s s s s 5 s s 5s s s s 8s g g When L, (a) When (b) When (c) When (d) When V L 6 D 6 D D dv dd D.5D. D 8, D 6, D, D 6, dv.58 6 board ftin. dd dv.56 8 board ftin. dd dv.5 0 board ftin. dd dv.56 8 board ftin. dd 8. f 5 6 f f 60 5
34 Review Eercises for Chapter f 86. f f f f f 6 6 f 8 5 f f f 80 6 f st t t t vt st t t at st 6t 6 t At 8,, At 0,, (a) (b) P R C p C ,500, ,500,000 P (c) P80,000 $9 per unit 5,000, ,000 The maimum profit occurs for 5,50, which means p 0.005,50 $0.50.
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