97909_03_ch03_p172-181.

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SECTION 3.1 DERIVATIVES OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS 181

v EXAMPLE 8 If f 共x兲 苷 e x ⫺ x, find f ⬘ and f ⬙. Compare the graphs of f and f ⬘.

3 SOLUTION Using the Difference Rule, we have

d x d x d
f ⬘共x兲 苷 共e ⫺ x兲 苷 共e 兲 ⫺ 共x兲 苷 e x ⫺ 1
f dx dx dx

fª
In Section 2.8 we defined the second derivative as the derivative of f ⬘, so

_1.5 1.5 d x d x d
f ⬙共x兲 苷 共e ⫺ 1兲 苷 共e 兲 ⫺ 共1兲 苷 e x
dx dx dx
_1
The function f and its derivative f ⬘ are graphed in Figure 8. Notice that f has a horizon-
FIGURE 8 tal tangent when x 苷 0; this corresponds to the fact that f ⬘共0兲 苷 0. Notice also that,
for x ⬎ 0, f ⬘共x兲 is positive and f is increasing. When x ⬍ 0, f ⬘共x兲 is negative and f is
decreasing.
y
3 EXAMPLE 9 At what point on the curve y 苷 e x is the tangent line parallel to the
(ln 2, 2) line y 苷 2x ?
2
SOLUTION Since y 苷 e x, we have y⬘ 苷 e x. Let the x-coordinate of the point in question
y=2x be a. Then the slope of the tangent line at that point is e a. This tangent line will be paral-
1
y=´
lel to the line y 苷 2x if it has the same slope, that is, 2. Equating slopes, we get

0 1 x ea 苷 2 a 苷 ln 2

FIGURE 9 Therefore the required point is 共a, e a 兲 苷 共ln 2, 2兲. (See Figure 9.)

3.1 Exercises

1. (a) How is the number e defined? 9. t共x兲 苷 x 2 共1 ⫺ 2x兲 10. h共x兲 苷 共x ⫺ 2兲共2x ⫹ 3兲
(b) Use a calculator to estimate the values of the limits
11. t共t兲 苷 2t ⫺3兾4 12. B共 y兲 苷 cy⫺6
2.7 h ⫺ 1 2.8 h ⫺ 1
lim and lim
hl0 h hl0 h 12
13. A共s兲 苷 ⫺ 14. y 苷 x 5兾3 ⫺ x 2兾3
s5
correct to two decimal places. What can you conclude
about the value of e? 15. R共a兲 苷 共3a ⫹ 1兲2 16. h共t兲 苷 s
4
t ⫺ 4et
2. (a) Sketch, by hand, the graph of the function f 共x兲 苷 e x, pay- 17. S共 p兲 苷 sp ⫺ p 18. y 苷 sx 共x ⫺ 1兲
ing particular attention to how the graph crosses the y-axis.
What fact allows you to do this? 4
19. y 苷 3e x ⫹ 20. S共R兲 苷 4␲ R 2
(b) What types of functions are f 共x兲 苷 e x and t共x兲 苷 x e ? 3
sx
Compare the differentiation formulas for f and t.
sx ⫹ x
(c) Which of the two functions in part (b) grows more rapidly 21. h共u兲 苷 Au 3 ⫹ Bu 2 ⫹ Cu 22. y 苷
when x is large? x2
x 2 ⫹ 4x ⫹ 3
3–32 Differentiate the function. 23. y 苷 24. t共u兲 苷 s2 u ⫹ s3u
sx
3. f 共x兲 苷 2 40 4. f 共x兲 苷 e 5
25. j共x兲 苷 x 2.4 ⫹ e 2.4 26. k共r兲 苷 e r ⫹ r e
5. f 共t兲 苷 2 ⫺ t 6. F 共x兲 苷 x
2 3 8
3 4
b c
27. H共x兲 苷 共x ⫹ x ⫺1兲3 28. y 苷 ae v ⫹ ⫹
7. f 共x兲 苷 x 3 ⫺ 4x ⫹ 6 8. f 共t兲 苷 1.4t 5 ⫺ 2.5t 2 ⫹ 6.7 v v2

; Graphing calculator or computer required 1. Homework Hints available at stewartcalculus.com

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182 CHAPTER 3 DIFFERENTIATION RULES

29. u 苷 s
5
t ⫹ 4 st 5 30. v 苷 冉 sx ⫹ 3
s
1
x
冊 2 47. The equation of motion of a particle is s 苷 t 3 ⫺ 3t , where s
is in meters and t is in seconds. Find
(a) the velocity and acceleration as functions of t,
A
31. z 苷 10 ⫹ Be y 32. y 苷 e x⫹1 ⫹ 1 (b) the acceleration after 2 s, and
y
(c) the acceleration when the velocity is 0.
48. The equation of motion of a particle is
33–34 Find an equation of the tangent line to the curve at the s 苷 t 4 ⫺ 2t 3 ⫹ t 2 ⫺ t, where s is in meters and t is in
given point. seconds.
33. y 苷 s
4
x, 共1, 1兲 34. y 苷 x 4 ⫹ 2x 2 ⫺ x, 共1, 2兲 (a) Find the velocity and acceleration as functions of t .
(b) Find the acceleration after 1 s.
; (c) Graph the position, velocity, and acceleration functions
35–36 Find equations of the tangent line and normal line to the on the same screen.
curve at the given point.
49. Boyle’s Law states that when a sample of gas is compressed
35. y 苷 x 4 ⫹ 2e x , 共0, 2兲 36. y 苷 x 2 ⫺ x 4, 共1, 0兲 at a constant pressure, the pressure P of the gas is inversely
proportional to the volume V of the gas.
(a) Suppose that the pressure of a sample of air that occupies
; 37–38 Find an equation of the tangent line to the curve at the 0.106 m 3 at 25⬚ C is 50 kPa. Write V as a function of P.
given point. Illustrate by graphing the curve and the tangent line (b) Calculate dV兾dP when P 苷 50 kPa. What is the meaning
on the same screen.
of the derivative? What are its units?
37. y 苷 3x 2 ⫺ x 3, 共1, 2兲 38. y 苷 x ⫺ sx , 共1, 0兲
; 50. Car tires need to be inflated properly because overinflation or
underinflation can cause premature treadware. The data in the
table show tire life L ( in thousands of miles) for a certain
; 39– 40 Find f ⬘共x兲. Compare the graphs of f and f ⬘ and use them
to explain why your answer is reasonable. type of tire at various pressures P ( in lb兾in2 ).

39. f 共x兲 苷 x 4 ⫺ 2x 3 ⫹ x 2 40. f 共x兲 苷 x 5 ⫺ 2x 3 ⫹ x ⫺ 1

P 26 28 31 35 38 42 45

L 50 66 78 81 74 70 59
; 41. (a) Use a graphing calculator or computer to graph the func-
tion f 共x兲 苷 x ⫺ 3x ⫺ 6x ⫹ 7x ⫹ 30 in the viewing
4 3 2

rectangle 关⫺3, 5兴 by 关⫺10, 50兴. (a) Use a graphing calculator or computer to model tire life
(b) Using the graph in part (a) to estimate slopes, make with a quadratic function of the pressure.
a rough sketch, by hand, of the graph of f ⬘. (See (b) Use the model to estimate dL兾dP when P 苷 30 and when
Example 1 in Section 2.8.) P 苷 40. What is the meaning of the derivative? What are
(c) Calculate f ⬘共x兲 and use this expression, with a graphing the units? What is the significance of the signs of the
device, to graph f ⬘. Compare with your sketch in part (b). derivatives?
; 42. (a) Use a graphing calculator or computer to graph the func- 51. Find the points on the curve y 苷 2x 3 ⫹ 3x 2 ⫺ 12x ⫹ 1
tion t共x兲 苷 e x ⫺ 3x 2 in the viewing rectangle 关⫺1, 4兴 where the tangent is horizontal.
by 关⫺8, 8兴.
(b) Using the graph in part (a) to estimate slopes, make 52. For what value of x does the graph of f 共x兲 苷 e x ⫺ 2x have a
a rough sketch, by hand, of the graph of t⬘. (See horizontal tangent?
Example 1 in Section 2.8.) 53. Show that the curve y 苷 2e x ⫹ 3x ⫹ 5x 3 has no tangent line
(c) Calculate t⬘共x兲 and use this expression, with a graphing with slope 2.
device, to graph t⬘. Compare with your sketch in part (b).
54. Find an equation of the tangent line to the curve y 苷 x sx
43– 44 Find the first and second derivatives of the function. that is parallel to the line y 苷 1 ⫹ 3x.
43. f 共x兲 苷 10x 10 ⫹ 5x 5 ⫺ x 44. G 共r兲 苷 sr ⫹ s
3
r 55. Find equations of both lines that are tangent to the curve
y 苷 1 ⫹ x 3 and parallel to the line 12x ⫺ y 苷 1.

; 56. At what point on the curve y 苷 1 ⫹ 2e ⫺ 3x is the tangent

x
; 45– 46 Find the first and second derivatives of the function.
Check to see that your answers are reasonable by comparing the line parallel to the line 3x ⫺ y 苷 5? Illustrate by graphing
graphs of f , f ⬘, and f ⬙. the curve and both lines.

45. f 共x兲 苷 2 x ⫺ 5x 3兾4 46. f 共x兲 苷 e x ⫺ x 3 57. Find an equation of the normal line to the parabola
y 苷 x 2 ⫺ 5x ⫹ 4 that is parallel to the line x ⫺ 3y 苷 5.

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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SECTION 3.1 DERIVATIVES OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS 183

58. Where does the normal line to the parabola y 苷 x ⫺ x 2 at the

point (1, 0) intersect the parabola a second time? Illustrate with
69. (a) For what values of x is the function f 共x兲 苷 x 2 ⫺ 9 ⱍ ⱍ
differentiable? Find a formula for f ⬘.
a sketch. (b) Sketch the graphs of f and f ⬘.
59. Draw a diagram to show that there are two tangent lines to the
parabola y 苷 x 2 that pass through the point 共0, ⫺4兲. Find the
70. Where is the function h共x兲 苷 x ⫺ 1 ⫹ x ⫹ 2 differenti- ⱍ ⱍ ⱍ ⱍ
able? Give a formula for h⬘ and sketch the graphs of h and h⬘.
coordinates of the points where these tangent lines intersect the
parabola. 71. Find the parabola with equation y 苷 ax 2 ⫹ bx whose tangent
line at (1, 1) has equation y 苷 3x ⫺ 2.
60. (a) Find equations of both lines through the point 共2, ⫺3兲 that
are tangent to the parabola y 苷 x ⫹ x. 2
72. Suppose the curve y 苷 x 4 ⫹ ax 3 ⫹ bx 2 ⫹ cx ⫹ d has a tan-
(b) Show that there is no line through the point 共2, 7兲 that is gent line when x 苷 0 with equation y 苷 2x ⫹ 1 and a tangent
tangent to the parabola. Then draw a diagram to see why. line when x 苷 1 with equation y 苷 2 ⫺ 3x. Find the values
61. Use the definition of a derivative to show that if f 共x兲 苷 1兾x, of a, b, c, and d.
then f ⬘共x兲 苷 ⫺1兾x 2. (This proves the Power Rule for the 73. For what values of a and b is the line 2x ⫹ y 苷 b tangent to
case n 苷 ⫺1.) the parabola y 苷 ax 2 when x 苷 2?
62. Find the nth derivative of each function by calculating the first
74. Find the value of c such that the line y 苷 2 x ⫹ 6 is tangent to
3
few derivatives and observing the pattern that occurs.
the curve y 苷 csx .
(a) f 共x兲 苷 x n (b) f 共x兲 苷 1兾x
75. Let
63. Find a second-degree polynomial P such that P共2兲 苷 5,
P⬘共2兲 苷 3, and P ⬙共2兲 苷 2.

64. The equation y ⬙ ⫹ y⬘ ⫺ 2y 苷 x is called a differential equa-2

f 共x兲 苷 再 x2
mx ⫹ b
if x 艋 2
if x ⬎ 2
tion because it involves an unknown function y and its deriv-
Find the values of m and b that make f differentiable
atives y⬘ and y ⬙. Find constants A, B, and C such that the
everywhere.
function y 苷 Ax 2 ⫹ Bx ⫹ C satisfies this equation. (Differen-
tial equations will be studied in detail in Chapter 9.) 76. A tangent line is drawn to the hyperbola xy 苷 c at a point P.
65. Find a cubic function y 苷 ax ⫹ bx ⫹ cx ⫹ d whose graph 3 2 (a) Show that the midpoint of the line segment cut from this
has horizontal tangents at the points 共⫺2, 6兲 and 共2, 0兲. tangent line by the coordinate axes is P.
(b) Show that the triangle formed by the tangent line and the
66. Find a parabola with equation y 苷 ax 2 ⫹ bx ⫹ c that has coordinate axes always has the same area, no matter where
slope 4 at x 苷 1, slope ⫺8 at x 苷 ⫺1, and passes through the P is located on the hyperbola.
point 共2, 15兲.
x 1000 ⫺ 1

再
67. Let 77. Evaluate lim .
xl1 x⫺1
x2 ⫹ 1 if x ⬍ 1
f 共x兲 苷
x⫹1 if x 艌 1 78. Draw a diagram showing two perpendicular lines that intersect
on the y-axis and are both tangent to the parabola y 苷 x 2.
Is f differentiable at 1? Sketch the graphs of f and f ⬘.
Where do these lines intersect?
68. At what numbers is the following function t differentiable?

再
79. If c ⬎ 2 , how many lines through the point 共0, c兲 are normal
1

2x if x 艋 0 lines to the parabola y 苷 x 2 ? What if c 艋 12 ?

t共x兲 苷 2x ⫺ x 2 if 0 ⬍ x ⬍ 2
80. Sketch the parabolas y 苷 x 2 and y 苷 x 2 ⫺ 2x ⫹ 2. Do you
2⫺x if x 艌 2
think there is a line that is tangent to both curves? If so, find its
Give a formula for t⬘ and sketch the graphs of t and t⬘. equation. If not, why not?

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.