SECTION 3.1 DERIVATIVES OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS 181
If , find and . Compare the graphs of and .
SOLUTION
Using the Difference Rule, we have
In Section 2.8 we defined the second derivative as the derivative of , so
The function f and its derivative are graphed in Figure 8. Notice that has a horizon- tal tangent when ; this corresponds to the fact that . Notice also that, for , is positive and is increasing. When , is negative and is decreasing.
At what point on the curve is the tangent line parallel to the
line ?
SOLUTION
Since , we have . Let the x-coordinate of the point in question be a. Then the slope of the tangent line at that point is . This tangent line will be paral- lel to the line if it has the same slope, that is, 2. Equating slopes, we get
Therefore the required point is . (See Figure 9.)
EXAMPLE 9EXAMPLE 8
v
!a, e
a" ! !ln 2, 2"
a ! ln 2 e
a! 2
y ! 2x
e
ay! ! e
xy ! e
xy ! 2x y ! e
xf !!x" f x " 0 f !!x" f
x # 0 x ! 0 f !!0" ! 0
f f !
f $!x" ! d
dx !e
x% 1" ! d
dx !e
x" % d
dx !1" ! e
xf ! f !!x" ! d
dx !e
x% x" ! d
dx !e
x" % d
dx !x" ! e
x% 1
f ! f f $
f ! f !x" ! e
x% x
FIGURE 8 3
_1
1.5 _1.5
f
fª
FIGURE 9
1 1
0 x
2 3 y
y=´
y=2x (ln 2, 2)
1. (a) How is the number edefined?
(b) Use a calculator to estimate the values of the limits
and
correct to two decimal places. What can you conclude about the value of e?
2. (a) Sketch, by hand, the graph of the function , pay- ing particular attention to how the graph crosses the y-axis.
What fact allows you to do this?
(b) What types of functions are and ? Compare the differentiation formulas for and t.
(c) Which of the two functions in part (b) grows more rapidly when x is large?
3–32 Differentiate the function.
3. 4.
5. 6.
7. f!x" ! x3%4x & 6 8. f!t" ! 1.4t5%2.5t2&6.7 F!x" !34x8
f!t" ! 2 %23t
f!x" !e5 f!x" ! 240
ft!x" ! xe f!x" !ex
f!x" !ex
h l 0lim
2.8h%1 lim h
h l 0
2.7h%1 h
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21. 22.
23. 24.
25. 26.
27. 28.
t!x" ! x2!1 % 2x" h!x" ! !x % 2"!2x & 3"
t!t" ! 2t%3#4 B!y" !cy%6
y ! aev& b v & c
v2 H!x" ! !x & x%1"3
k!r" !er&re j!x" ! x2.4&e2.4
t!u" !s2u&s3u y ! x2&4x & 3
sx
y ! sx & x x2 h!u" ! Au3&Bu2&Cu
S!R" ! 4'R2 y ! 3ex& 4
s3x
y !sx !x % 1"
S!p" !sp %p
h!t" !s4t %4et R!a" ! !3a & 1"2
y ! x5#3%x2#3 A!s" ! %12
s5
3.1 Exercises
; Graphing calculator or computer required 1.Homework Hints available at stewartcalculus.com 97909_03_ch03_p172-181 . qk:97909_03_ch03_p172-181 9/21/10 9:42 AM Page 181
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182 CHAPTER 3 DIFFERENTIATION RULES
29. 30.
31. 32.
33–34 Find an equation of the tangent line to the curve at the given point.
33. , 34. ,
35–36 Find equations of the tangent line and normal line to the curve at the given point.
35. , 36. ,
;37–38 Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen.
37. , 38. ,
;39–40 Find . Compare the graphs of and and use them to explain why your answer is reasonable.
39. 40.
;41. (a) Use a graphing calculator or computer to graph the func-
tion in the viewing
rectangle by .
(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of . (See Example 1 in Section 2.8.)
(c) Calculate and use this expression, with a graphing device, to graph . Compare with your sketch in part (b).
;42. (a) Use a graphing calculator or computer to graph the func-
tion in the viewing rectangle
by .
(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of . (See Example 1 in Section 2.8.)
(c) Calculate and use this expression, with a graphing device, to graph . Compare with your sketch in part (b).
43–44 Find the first and second derivatives of the function.
43. 44.
;45–46 Find the first and second derivatives of the function.
Check to see that your answers are reasonable by comparing the graphs of , , and .
45. f!x" ! 2x ! 5x3#4 46. f!x" !ex!x3 f f " f #
f!x" ! 10x10$5x5!x G!r" !sr $ s3r t"
t"!x"
t"
$!8, 8%t!x" !ex!3x2 $!1, 4%
f "
f "!x"
f "
$!3, 5% $!10, 50%
f!x" ! x4!3x3!6x2$7x $ 30
f!x" ! x4!2x3$x2 f!x" ! x5!2x3$x ! 1
f "!x" f f "
y ! 3x2!x3 !1, 2" y ! x !sx !1, 0"
y ! x4$2ex !0, 2" y ! x2!x4 !1, 0"
y !s4x !1, 1" y ! x4$2x2!x !1, 2"
z ! A
y10 $Bey y !ex$1$1
u!s5t $4st5 v!
&
sx $ s31x'
2 47. The equation of motion of a particle is , whereis in meters and is in seconds. Find
(a) the velocity and acceleration as functions of , (b) the acceleration after 2 s, and
(c) the acceleration when the velocity is 0.
48. The equation of motion of a particle is
, where is in meters and is in seconds.
(a) Find the velocity and acceleration as functions of . (b) Find the acceleration after 1 s.
; (c) Graph the position, velocity, and acceleration functions on the same screen.
49. Boyle’s Law states that when a sample of gas is compressed at a constant pressure, the pressure of the gas is inversely proportional to the volume of the gas.
(a) Suppose that the pressure of a sample of air that occupies at is . Write as a function of . (b) Calculate when . What is the meaning
of the derivative? What are its units?
;50. Car tires need to be inflated properly because overinflation or underinflation can cause premature treadware. The data in the table show tire life ( in thousands of miles) for a certain type of tire at various pressures ( in ).
(a) Use a graphing calculator or computer to model tire life with a quadratic function of the pressure.
(b) Use the model to estimate when and when . What is the meaning of the derivative? What are the units? What is the significance of the signs of the derivatives?
51. Find the points on the curve where the tangent is horizontal.
52. For what value of does the graph of have a horizontal tangent?
53. Show that the curve has no tangent line with slope 2.
54. Find an equation of the tangent line to the curve that is parallel to the line .
55. Find equations of both lines that are tangent to the curve and parallel to the line .
;56. At what point on the curve is the tangent line parallel to the line ? Illustrate by graphing the curve and both lines.
57. Find an equation of the normal line to the parabola that is parallel to the line .
y ! x2!5x $ 4 x ! 3y ! 5
t s!t3!3t s
3x ! y ! 5y ! 1 $ 2ex!3x
y ! 1 $ x3 12x ! y ! 1
y ! 1 $ 3x y ! xsx
y ! 2ex$3x $ 5x3
x f!x" !ex!2x
y ! 2x3$3x2!12x $ 1 P ! 40
dL#dP P ! 30 P lb#in2 L
dV#dP P ! 50 kPa
0.106 m3 25%C 50 kPa V P
V P
t
s!t4!2t3$t2!t s t
t
P 26 28 31 35 38 42 45
L 50 66 78 81 74 70 59
97909_03_ch03_p182-191 . qk:97909_03_ch03_p182-191 9/21/10 9:51 AM Page 182
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.
