The Fundamental Theorem Theory and Examples If ƒ is continuous, we expect
69. The Fundamental Theorem Theory and Examples If ƒ is continuous, we expect
61. Show that if k is a positive constant, then the area between the
1 x+h
x -axis and one arch of the curve y= sin kx is 2 >k .
h :0 h L ƒstd dt x
lim
62. Find to equal ƒ(x), as in the proof of Part 1 of the Fundamental Theo-
1 x t 2 rem. For instance, if ƒstd = cos t , then
63. Suppose Find 2 1 x 1 ƒstd dt = x - 2x + 1 . ƒ(x). 64. x Find ƒ(4) if 1
0 ƒstd dt = x cos px .
The right-hand side of Equation (7) is the difference quotient for the derivative of the sine, and we expect its limit as
h:0 Find the linearization of
x+ 1 Graph cos x for -p … x … 2p . Then, in a different color if ƒsxd = 2 -
9 dt
L 2 1+t
possible, graph the right-hand side of Equation (7) as a function of x for h=
2, 1, 0.5 , and 0.1. Watch how the latter curves con- at x= 1.
verge to the graph of the cosine as h:0.
66. Find the linearization of
T 70. Repeat Exercise 69 for ƒstd = 3t 2 . What is
Graph ƒsxd = 3x 2 for -1…x…1. Then graph the quotient Suppose that ƒ has a positive derivative for all values of x and that ssx + hd 3 -x 3 d as a function of x for and h= 1, 0.5, 0.2 , 0.1. ƒs1d = 0 . Which of the following statements must be true of the
>h
Watch how the latter curves converge to the graph of 3x 2 as function
h:0.
g sxd =
ƒstd dt ? L
0 COMPUTER EXPLORATIONS
Fsxd = Give reasons for your answers. x In Exercises 71–74, let 1 a ƒstd dt for the specified function ƒ a. g is a differentiable function of x.
and interval [a, b]. Use a CAS to perform the following steps and an- swer the questions posed.
b. g is a continuous function of x.
a. Plot the functions ƒ and F together over [a, b].
c. The graph of g has a horizontal tangent at x= 1. b. Solve the equation F¿sxd = 0. What can you see to be true about d. g has a local maximum at x= 1. the graphs of ƒ and F at points where F¿sxd = 0? Is your
e. g has a local minimum at x= 1. observation borne out by Part 1 of the Fundamental Theorem f. The graph of g has an inflection point at x= 1. coupled with information provided by the first derivative? g. The graph of dg >dx crosses the x-axis at x= 1. Explain your answer.
Suppose that ƒ has a negative derivative for all values of x and c. Over what intervals (approximately) is the function F increasing that ƒs1d = 0 . Which of the following statements must be true of
and decreasing? What is true about ƒ over those intervals? the function
d. Calculate the derivative ƒ¿ and plot it together with F. What can
you see to be true about the graph of F at points where hsxd =
ƒstd dt ? ƒ¿sxd = 0 ? Is your observation borne out by Part 1 of the L 0 Fundamental Theorem? Explain your answer.
368
Chapter 5: Integration
71. ƒsxd = x 3 - 4x 2 + 3x, [0, 4] d. Using the information from parts (a)–(c), draw a rough hand- sketch of y = Fsxd over its domain. Then graph F(x) on your
72. ƒsxd = 2x 4 - 17x 3 + 46x 2 - 43x + 12,
c0, 9
2d CAS to support your sketch.
75.
x a= 1, usxd = x 2 , ƒsxd = 21 - x 2
73. ƒsxd = sin 2x cos , [0, 2p] 3 76. a= 0, usxd = x 2 , ƒsxd = 21 - x 2 74. ƒsxd = x cos px , 2 [0, 2p] 77. a= 0, usxd = 1 - x, ƒsxd = x - 2x - 3
- 2x - 3 ƒ. Use a CAS to perform the following steps and answer the questions
u (x)
78. a= 0, usxd = 1 - x 2 , ƒsxd = x 2
In Exercises 75–78, let Fsxd = 1 a ƒstd dt for the specified a, u, and
In Exercises 79 and 80, assume that f is continuous and u(x) is twice- posed.
differentiable.
a. Find the domain of F.
79. d
usxd
ƒstd dt and check your answer using a CAS. Calculate F¿sxd and determine its zeros. For what points in its
domain is F increasing? decreasing?
Calculate F–sxd and determine its zero. Identify the local
2 ƒstd dt dx and check your answer using a CAS. L a extrema and the points of inflection of F.
Calculate
Chapter 5: Integration