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