156
CHAPTER 20
20.37
Verify the Mean-Value Theorem for integrals, for the function f(x) = x
2 + 5 on [0, 3].
and
But, 8 = x
2 + 5 when x =
20.38
Evaluate
Let u = 2x + l, jt = («-l)/2, du = 2dx. Then
20.39
Evaluate
Let M = sin x, du = cos x dx. Then
20.40
Using only geometric reasoning, calculate the average value of /(.v) =
on [0,2].
To find
Thus, y2 = -(x-l)2 + l, (*-l)2 + y2 = l.This is the equation of the circle with center at (1,0) and radius
let
y =
1. Hence, the graph of y = f(x) between * = 0 and x = 2 is the upper half of that circle. So, the
integral
is the area Till of that semicircle, and, therefore, the average value is 7r/4.
20.41
If, in a period of time T, an object moves along the *-axis from x, to x 2 , find a formula for its average velocity.
Let the initial and final time be t t and t 2 , with 7"= /, — /,. The average velocity is
Thus, as usual, the average velocity is the distance (more precisely, the displacement) divided by the time.
20.42
Prove that, if/is continuous on [a, b], D,[j*f(t) dt] =f(x).
Then
Let
By the MeanValue Theorem for integrals, the last integral is
for some x* between x and x + Ax. Hence,
and
But as
and, by the continuity of/,
20.43
Find
20.44
Find
[by Problem 20.42j.
by Problems 20.42 and 20.43 and the Chain Rule.
20.45
Calculate
By Problem 20.42, the derivative is
20.46
Calculate
By Problem 20.43, the derivative is —sin
3 x.
y
2 = -(x
2 - 2x) = -[(* - I)
2 - 1)1 = -(x -I)
2 + 1.
sin
5 x cos x dx.
sin
5 x cos x. dx = w
5 du =