7.6 The Fundamental Theorem of Calculus 419{x^2 sin (~) if x # 0 }
- In Chapter 6 we proved that the function f(x) = x
0 ifx=O
is differentiable everywhere, but its derivative f' is discontinuous at 0. (See
Exercise 6.2.17.) Prove that f' is integrable on [0,2/7r] and find f~/11: f'.
[Explain why FTC-I cannot be used here.]
{x^2 sin (I_) if x # 0 }
- As noted in Exercise 5, the function f(x) = x^2 is
0 ifx=O
differentiable everywhere. Let g(x) = f'(x), Vx E JR. Then g is a function
that has an antiderivative everywhere. Nevertheless, show that g is not
integrable on [O, l]. (See Exercise 6.4.31.) Thus, a function can have an
antiderivative over an interval and not be Riemann integrable there.^14 - Prove that Vx E JR, f~ 1 sgn = lx l - 1. [Thus, f: f can exist 'Vx E [a, b],
even when f has no antiderivative on [a, b].J - Consider the function f(x) = { : ~~ ~: ~ < 2}. Find a formula for
3ifx~2
fox f that is valid for all -oo:::;; x:::;; +oo. (See Example 7.6.7.) - Suppose f is integrable on [a, b]. Prove that
(a) if f(x) ~ 0 on [a, b], then F(x) = f: f is monotone increasing there.
(b) if f(x) :::;; 0 on [a, b], then F(x) = f: f is monotone decreasing there. - Suppose f and F are continuous on [a, b] and F(a) = 0. Prove that the
following are equivalent:
(a) F' = f on [a,b].
(b) F(x) =fax f, 'Vx E [a, b]. - Apply the second form of the Fundamental Theorem of Calculus and the
chain rule to find a formula for each of the following, assuming that f is
continuous and g, h are differentiable on the appropriate intervals:
d 1a d r(x)
(a) dx x f (b) dx } a f
(c) -d la f
dx g(x)d 1h(x)
(d) - f
dx g(x)- For an example of a bounded function that has an antiderivative everywhere on [a , b] but
is not Riemann integrable there, see [49], p.107, Example 35. See also [131], Section 9.7.