328 Functions, graphs, and numbers
where fg(x) dx stands for some particular function whose derivative is 9(x), and
then so determine c that f(a) = d. We can also determine f from the formula(3) f(x) = f(a) +f aXf' (t) dt
in which the integral is a Riemann integral. Determine f in two different (or
superficially different) ways by using (2) and by using (3), and make the results
agree, when(a) f'(x) = 2x, f(2) = 3 (b) f'(x) = sin ax, f(O) = 0
(c) f'(x) = cos ax, f(0) = 0 (d) f'(x) = eaX, f(0) = 1(e) f(x) =z f(2)= 3 (f) f' (x) = /, f(4) = 0
8 Prove that if u and v are functions that have continuous derivatives over
an interval I containing a and x, thenfax u(t)v'(t) dt= u(t)v(t)]a-f ax n(t)u'(t) dt.
Hint: Let FI(x) and F2(x) denote the left and right sides of the formula. Then
show that FI(a) = F2(a) and that Fi(x) = F'2(x) when x is in I.
9 Iff(0) =0and
fi(x) x27
27 + X27'the result of writing formula (2) of Problem 7 as an aid to finding f(1) is rather
(or more) futile, but we can write a version of (3) and undertake to estimate
f(l). Do it.
10 Sketch a graph over the interval 0 5 x S 1 of the function f for which
f(x) = x2. Let e = g. Use your eyes to select a 3 > 0 such that If(X2) - f(x1)j
< e whenever 0 5x1 <= 1, 0 5x2 5 1, and 1x2 - x11 < S. Note that if this f
were the only continuous function, we would not need to work so long to prove
Theorem 5.58.
11 Supposing that f has a continuous derivative over the interval a < x <= b,
show that the functions F and C for whichF(x) = f(a) + fax [1 + f'(t)+ if'(t)!] dtG(x) = fax [1- f' (t) + If'(t)I] dtare both increasing over the interval a 5 x <= b andf(x) = F(x) - G(x).Remark: This problem contains an important idea. It is sometimes useful to
know about the possibility of representing a given function as the difference of
two increasing functions.(^12) Prove the following theorem, which is known as an extended (not gen-
eralized) mean-value theorem or as a Taylor theorem.