Chapter 7.2, Problem 78E

Looking ahead (to Chapter 9) Suppose that a function f has derivatives of all orders near x = 0. By the Fundamental Theorem of Calculus,

$f(x)-f(0)={\displaystyle {\int }_0^x{f}^{\prime }(t)dt.}$

1. a. Evaluate the integral using integration by parts to show that

$f(x)=f(0)+x{f}^{\prime }(0)+{\displaystyle {\int }_0^x{f}^{″}(t)(x-t)dt.}$

1. b. Show that integrating by parts n times gives

$\begin{array}{lll}f(x)\hfill & =\hfill & f(0)+f^{\prime }(0)+\frac{1}{2!}{x}^2{f}^{″}(0)+\cdot \cdot \cdot +\frac{1}{n!}{x}^n{f}^{(n)}(0)\hfill \\ \hfill & \hfill & +\frac{1}{n!}{\displaystyle {\int }_0^x{f}^{(n+1)}(t){(x-t)}^{n}dt+\cdot \cdot \cdot .}\hfill \end{array}$

This expression, called the Taylor series for f at x = 0, is revisited in Chapter 9.