Chapter 11.1, Problem 82E

a.

To determine

To show: The Fundamental Theorem of Calculus can be written in the form $f\left(x\right)=f\left(a\right)+{\displaystyle {\int }_a^x{f}^{\prime }\left(t\right)}dt$.

b.

To determine

To show: The result $f\left(x\right)=f\left(a\right)+\left(x-a\right)f^{\prime }\left(a\right)+{\displaystyle {\int }_a^x\left(x-t\right)f^{″}\left(t\right)dt}$ by using integration by parts.

c.

To determine

To show: The n integration by parts gives

$f\left(x\right)=f\left(a\right)+\frac{f^{\prime }\left(a\right)}{1!}\left(x-a\right)+\frac{f^{″}\left(a\right)}{2!}{\left(x-a\right)}^2+\cdots +\frac{f^n\left(t\right)}{n!}{\left(x-t\right)}^n+{\displaystyle {\int }_a^x\frac{f^{n+1}\left(t\right)}{n!}{\left(x-t\right)}^{n}dt.}$

d.

To determine

To show: The $R_n$ can be expressed in the form $R_n\left(x\right)=\frac{f^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)!}{\left(x-a\right)}^{n+1}$ where c is between a and x.