Chapter 7.4, Problem 38E

To determine

(a)

To find:

Theinverse function of $t\left(F\right)$.

To determine

(b)

To justify:

The approximation $M=\frac{1}{N}{\displaystyle \sum _{j=1}^{N}t\left(\frac{j}{N}\right)}$.

To determine

(c)

To verify:

The givenformula ${\displaystyle \int \ln xdx=x\ln x-x}$ and show that

$M=\underset{c\to 0}{\lim }\left(\frac{1}{k}+\frac{1}{k}\left(c\ln c-c\right)\right)$ for $c>0$.

To determine

(d)

To verify:

The given formula ${\displaystyle \int \ln xdx=x\ln x-x}$ and show that

$M=\underset{c\to 0}{\lim }\left(\frac{1}{k}+\frac{1}{k}\left(c\ln c-c\right)\right)$ for $c>0$.