Chapter 4.2, Problem 45E

(a)

To determine

To Graph: Triangle made by binding of graphs of $y=x,y=0\text{ and }x=2$

(b)

To determine

To Prove: $0<1\left(\frac{2}{n}\right)<\mathrm{...}<\left(n-1\right)\left(\frac{2}{n}\right)<n\left(\frac{2}{n}\right)$ are the endpoints.

(c)

To determine

To Prove: The relation $s\left(n\right)={\displaystyle \sum _{i=1}^n\left[\left(i-1\right)\left(\frac{2}{n}\right)\right]\left(\frac{2}{n}\right)}$

(d)

To determine

To Prove: The relation $S\left(n\right)={\displaystyle \sum _{i=1}^n\left[\left(i\right)\left(\frac{2}{n}\right)\right]\left(\frac{2}{n}\right)}$

(e)

To determine

The values for $s\left(n\right)={\displaystyle \sum _{i=1}^n\left[\left(i-1\right)\left(\frac{2}{n}\right)\right]}\left(\frac{2}{n}\right)$ and $S\left(n\right)={\displaystyle \sum _{i=1}^n\left[\left(i\right)\left(\frac{2}{n}\right)\right]}\left(\frac{2}{n}\right)$ at $n=5,10,50,100$

(f)

To determine

To Prove: That $\underset{n\to \infty }{\lim }s\left(n\right)=\underset{n\to \infty }{\lim }S\left(n\right)=2$