Chapter 7.3, Problem 38E

To determine

(a)

To verify:

The initial value theorem $f(0)=\underset{s\to \infty }{\lim }sL\left\{f\right\}\left(s\right)$ in function $f\left(t\right)=1$.

To determine

(b)

To verify:

The initial value theorem $f(0)=\underset{s\to \infty }{\lim }sL\left\{f\right\}\left(s\right)$ in function $f\left(t\right)=e^{\text{t}}$.

To determine

(c)

To verify:

The initial value theorem $f(0)=\underset{s\to \infty }{\lim }sL\left\{f\right\}\left(s\right)$ in function $f\left(t\right)=e^{-\text{t}}$.

To determine

(d)

To verify:

The initial value theorem $f(0)=\underset{s\to \infty }{\lim }sL\left\{f\right\}\left(s\right)$ in function $f\left(t\right)=\cos t$.

To determine

(e)

To verify:

The initial value theorem $f(0)=\underset{s\to \infty }{\lim }sL\left\{f\right\}\left(s\right)$ in function $f\left(t\right)=\sin t$.

To determine

(f)

To verify:

The initial value theorem $f(0)=\underset{s\to \infty }{\lim }sL\left\{f\right\}\left(s\right)$ in function $f\left(t\right)=t^2$.

To determine

(g)

To verify:

The initial value theorem $f(0)=\underset{s\to \infty }{\lim }sL\left\{f\right\}\left(s\right)$ in function $f\left(t\right)=t\cos t$.