Question

In problemsfind either f(S)or f(t),as indicated$\mathit{L}\mathbf{\left\{}{\mathbf{t}}^{\mathbf{3}}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}\mathbf{\right\}}$

Short answer

find either f(S)or f(t),as indicated

$\begin{array}{rcl}L\left\{t^3{e}^{-2t}\right\}& =& \underset{s\to s=(-2)}{\lim }L\left\{t^3\right\}\\ & =& \frac{6}{{\left(s+2\right)}^4}\end{array}$

Step-by-step solution

Step1: definition of indicated

The definition of indicate is to refer to, point to, or show something, or to make a brief statement acknowledging something. When you point to Susie, you are indicating.

$\begin{array}{rcl}& & L\left\{t^3{e}^{-2t}\right\}\\ & =& \underset{s\to s=(-2)}{\lim }L\left\{t^3\right\}\\ & =& \underset{s\to s=(-2)}{\lim }\frac{3!}{s^4}\\ & =& \frac{6}{{\left(s+2\right)}^4}\\ & & \\ & & \end{array}$

Step: first translation theorem.  

if $L\left\{f\left(t\right)\right\}=f\left(S\right)$and a is a any real number. then

$L\left\{e^{at}f\left(t\right)\right\}=F\left(s-a\right)$

Hence,$\frac{6}{{\left(s+2\right)}^4}$