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# Suppose that $0 \le f(t) \le Me^{at}$ and $0 \le f^\prime (t) \le Ke^{at}$ for $t \ge 0$, where $f^\prime$ is continuous. If the Laplace transform of $f(t)$ is $F(s)$ and the Laplace transform of $f^\prime (t)$ is $G(s)$ , show that$G(s) = s F(s) - f(0)$ $s > a$

## $\lim _{t \rightarrow \infty} f(t) e^{-s t}=0$ for $s>a \Rightarrow G(s)=0-f(0)+s F(s)=s F(s)-f(0)$ for $s>a$

#### Topics

Integration Techniques

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##### Kristen K.

University of Michigan - Ann Arbor

##### Samuel H.

University of Nottingham

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### Video Transcript

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#### Topics

Integration Techniques

##### Kristen K.

University of Michigan - Ann Arbor

##### Samuel H.

University of Nottingham

Lectures

Join Bootcamp