### Convolution of Two FunctionsIn this exercise, you are tasked with finding the convolution of the functions $ f(t) = e^{-2t} $ and \[g(t) = \begin{cases} 4, & 0 \leq t < 5 \\0, & t \geq 5\end{cases}\]#### Problem StatementCompute the convolution $ (f * g)(t) $.#### ExplanationThe function $ f(t) = e^{-2t} $ is an exponentially decaying function. The function $ g(t) $ is a piecewise function that takes the value 4 for $ 0 \leq t < 5 $ and 0 for $ t \geq 5 $.Convolution involves integrating the product of these functions, considering one of them as a shifted version, usually expressed as:\[(f * g)(t) = \int_{0}^{t} f(\tau) \cdot g(t-\tau) \, d\tau\]Here, integrate over the appropriate intervals based on the definitions of $ g(t) $.This process provides insight into how the shape of one function influences another when they are combined over time.Fill in the result of the convolution in the space provided:\[(f * g)(t) = \underline{\hspace{5cm}}\]

Given data: ft=e-2t gt=4, 0≤t<50, t≥5 The Convolution is given by: f*gt=∫r=0tft-rgr dr…

Answered: 4, 0 5 (f * 9)(t) =

Math

Advanced Math

4, 0 5 (f * 9)(t) =

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

### Convolution of Two Functions

In this exercise, you are tasked with finding the convolution of the functions $ f(t) = e^{-2t} $ and

\[
g(t) =
\begin{cases}
4, & 0 \leq t < 5 \\
0, & t \geq 5
\end{cases}
\]

#### Problem Statement

Compute the convolution $ (f * g)(t) $.

#### Explanation

The function $ f(t) = e^{-2t} $ is an exponentially decaying function. The function $ g(t) $ is a piecewise function that takes the value 4 for $ 0 \leq t < 5 $ and 0 for $ t \geq 5 $.

Convolution involves integrating the product of these functions, considering one of them as a shifted version, usually expressed as:

\[
(f * g)(t) = \int_{0}^{t} f(\tau) \cdot g(t-\tau) \, d\tau
\]

Here, integrate over the appropriate intervals based on the definitions of $ g(t) $.

This process provides insight into how the shape of one function influences another when they are combined over time.

Fill in the result of the convolution in the space provided:

\[
(f * g)(t) = \underline{\hspace{5cm}}
\]

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