Math Problem Statement

Consider the function: \[ f(t) = \begin{cases} 2, & t < 2, \\ t - 2, & 2 \leq t < 7, \\ 4, & t \geq 7. \end{cases} \] (a) Represent \( f(t) \) using a combination of Heaviside step functions. Use \( h(t - c) \) for the Heaviside function \( h_c(t) \). (b) Find the Laplace transform \( F(s) = \mathcal{L}\{f(t)\} \).

Solution

Let's tackle this step-by-step.

Part (a): Represent f(t)f(t) using Heaviside step functions

The function f(t)f(t) is defined in pieces, so we need to rewrite it using the Heaviside step function h(t−c)h(t - c), which "activates" the function at t=ct = c.

Given:

2, & t < 2, \\ t - 2, & 2 \leq t < 7, \\ 4, & t \geq 7. \end{cases}$$ We can represent each piece with a Heaviside function that "turns on" the appropriate parts of $$f(t)$$ at each boundary. 1. For $$t < 2$$: $$f(t) = 2$$. 2. For $$2 \leq t < 7$$: $$f(t) = t - 2$$. 3. For $$t \geq 7$$: $$f(t) = 4$$. The Heaviside function representation of $$f(t)$$ is: $$f(t) = 2 + (t - 2 - 2) h(t - 2) + (4 - (t - 2)) h(t - 7)$$ Simplifying this, we get: \[ f(t) = 2 + (t - 4) h(t - 2) +

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Heaviside Step Functions
Laplace Transform

Formulas

Heaviside function: h(t - c) = 0 for t < c, h(t - c) = 1 for t ≥ c
Laplace transform of a Heaviside function multiplied by a shifted function

Theorems

Properties of Laplace Transforms
Piecewise Function Representation

Suitable Grade Level

Undergraduate - Calculus and Differential Equations

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