Math Problem Statement
Consider the function: f(t) = { 2, t < 2; t - 2, 2 ≤ t < 7; 4, t ≥ 7 }. 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) = L{f(t)}.
Solution
Let's solve this step by step.
Part (a): Represent using Heaviside step functions
The function is defined in piecewise form:
\begin{cases} 2, & t < 2 \\ t - 2, & 2 \le t < 7 \\ 4, & t \ge 7 \end{cases}$$ To represent $$f(t)$$ using the Heaviside step function $$h(t - c)$$, we rewrite each part of the function in terms of shifts at points $$t = 2$$ and $$t = 7$$: 1. **For $$t < 2$$:** $$f(t) = 2$$. 2. **For $$2 \le t < 7$$:** $$f(t)$$ changes to $$t - 2$$, so we adjust by adding $$(t - 2 - 2)h(t - 2)$$. 3. **For $$t \ge 7$$:** $$f(t)$$ becomes 4, so we need to adjust for this change at $$t = 7$$. The Heaviside function representation is: $$f(t) = 2 + (t - 2 - 2) h(t - 2) + (4 - (t - 2)) h(t - 7)$$ Simplify the terms to get: $$f(t) = 2 + (t - 2)h(t - 2) + (4 - (t - 2)) h(t - 7)$$ ### Part (b): Find the Laplace transform $$F(s) = \mathcal{L}\{f(t)\}$$ The Laplace transform of $$f(t)$$, represented with Heaviside functions, can be calculated as: \[ F(s)Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Heaviside Step Functions
Laplace Transform
Formulas
Heaviside function: h(t - c)
Laplace transform of Heaviside function h(t - c): L{f(t)h(t - c)} = e^{-cs} * L{f(t + c)}
Theorems
Laplace Transform
Suitable Grade Level
Undergraduate (Calculus and Differential Equations)
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