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}. Represent f(t) using a combination of Heaviside step functions. Use h(t - c) for the Heaviside function h_c(t). Find the Laplace transform F(s) = \mathcal{L} \{ f(t) \}.
Solution
Let's break down this problem step by step.
Given Function
The function is defined in pieces:
\begin{cases} 2, & t < 2 \\ t - 2, & 2 \leq t < 7 \\ 4, & t \geq 7 \end{cases}$$ ### Part (a): Represent $$f(t)$$ Using Heaviside Step Functions To express $$f(t)$$ in terms of Heaviside functions $$h(t - c)$$, let's represent each change in the function at the points $$t = 2$$ and $$t = 7$$. 1. **Base function**: For $$t < 2$$, $$f(t) = 2$$. 2. **Change at $$t = 2$$**: When $$t \geq 2$$, the function changes to $$t - 2$$. We can represent this change as: $$(t - 2 - 2) h(t - 2) = (t - 2) h(t - 2)$$ 3. **Change at $$t = 7$$**: When $$t \geq 7$$, the function changes to 4. We can express this by subtracting $$(t - 2)$$ and adding 4 at $$t = 7$$: $$(4 - (t - 2)) h(t - 7) = (4 - t + 2) h(t - 7)$$ Thus, the Heaviside representation of $$f(t)$$ is: $$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) \}$$ To find $$F(s)$$, we will take the Laplace transform of each term. 1. **Laplace of $$2$$**: $$\mathcal{L}\{2\} = \frac{2}{s}$$ 2. **Laplace of $$(t - 2) h(t - 2)$$**: Using the shift property, $$\mathcal{L}\{(t - a) h(t - a)\} = \frac{e^{-as}}{s^2}$$, we get: $$\mathcal{L}\{(t - 2) h(t - 2)\} = \frac{e^{-2s}}{s^2}$$ 3. **Laplace of $$(4 - (t - 2)) h(t - 7)$$**: Simplify this term as \( (4 - t + 2) = 4 - t + 2 \Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Heaviside step functions
Laplace transform
Formulas
Heaviside step function representation of piecewise functions
Laplace transform of Heaviside functions
Theorems
Laplace transform shifting theorem
Suitable Grade Level
Undergraduate level
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