Math Problem Statement
Consider the function:
\( f(t) = \begin{cases} t^2, & 0 \leq t < 2 \\ 7, & t \geq 2 \end{cases}
\)
(a) Represent \( f(t) \) using a combination of Heaviside step functions. Use \( h(t - c) \) for the Heaviside function.
(b) Find the Laplace transform \( F(s) = \mathcal{L} \{ f(t) \} \).
Solution
Let's work through each part of this problem step-by-step.
Part (a): Represent using the Heaviside step function
The function is defined as:
\begin{cases} t^2 & \text{for } 0 \leq t < 2, \\ 7 & \text{for } t \geq 2. \end{cases}$$ To represent $$f(t)$$ using the Heaviside step function $$h(t - c)$$, we can rewrite $$f(t)$$ in a piecewise form that transitions from $$t^2$$ to 7 at $$t = 2$$. The Heaviside step function $$h(t - 2)$$ "activates" at $$t = 2$$. We can use it to modify $$f(t)$$ as follows: $$f(t) = t^2 + (7 - t^2) \cdot h(t - 2)$$ Here's how this expression works: - For $$0 \leq t < 2$$, $$h(t - 2) = 0$$, so $$f(t) = t^2$$. - For $$t \geq 2$$, $$h(t - 2) = 1$$, making $$f(t) = t^2 + (7 - t^2) = 7$$. So, the answer to part (a) is: $$f(t) = t^2 + (7 - t^2) \cdot h(t - 2).$$ ### Part (b): Find the Laplace transform $$F(s) = \mathcal{L}\{f(t)\}$$ Now, let's find the Laplace transform of $$f(t)$$ as defined in part (a). #### Step-by-Step Laplace Transform Calculation 1. **Laplace transform of $$t^2$$**: $$\mathcal{L}\{t^2\} = \frac{2}{s^3}.$$ 2. **Laplace transform of $$(7 - t^2) \cdot h(t - 2)$$**: Using the **second shifting theorem** (i.e., $$\mathcal{L}\{f(t - a) h(t - a)\} = e^{-as} \mathcal{L}\{f(t)\}$$ where $$f(t) = 7 - t^2$$ and $$a = 2$$), we get: $$\mathcal{L}\{(7 - t^2) h(t - 2)\} = e^{-2s} \cdot \mathcal{L}\{7 - (t + 2)^2\}.$$ Let's break this down: 1. \( 7 - (t + 2)^2 = 7 - (t^2 + 4t + 4) = -t^2 - 4t + 3Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Heaviside Step Function
Laplace Transform
Formulas
Heaviside function representation
Laplace transform using second shifting theorem
Theorems
Second Shifting Theorem
Suitable Grade Level
Undergraduate Mathematics
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