Math Problem Statement
Find the Laplace transform F(s) = L{f(t)} of the function f(t) by splitting the integral into three pieces, where f(t) = {-3t for 0 <= t < 3, 2 - 4t for 3 <= t < 4, and 4 for t >= 4}.
Solution
To find the Laplace transform of the given piecewise function , we can split the integral into three parts according to the intervals provided:
Given:
-3t & 0 \leq t < 3 \\ 2 - 4t & 3 \leq t < 4 \\ 4 & t \geq 4 \end{cases}$$ ### Step 1: Laplace Transform Setup The Laplace transform $$\mathcal{L} \{ f(t) \}$$ is calculated as: $$F(s) = \int_0^{\infty} f(t) e^{-st} \, dt$$ We break it up based on the intervals: $$F(s) = \int_0^3 (-3t)e^{-st} \, dt + \int_3^4 (2 - 4t)e^{-st} \, dt + \int_4^{\infty} 4e^{-st} \, dt$$ ### Step 2: Calculate Each Integral Separately #### Integral 1: $$\int_0^3 (-3t)e^{-st} \, dt$$ 1. **Set up the integral**: $$\int_0^3 (-3t)e^{-st} \, dt = -3 \int_0^3 t e^{-st} \, dt$$ 2. **Use integration by parts** with $$u = t$$ and $$dv = e^{-st} dt$$. 3. Solve to find this term. #### Integral 2: $$\int_3^4 (2 - 4t)e^{-st} \, dt$$ 1. **Set up the integral**: $$\int_3^4 (2 - 4t)e^{-st} \, dt = 2 \int_3^4 e^{-st} \, dt - 4 \int_3^4 t e^{-st} \, dt$$ 2. Separate the terms and solve each using appropriate techniques. #### Integral 3: $$\int_4^{\infty} 4e^{-st} \, dt$$ 1. **Set up the integral**: $$\int_4^{\infty} 4e^{-st} \, dt = 4 \int_4^{\infty} e^{-st} \, dt$$ 2. This is a standard Laplace transform of a constant starting at $$t = 4$$, and can be solved accordingly. ### Solution After evaluating each integral, enter the expressions in the form: $$F(s) = \text{(Expression for 1st integral)} + \text{(Expression for 2nd integral)} + \text{(Expression for 3rd integral)}$$ Would you like further step-by-step calculations on any specific interval? --- 1. How do we apply Laplace transforms to piecewise functions? 2. What is the role of the Heaviside function in transforming piecewise functions? 3. How does integration by parts help in solving Laplace transforms? 4. What are common Laplace transforms for functions involving constants and linear terms? 5. Why do we split the integral by intervals in this piecewise function? **Tip:** When dealing with piecewise functions, remember to split each interval and apply transformations independently to simplify the overall process.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Laplace Transform
Piecewise Functions
Integration by Parts
Formulas
Laplace Transform formula: F(s) = L{f(t)} = ∫_0^∞ f(t)e^(-st) dt
Integration by parts formula: ∫ u dv = uv - ∫ v du
Theorems
Laplace Transform for piecewise functions
Suitable Grade Level
Undergraduate - Calculus