Math Problem Statement
Find the Laplace Transform of f(t)=0 0≤t<3 = 4 t≥3
Solution
The Laplace Transform of a piecewise function is found by taking the transform of each piece of the function within its respective interval. The given function is:
\begin{cases} 0, & 0 \leq t < 3 \\ 4, & t \geq 3 \end{cases}$$ ### Step 1: Express the function using the Heaviside (Unit Step) function To handle piecewise functions, we can rewrite $$f(t)$$ using the Heaviside step function $$u(t)$$. The function $$f(t)$$ can be expressed as: $$f(t) = 4 \cdot u(t - 3)$$ Where $$u(t - 3)$$ shifts the function so that it becomes active (non-zero) at $$t = 3$$. ### Step 2: Apply the Laplace Transform The Laplace Transform of $$u(t - a) \cdot f(t - a)$$ is given by: $$\mathcal{L}\{u(t - a) \cdot f(t - a)\} = e^{-as} \cdot \mathcal{L}\{f(t)\}$$ In this case, $$f(t - 3) = 4$$, and the Laplace Transform of a constant $$4$$ is: $$\mathcal{L}\{4\} = \frac{4}{s}$$ Thus, the Laplace Transform of $$f(t)$$ is: $$\mathcal{L}\{f(t)\} = 4 \cdot \frac{e^{-3s}}{s}$$ ### Final Answer: The Laplace Transform of the given function is: $$\mathcal{L}\{f(t)\} = \frac{4e^{-3s}}{s}$$ Do you have any questions or would you like further details? Here are 5 related questions: 1. How do you find the Laplace Transform of other step functions? 2. How does the Laplace Transform handle discontinuous functions in general? 3. What are the properties of the Heaviside function in Laplace Transforms? 4. How can Laplace Transforms be used to solve differential equations? 5. What is the inverse Laplace Transform of $$\frac{4e^{-3s}}{s}$$? **Tip:** Remember that the Heaviside function simplifies the analysis of piecewise functions in Laplace Transforms.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Laplace Transform
Piecewise Functions
Heaviside Step Function
Formulas
Laplace Transform of f(t): \( \mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) dt \)
Heaviside Step Function: \( u(t - a) \)
Laplace Transform of Heaviside Function: \( \mathcal{L}\{u(t - a)f(t - a)\} = e^{-as} \cdot \mathcal{L}\{f(t)\} \)
Theorems
Properties of Laplace Transforms
Heaviside Step Function in Laplace Transform
Suitable Grade Level
Undergraduate (Calculus Level)
Related Recommendation
Laplace Transform of a Piecewise Function Using Heaviside and Time Shift
Laplace Transform of a Piecewise Function Involving the Heaviside Function
Laplace Transform of Piecewise Function f(t) = 0 for t < 4 and t^2 - 8t + 22 for t ≥ 4
Laplace Transform of a Piecewise Function f(t) with Shift at t=3
Laplace Transform of a Piecewise Function with Defined Intervals