Math Problem Statement
show steps to take the laplace transform of a piecewise function that is u(t)=t from 0 to 1 u(t)=1 from 1 to 2 and u(t)=3-t from 2 to 3 and u(t) = 0 otherwise
Solution
To take the Laplace transform of a piecewise function, you need to express the function using Heaviside step functions (also called unit step functions), and then apply the Laplace transform to each term individually.
Let be the given piecewise function:
\begin{cases} t, & 0 \leq t < 1 \\ 1, & 1 \leq t < 2 \\ 3 - t, & 2 \leq t < 3 \\ 0, & t \geq 3 \end{cases}$$ ### Steps to take the Laplace transform of the piecewise function: #### 1. **Rewrite the function using the Heaviside function:** The Heaviside function $$u_c(t)$$ is defined as: $$u_c(t) = \begin{cases} 0, & t < c \\ 1, & t \geq c \end{cases}$$ Using Heaviside functions, you can express the piecewise function as: $$u(t) = t u_0(t) - t u_1(t) + u_1(t) - u_2(t) + (3 - t) u_2(t) - (3 - t) u_3(t)$$ Simplify the expression by grouping terms: $$u(t) = t u_0(t) + (1 - t) u_1(t) + (3 - t) u_2(t)$$ #### 2. **Break the function into intervals using the Heaviside function:** This can now be written in terms of simpler Laplace transforms: $$u(t) = \begin{cases} t, & 0 \leq t < 1 \\ 1, & 1 \leq t < 2 \\ 3 - t, & 2 \leq t < 3 \end{cases}$$Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Laplace Transform
Piecewise Functions
Heaviside Step Function
Formulas
Laplace transform of a function: L{f(t)} = ∫(0 to ∞) e^(-st) f(t) dt
Heaviside function: u_c(t) = 0 for t < c, 1 for t ≥ c
Theorems
Superposition Theorem in Laplace Transforms
Suitable Grade Level
Undergraduate (Advanced Calculus/Engineering Math)