Math Problem Statement

give manually soved step by step solution in detail

Give manually solved step-by-step solution in detail for the provided problem involving Heaviside step functions and Laplace transforms.

Solution

To solve this problem, let's go through it in detail.

Problem Summary

  1. We are given a piecewise function f(t)f(t) represented in the graph.
  2. The task has two parts:
    • Part (a): Express f(t)f(t) using Heaviside step functions.
    • Part (b): Find the Laplace transform F(s)=L{f(t)}F(s) = \mathcal{L}\{f(t)\}.

Solution

Step 1: Analyzing the Function f(t)f(t) from the Graph

From the graph:

  • f(t)f(t) starts at t=0t = 0 and is f(0)=0f(0) = 0.
  • From t=0t = 0 to t=4t = 4, the function increases linearly to f(4)=4f(4) = 4.
  • From t=4t = 4 to t=8t = 8, f(t)f(t) is constant at 4.
  • For t>8t > 8, f(t)f(t) drops back to 0.

This suggests that f(t)f(t) can be defined in three sections:

  1. f(t)=t4f(t) = \frac{t}{4} for 0≤t<40 \leq t < 4.
  2. f(t)=4f(t) = 4 for 4≤t<84 \leq t < 8.
  3. f(t)=0f(t) = 0 for t≥8t \geq 8.

Step 2: Expressing f(t)f(t) Using Heaviside Step Functions

The Heaviside function, h(t−a)h(t - a), is defined as:

\begin{cases} 0 & \text{for } t < a \\ 1 & \text{for } t \geq a \end{cases}$$ We can use this to "turn on" or "turn off" parts of the function at specific points. 1. **First segment (from $$0$$ to $$4$$):** $$f(t) = \frac{t}{4}$$ for $$0 \leq t < 4$$. - This part can be written as $$\frac{t}{4} \cdot h(t)$$. 2. **Second segment (from $$4$$ to $$8$$):** $$f(t) = 4$$ for $$4 \leq t < 8$$. - We subtract the value at $$t = 4$$ and add it back with a constant to match this section: $$4 \cdot h(t - 4) - 4 \cdot h(t - 8)$$ 3. **Third segment (for $$t \geq 8$$):** The function becomes zero, which we accomplish by subtracting the previous terms after $$t = 8$$. Thus, we can combine these terms to represent $$f(t)$$ as: \[ f(t) = \frac{t}{4} \cdot h(t) + (4 - \frac{t}{4}) h(t - 4) -

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Heaviside Step Functions
Laplace Transform

Formulas

Heaviside step function: h(t - a)
Laplace Transform: L{f(t)} = F(s)

Theorems

Laplace Transform of Piecewise Functions
Shifting Theorem for Laplace Transforms

Suitable Grade Level

College Level

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