Math Problem Statement
Solution
To represent the given piecewise function using the Heaviside step functions, we can decompose each segment by turning on and off the function parts using the Heaviside function , where is 0 for and 1 for .
The function is given as:
0 & \text{if } t < 3, \\ 2t - 2 & \text{if } 3 \leq t < 4, \\ 6 & \text{if } 4 \leq t < 8, \\ 0 & \text{if } t \geq 8. \end{cases}$$ ### Step-by-Step Conversion to Heaviside Function Notation 1. **For the interval $$3 \leq t < 4$$:** The function is $$2t - 2$$. - This can be represented as $$(2t - 2) \cdot h(t - 3) \cdot (1 - h(t - 4))$$. 2. **For the interval $$4 \leq t < 8$$:** The function value is constant at 6. - This can be represented as $$6 \cdot h(t - 4) \cdot (1 - h(t - 8))$$. 3. **Combining the terms:** To express $$f(t)$$ in a single expression using Heaviside functions, we add each term: $$f(t) = (2t - 2) \cdot h(t - 3) \cdot (1 - h(t - 4)) + 6 \cdot h(t - 4) \cdot (1 - h(t - 8)).$$ ### Final Representation Thus, the function $$f(t)$$ can be expressed as: $$f(t) = (2t - 2) \cdot h(t - 3) \cdot (1 - h(t - 4)) + 6 \cdot h(t - 4) \cdot (1 - h(t - 8)).$$ This expression fully represents $$f(t)$$ in terms of Heaviside step functions. Would you like further clarification on the Heaviside function properties, or how each term was derived? Here are some follow-up questions to deepen understanding: 1. How does the Heaviside function help in representing piecewise functions? 2. Can the above expression be simplified further for easier interpretation? 3. How would the function change if an additional interval was added? 4. What would happen if the boundaries of the intervals changed? 5. How could you represent a more complex function with multiple intervals using Heaviside functions? **Tip:** The Heaviside step function is useful for turning portions of a function on and off, which is particularly helpful in transforming piecewise-defined functions into a single expression.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Heaviside Step Function
Function Transformation
Formulas
Heaviside function: h(t - c) = 0 for t < c and 1 for t ≥ c
Piecewise function decomposition with Heaviside functions
Theorems
-
Suitable Grade Level
Undergraduate
Related Recommendation
Laplace Transform of a Piecewise Function Using Heaviside Step Functions
Laplace Transform and Heaviside Step Function for Piecewise Function
Heaviside Step Functions and Laplace Transform for Piecewise Function \( f(t) \)
Representing f(t) with Heaviside Step Functions
Using Heaviside Functions and Laplace Transforms for a Piecewise Function