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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
What does the First Shifting Theorem state?
💡 Hint: Remember the relationship between multiplication by e and shifting.
Question 2
Easy
If \( f(t) = t \), what is \( \mathcal{L}\{e^{3t} t\} \)?
💡 Hint: Determine the basic Laplace Transform of t first.
Practice 4 more questions and get performance evaluation
Interactive Quizzes
Engage in quick quizzes to reinforce what you've learned and check your comprehension.
Question 1
What is the First Shifting Theorem's main result?
- \\( \\mathcal{L}\\{f(t)\\} = F(s) \\)
- \\( \\mathcal{L}\\{e^{at} f(t)\\} = F(s + a) \\)
- \\( \\mathcal{L}\\{e^{at} f(t)\\} = F(s - a) \\)
💡 Hint: Focus on the direction of the shift.
Question 2
True or False: The condition \( s > a \) must be satisfied for the theorem to apply.
- True
- False
💡 Hint: Think about the implications of the exponential function.
Solve 1 more question and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
Derive the Laplace Transform for \( e^{2t} \cos(5t) \) using the First Shifting Theorem.
💡 Hint: Identify the base transform first, then apply the theorem.
Question 2
If \( g(t) = e^{t} \sin(t) \), find \( \mathcal{L}\{g(t)\} \).
💡 Hint: Remember to isolate the transform of \\( \\sin(t) \\) before applying the exponent.
Challenge and get performance evaluation