Practice Topic 3: First Shifting Theorem (1.1) - Topic 3: First Shifting Theorem

Practice: Topic 3: First Shifting Theorem

Lesson 1.1Topic 3: First Shifting Theorem

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6 sample questions

Question 1.
What is the First Shifting Theorem?
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Think about how exponential functions influence the transforms.
Question 2.
Identify a condition for applying the theorem.
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Check the relationship between s and a.
3Question 3.
What is the result of applying the First Shifting Theorem to $f(t) = \sin(b t)$ ?
- A. \$ \\mathcal{L}\\{e^{at} sin(bt)\\} = \\frac{b}{(s-a)^2 + b^2} \$
- B. \$ \\mathcal{L}\\{e^{at} sin(bt)\\} = \\frac{a}{(s+a)^2 + b^2} \$
- C. \$ \\mathcal{L}\\{e^{at} sin(bt)\\} = \\frac{b}{(s + a)^2 + b^2} \$
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Recall the basic formula for sin under the shift.
4Question 4.
True or False: The First Shifting Theorem can only be applied when s > a.
- True
- False
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Consider the implications of the theorem's requirements.
Question 5.
Derive the Laplace Transform of $e^{0.5t} * t^2$ using the First Shifting Theorem.
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Start with the basic transform and replace \$ s \$ in the final result.
Question 6.
Find a Laplace Transform for $e^{4t} * an(t)$ using theoretical explanations.
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Consider the behavior of tan in Laplace transforms and apply the theorem correctly.

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