Practice Summary (1.10) - Topic 3: First Shifting Theorem - Mathematics - iii (Differential Calculus) - Vol 1

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Lesson 1.10Topic 3: First Shifting Theorem

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6 sample questions

Question 1.
What does the First Shifting Theorem state?
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Remember the relationship between multiplication by e and shifting.
Question 2.
If $f(t) = t$ , what is $\mathcal{L}\{e^{3t} t\}$ ?
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Determine the basic Laplace Transform of t first.
3Question 3.
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) \$
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Focus on the direction of the shift.
4Question 4.
True or False: The condition $s > a$ must be satisfied for the theorem to apply.
- True
- False
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Think about the implications of the exponential function.
Question 5.
Derive the Laplace Transform for $e^{2t} \cos(5t)$ using the First Shifting Theorem.
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Identify the base transform first, then apply the theorem.
Question 6.
If $g(t) = e^{t} \sin(t)$ , find $\mathcal{L}\{g(t)\}$ .
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Remember to isolate the transform of \$ \\sin(t) \$ before applying the exponent.

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