6.3 - Proof of the Theorem
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Practice Questions
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What is the definition of the Laplace Transform?
💡 Hint: What does it do to a time domain function?
What theorem allows the interchange of integration order?
💡 Hint: Recall its significance in multivariable calculus.
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Interactive Quizzes
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What is the formula for the Laplace Transform of an integral?
💡 Hint: Recall how integration affects the transformation.
True or False: The Laplace Transform can only be applied to continuous functions.
💡 Hint: Think about the types of functions we defined earlier.
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Challenge Problems
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Prove the theorem that L{f(t)} = F(s) implies L{∫f(τ)dτ} = F(s)/s by evaluating in detail each step including usage of Fubini's theorem.
💡 Hint: Carefully track changes in limits during integration and remind yourself of the definition throughout.
How would you apply the theorem to analyze a capacitor charging described by a differential equation?
💡 Hint: Remember to define your variables clearly before applying any transforms.
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