Calculus + Induction

Here’s the question we considered on Thursday. Now that you’ve got a solution for parts 1 and 2, complete part 3 as a homework assignment due on Tuesday, the 25th of October.

The function \(f\)is defined by \(f(x) = e^x \sin x\).

  1. Show that \(f”(x) = 2e^x \sin \left(x+\frac{\pi}{2}\right)\).
  2. Obtain a similar expression for \(f^{(4)}(x)\).
  3. Suggest an expression for \(f^{(2n)}(x), n \in \mathbb{Z}^+\), and prove your conjecture using mathematical induction.

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