3 Prove by mathematical induction that, for every positive integer \(n\),
$$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \mathrm { e } ^ { x } \sin x \right) = ( \sqrt { } 2 ) ^ { n } \mathrm { e } ^ { x } \sin \left( x + \frac { 1 } { 4 } n \pi \right)$$
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Question 3:
Answer Marks
Guidance
Answer/Working Marks
Guidance
\(\frac{d}{dx}(e^x \sin x) = e^x \sin x + e^x \cos x\) B1
Differentiates once
\(= \sqrt{2}e^x\left(\frac{1}{\sqrt{2}}\sin x + \frac{1}{\sqrt{2}}\cos x\right)\) M1
Rearranges
\(= \sqrt{2}e^x\sin\left(x + \frac{1}{4}\pi\right) \Rightarrow H_1\) true A1
Shows true for \(n=1\)
\(H_k: \frac{d^k}{dx^k}(e^x \sin x) = (\sqrt{2})^k e^x \sin\left(x + \frac{1}{4}k\pi\right)\) B1
States inductive hypothesis (may be seen by implication)
\(\frac{d^{k+1}}{dx^{k+1}} = (\sqrt{2})^k\left(\sin\left(x+\frac{1}{4}k\pi\right)e^x + e^x\cos\left(x+\frac{1}{4}k\pi\right)\right)\) M1
Differentiates
\(= (\sqrt{2})^{k+1}e^x\left(\frac{1}{\sqrt{2}}\sin\left(x+\frac{1}{4}k\pi\right) + \frac{1}{\sqrt{2}}\cos\left(x+\frac{1}{4}k\pi\right)\right)\) A1
Rearranges
\(= (\sqrt{2})^{k+1}e^x\sin\left(x+\frac{1}{4}(k+1)\pi\right) \Rightarrow H_{k+1}\) true A1
Shows \(H_k \Rightarrow H_{k+1}\)
\(\therefore\) by PMI, true for all positive integers \(n\) (CWO) States conclusion; total [7]
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## Question 3:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{d}{dx}(e^x \sin x) = e^x \sin x + e^x \cos x$ | B1 | Differentiates once |
| $= \sqrt{2}e^x\left(\frac{1}{\sqrt{2}}\sin x + \frac{1}{\sqrt{2}}\cos x\right)$ | M1 | Rearranges |
| $= \sqrt{2}e^x\sin\left(x + \frac{1}{4}\pi\right) \Rightarrow H_1$ true | A1 | Shows true for $n=1$ |
| $H_k: \frac{d^k}{dx^k}(e^x \sin x) = (\sqrt{2})^k e^x \sin\left(x + \frac{1}{4}k\pi\right)$ | B1 | States inductive hypothesis (may be seen by implication) |
| $\frac{d^{k+1}}{dx^{k+1}} = (\sqrt{2})^k\left(\sin\left(x+\frac{1}{4}k\pi\right)e^x + e^x\cos\left(x+\frac{1}{4}k\pi\right)\right)$ | M1 | Differentiates |
| $= (\sqrt{2})^{k+1}e^x\left(\frac{1}{\sqrt{2}}\sin\left(x+\frac{1}{4}k\pi\right) + \frac{1}{\sqrt{2}}\cos\left(x+\frac{1}{4}k\pi\right)\right)$ | A1 | Rearranges |
| $= (\sqrt{2})^{k+1}e^x\sin\left(x+\frac{1}{4}(k+1)\pi\right) \Rightarrow H_{k+1}$ true | A1 | Shows $H_k \Rightarrow H_{k+1}$ |
| $\therefore$ by PMI, true for all positive integers $n$ (CWO) | | States conclusion; total [7] |
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3 Prove by mathematical induction that, for every positive integer $n$,
$$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \mathrm { e } ^ { x } \sin x \right) = ( \sqrt { } 2 ) ^ { n } \mathrm { e } ^ { x } \sin \left( x + \frac { 1 } { 4 } n \pi \right)$$
\hfill \mbox{\textit{CAIE FP1 2013 Q3 [7]}}