5 It is given that \(y = ( 1 + x ) ^ { 2 } \ln ( 1 + x )\). Find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\).
Prove by mathematical induction that, for every integer \(n \geqslant 3\),
$$\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } } = ( - 1 ) ^ { n - 1 } \frac { 2 ( n - 3 ) ! } { ( 1 + x ) ^ { n - 2 } }$$
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Question 5:
Part (i):
Answer Marks
Guidance
Answer/Working Marks
Guidance
\(y' = 2(1+x)\ln(1+x)+(1+x)\) B1
Differentiates once
\(y'' = 2\ln(1+x)+3\) B1
Differentiates twice
\(y''' = \frac{2}{1+x}\) B1
Differentiates three times; allow B1\(\checkmark\) if constant term in previous line incorrect
Part (ii):
Answer Marks
Guidance
Answer/Working Marks
Guidance
\(\frac{d^3y}{dx^3} = \frac{(-1)^2\cdot 2\cdot 0!}{1+x} = \frac{2}{1+x} \Rightarrow H_3\) is true B1
Proves base case
\(H_k: \frac{d^k y}{dx^k} = \frac{(-1)^{k-1}\cdot 2\cdot(k-3)!}{(1+x)^{k-2}}\) for some \(k\) B1
States inductive hypothesis
\(\frac{d^{k+1}y}{dx^{k+1}} = (-1)^{k-1}\cdot 2(k-3)!\cdot(-1)(k-2)(1+x)^{-(k-1)}\) M1
Differentiates
\(= \frac{(-1)^k\cdot 2\cdot(k-2)!}{(1+x)^{k-1}} \Rightarrow H_{k+1}\) is true A1
Proves inductive step
Hence by PMI \(H_n\) is true for all integers \(\geq 3\) A1
States conclusion
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## Question 5:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y' = 2(1+x)\ln(1+x)+(1+x)$ | B1 | Differentiates once |
| $y'' = 2\ln(1+x)+3$ | B1 | Differentiates twice |
| $y''' = \frac{2}{1+x}$ | B1 | Differentiates three times; allow B1$\checkmark$ if constant term in previous line incorrect |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{d^3y}{dx^3} = \frac{(-1)^2\cdot 2\cdot 0!}{1+x} = \frac{2}{1+x} \Rightarrow H_3$ is true | B1 | Proves base case |
| $H_k: \frac{d^k y}{dx^k} = \frac{(-1)^{k-1}\cdot 2\cdot(k-3)!}{(1+x)^{k-2}}$ for some $k$ | B1 | States inductive hypothesis |
| $\frac{d^{k+1}y}{dx^{k+1}} = (-1)^{k-1}\cdot 2(k-3)!\cdot(-1)(k-2)(1+x)^{-(k-1)}$ | M1 | Differentiates |
| $= \frac{(-1)^k\cdot 2\cdot(k-2)!}{(1+x)^{k-1}} \Rightarrow H_{k+1}$ is true | A1 | Proves inductive step |
| Hence by PMI $H_n$ is true for all integers $\geq 3$ | A1 | States conclusion |
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5 It is given that $y = ( 1 + x ) ^ { 2 } \ln ( 1 + x )$. Find $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$.
Prove by mathematical induction that, for every integer $n \geqslant 3$,
$$\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } } = ( - 1 ) ^ { n - 1 } \frac { 2 ( n - 3 ) ! } { ( 1 + x ) ^ { n - 2 } }$$
\hfill \mbox{\textit{CAIE FP1 2013 Q5 [8]}}