CAIE Further Paper 2 2023 November — Question 3 6 marks

Exam BoardCAIE
ModuleFurther Paper 2 (Further Paper 2)
Year2023
SessionNovember
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTaylor series
TypeComposition of two standard series
DifficultyChallenging +1.2 This requires composing the standard series for e^x with the series for tanh^(-1)(u), then expanding and collecting terms. While it involves multiple steps (substitution, series manipulation, and careful algebra), it's a fairly standard Further Maths technique with no novel insight required. The question guides students by specifying the form of the answer, making it moderately above average difficulty.
Spec4.07e Inverse hyperbolic: definitions, domains, ranges4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n
3 Find the first three terms in the Maclaurin's series for \(\tanh ^ { - 1 } \left( \frac { 1 } { 2 } e ^ { x } \right)\) in the form \(\frac { 1 } { 2 } \ln a + b x + c x ^ { 2 }\), giving the exact values of the constants \(a , b\) and \(c\).
Question 3:
AnswerMarks Guidance
AnswerMarks Guidance
\(\frac{dy}{dx} = \frac{-\frac{1}{2}e^x}{1-\frac{1}{4}e^{2x}}\)B1
\(\frac{d^2y}{dx^2} = \frac{\left(1-\frac{1}{4}e^{2x}\right)\left(\frac{1}{2}e^x\right)-\frac{1}{2}e^x\left(-\frac{1}{2}e^{2x}\right)}{\left(1-\frac{1}{4}e^{2x}\right)^2}\)B1
\(f'(0) = \frac{2}{3}\), \(f''(0) = \frac{10}{9}\)M1 Evaluates derivatives at \(x=0\)
\(f(0) = \tanh^{-1}\left(\frac{1}{2}\right) = \frac{1}{2}\ln\left(\frac{3}{2}\times 2\right)\)M1 Uses logarithmic form of \(\tanh^{-1}\)
\(\frac{1}{2}\ln 3 + \frac{2}{3}x + \frac{5}{9}x^2\)M1 A1 Applies \(f(x) = f(0) + f'(0)x + \frac{1}{2!}f''(0)x^2\)
6 
## Question 3:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = \frac{-\frac{1}{2}e^x}{1-\frac{1}{4}e^{2x}}$ | B1 | |
| $\frac{d^2y}{dx^2} = \frac{\left(1-\frac{1}{4}e^{2x}\right)\left(\frac{1}{2}e^x\right)-\frac{1}{2}e^x\left(-\frac{1}{2}e^{2x}\right)}{\left(1-\frac{1}{4}e^{2x}\right)^2}$ | B1 | |
| $f'(0) = \frac{2}{3}$, $f''(0) = \frac{10}{9}$ | M1 | Evaluates derivatives at $x=0$ |
| $f(0) = \tanh^{-1}\left(\frac{1}{2}\right) = \frac{1}{2}\ln\left(\frac{3}{2}\times 2\right)$ | M1 | Uses logarithmic form of $\tanh^{-1}$ |
| $\frac{1}{2}\ln 3 + \frac{2}{3}x + \frac{5}{9}x^2$ | M1 A1 | Applies $f(x) = f(0) + f'(0)x + \frac{1}{2!}f''(0)x^2$ |
| | **6** | |

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3 Find the first three terms in the Maclaurin's series for $\tanh ^ { - 1 } \left( \frac { 1 } { 2 } e ^ { x } \right)$ in the form $\frac { 1 } { 2 } \ln a + b x + c x ^ { 2 }$, giving the exact values of the constants $a , b$ and $c$.\\

\hfill \mbox{\textit{CAIE Further Paper 2 2023 Q3 [6]}}
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