| Exam Board | CAIE |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2020 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Exponential or trigonometric base functions |
| Difficulty | Standard +0.3 This is a guided, multi-part question on Maclaurin series with explicit scaffolding. Part (a) walks through the derivative using logarithmic differentiation, part (b) is immediate from (a), and part (c) applies the standard Maclaurin formula. While Taylor series is a Further Maths topic, the question requires only routine application of given hints with no problem-solving or insight needed. |
| Spec | 1.07l Derivative of ln(x): and related functions4.08a Maclaurin series: find series for function |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\ln y = x\ln 2 \Rightarrow \frac{1}{y}\frac{dy}{dx} = \ln 2\) | M1 A1 | |
| \(\frac{dy}{dx} = y\ln 2 = 2^x \ln 2\) | A1 | AG |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{d^2y}{dx^2} = 2^x(\ln 2)^2\) | B1 | |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y(0) = 1\), \(y'(0) = \ln 2\), \(y''(0) = (\ln 2)^2\) | B1 | |
| \(2^x = y(0) + y'(0)x + \frac{y''(0)}{2!}x^2 + \ldots = 1 + (\ln 2)x + \frac{(\ln 2)^2}{2}x^2 + \ldots\) | M1 A1 |
**Question 2(a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\ln y = x\ln 2 \Rightarrow \frac{1}{y}\frac{dy}{dx} = \ln 2$ | M1 A1 | |
| $\frac{dy}{dx} = y\ln 2 = 2^x \ln 2$ | A1 | **AG** |
| | **3** | |
---
**Question 2(b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{d^2y}{dx^2} = 2^x(\ln 2)^2$ | B1 | |
| | **1** | |
## Question 2(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y(0) = 1$, $y'(0) = \ln 2$, $y''(0) = (\ln 2)^2$ | B1 | |
| $2^x = y(0) + y'(0)x + \frac{y''(0)}{2!}x^2 + \ldots = 1 + (\ln 2)x + \frac{(\ln 2)^2}{2}x^2 + \ldots$ | M1 A1 | |
---2 It is given that $y = 2 ^ { x }$.
\begin{enumerate}[label=(\alph*)]
\item By differentiating $\ln y$ with respect to $x$, show that $\frac { \mathrm { dy } } { \mathrm { dx } } = 2 ^ { \mathrm { x } } \ln 2$.
\item Write down $\frac { d ^ { 2 } y } { d x ^ { 2 } }$.
\item Hence find the first three terms in the Maclaurin's series for $2 ^ { X }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 2 2020 Q2 [7]}}