| Exam Board | CAIE |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2020 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Use series to approximate integral |
| Difficulty | Standard +0.3 This is a straightforward application of standard Further Maths techniques: finding a Maclaurin series by differentiation (routine for e^(-x²)), then integrating term-by-term with simple limits. The small upper limit (1/5) makes arithmetic manageable. While it requires multiple steps, each is mechanical with no conceptual challenges or novel insights required. |
| Spec | 4.08a Maclaurin series: find series for function4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f'(x) = -2xe^{-x^2}\) | B1 | Finds first derivative |
| \(f''(x) = 4x^2e^{-x^2} - 2e^{-x^2}\) | B1 | Finds second derivative |
| \(f(0) = 1 \quad f'(0) = 0 \quad f''(0) = -2\) | M1 | Evaluates derivatives at zero |
| \(e^{-x^2} = 1 - x^2\) | M1 A1 | |
| Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\displaystyle\int_0^1 5 - x^2 \, dx = \left[x - \frac{1}{3}x^3\right]_0^{\frac{1}{5}} = \frac{74}{375}\) | M1 A1 | Substitutes \(1 - x^2\) or better |
| Total: 2 |
**Question 1:**
**Part (a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f'(x) = -2xe^{-x^2}$ | B1 | Finds first derivative |
| $f''(x) = 4x^2e^{-x^2} - 2e^{-x^2}$ | B1 | Finds second derivative |
| $f(0) = 1 \quad f'(0) = 0 \quad f''(0) = -2$ | M1 | Evaluates derivatives at zero |
| $e^{-x^2} = 1 - x^2$ | M1 A1 | |
| **Total: 5** | | |
**Part (b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\displaystyle\int_0^1 5 - x^2 \, dx = \left[x - \frac{1}{3}x^3\right]_0^{\frac{1}{5}} = \frac{74}{375}$ | M1 A1 | Substitutes $1 - x^2$ or better |
| **Total: 2** | | |1
\begin{enumerate}[label=(\alph*)]
\item By differentiating $\mathrm { e } ^ { - x ^ { 2 } }$, find the Maclaurin's series for $\mathrm { e } ^ { - x ^ { 2 } }$ up to and including the term in $x ^ { 2 }$.
\item Deduce an approximation to $\int _ { 0 } ^ { \frac { 1 } { 5 } } \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x$, giving your answer as a rational fraction in its lowest terms.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 2 2020 Q1 [7]}}