| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2009 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Solve exponential equation via iteration |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on standard C3 topics. Part (a) requires routine differentiation and solving f'(x)=0. Parts (b) and (c) involve mechanical application of a given iterative formula and interval verification—both are textbook exercises requiring careful arithmetic but no problem-solving insight or novel approaches. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| \(f'(x) = 3e^x + 3xe^x\) | M1 A1 | |
| \(3e^x + 3xe^x = 3e^x(1+x) = 0\) | M1 A1 | |
| \(x = -1\) | M1 A1 | |
| \(f(-1) = -3e^{-1} - 1\) | B1 | (5) |
| Answer | Marks | Guidance |
|---|---|---|
| \(x_1 = 0.2596\) | B1 | |
| \(x_2 = 0.2571\) | B1 | |
| \(x_3 = 0.2578\) | B1 | (3) |
| Answer | Marks |
|---|---|
| Choosing \((0.25755, 0.25765)\) or an appropriate tighter interval | M1 |
| \(f(0.25755) = -0.000379...\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Change of sign (and continuity) ⟹ root ∈ \((0.25755, 0.25765)\) | cso | A1 |
**(a)**
$f'(x) = 3e^x + 3xe^x$ | M1 A1
$3e^x + 3xe^x = 3e^x(1+x) = 0$ | M1 A1
$x = -1$ | M1 A1
$f(-1) = -3e^{-1} - 1$ | B1 | (5)
**(b)**
$x_1 = 0.2596$ | B1
$x_2 = 0.2571$ | B1
$x_3 = 0.2578$ | B1 | (3)
**(c)**
Choosing $(0.25755, 0.25765)$ or an appropriate tighter interval | M1
$f(0.25755) = -0.000379...$ | A1
$f(0.25765) = 0.000109...$
Change of sign (and continuity) ⟹ root ∈ $(0.25755, 0.25765)$ | cso | A1 | (3) [11]
(⟹ $x = 0.2576$, is correct to 4 decimal places)
Note: $x = 0.25762765...$ is accurate
---7.
$$f ( x ) = 3 x e ^ { x } - 1$$
The curve with equation $y = \mathrm { f } ( x )$ has a turning point $P$.
\begin{enumerate}[label=(\alph*)]
\item Find the exact coordinates of $P$.
The equation $\mathrm { f } ( x ) = 0$ has a root between $x = 0.25$ and $x = 0.3$
\item Use the iterative formula
$$x _ { n + 1 } = \frac { 1 } { 3 } \mathrm { e } ^ { - x _ { n } }$$
with $x _ { 0 } = 0.25$ to find, to 4 decimal places, the values of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$.
\item By choosing a suitable interval, show that a root of $\mathrm { f } ( x ) = 0$ is $x = 0.2576$ correct to 4 decimal places.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2009 Q7 [11]}}