| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Rearrange to iterative form |
| Difficulty | Standard +0.3 This is a straightforward multi-part question combining standard differentiation (finding a tangent), algebraic manipulation to derive an iteration formula, and mechanical application of fixed-point iteration. Part (a) is routine C3 differentiation; part (b) requires rearranging f'(x)=0 which is shown step-by-step; part (c) is calculator work. Slightly above average due to the iteration topic being less common than basic calculus, but all steps are procedural with no novel insight required. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07m Tangents and normals: gradient and equations1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
\includegraphics{figure_2}
Figure 2 shows part of the curve $C$ with equation $y = f(x)$, where
$$f(x) = 0.5e^x - x^2.$$
The curve $C$ cuts the $y$-axis at $A$ and there is a minimum at the point $B$.
\begin{enumerate}[label=(\alph*)]
\item Find an equation of the tangent to $C$ at $A$. [4]
\end{enumerate}
The $x$-coordinate of $B$ is approximately $2.15$. A more exact estimate is to be made of this coordinate using iterations $x_{n+1} = \ln g(x_n)$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that a possible form for $g(x)$ is $g(x) = 4x$. [3]
\item Using $x_{n+1} = \ln 4x_n$, with $x_0 = 2.15$, calculate $x_1$, $x_2$ and $x_3$. Give the value of $x_3$ to 4 decimal places. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q9 [9]}}