Question 8 - A-Level Maths

Exam Board	AQA
Module	FP1 (Further Pure Mathematics 1)
Year	2006
Session	June
Topic	Newton-Raphson method
Type	Intersection of curves via iteration

The function f is defined for all real values of $x$ by $$\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 1$$
1. Express $\mathrm { f } ( 1 + h ) - \mathrm { f } ( 1 )$ in the form $$p h + q h ^ { 2 } + r h ^ { 3 }$$ where $p , q$ and $r$ are integers.
2. Use your answer to part (a)(i) to find the value of $f ^ { \prime } ( 1 )$.
The diagram shows the graphs of $$y = \frac { 1 } { x ^ { 2 } } \quad \text { and } \quad y = x + 1 \quad \text { for } \quad x > 0$$
\includegraphics[max width=\textwidth, alt={}]{e44987a7-2cdf-442a-aecb-abd3e889ecd4-5_643_791_1160_596}
The graphs intersect at the point $P$.
1. Show that the $x$-coordinate of $P$ satisfies the equation $\mathrm { f } ( x ) = 0$, where f is the function defined in part (a).
2. Taking $x _ { 1 } = 1$ as a first approximation to the root of the equation $\mathrm { f } ( x ) = 0$, use the Newton-Raphson method to find a second approximation $x _ { 2 }$ to the root.
  (3 marks)
The region enclosed by the curve $y = \frac { 1 } { x ^ { 2 } }$, the line $x = 1$ and the $x$-axis is shaded on the diagram. By evaluating an improper integral, find the area of this region.
(3 marks)