Using iteration in a longer problem | CAIE 9709 Pure Mathematics 2 Notes

Chapters

Pure Mathematics 2

Mathematics9709

Carry one variable through the whole problem

Define the required variable and note its allowed interval, units or geometric meaning.
Use the earlier topic to form the equation. For a stationary point, this usually means setting $\dfrac{dy}{dx}=0$ .
Rearrange to the exact form $x=F(x)$ requested by the question and locate the relevant root.
Iterate from the stated starting value, keeping full calculator precision and showing the required terms.
Return to the context: state whether the result is an $x$ -coordinate, angle, length or another quantity, and reject any value outside the allowed interval.

Finish a differentiation problem with iteration

Worked example

Find the x-coordinate of a maximum point

In 9709/22/M/J/24 Q6, the curve is:

y=\frac{\ln(2x+1)}{x+3}

Write the numerator as $u=\ln(2x+1)$ and the denominator as $v=x+3$ . The chain rule gives $u'=2/(2x+1)$ , while $v'=1$ . Substituting these into the quotient rule $\,dy/dx=(vu'-uv')/v^2$ gives the derivative.

At the maximum, $dy/dx=0$ . On the relevant positive interval the denominator $v^2$ is not zero, so the numerator $vu'-uv'$ must be zero. Substitution gives:

\frac{2(x+3)}{2x+1}=\ln(2x+1)

x=\frac{x+3}{\ln(2x+1)}-0.5

The sign check from the first lesson places the root between 2.5 and 3. Using 2.5 as a starting value gives:

Value	6 significant figures
$x_0$	$2.50000$
$x_1$	$2.56961$
$x_2$	$2.56917$
$x_3$	$2.56917$

Therefore the $x$ -coordinate of the maximum point is $2.569$ to 4 significant figures. (9709/22/M/J/24 Q6(a)–(d))

Keep the numerical answer tied to the context

A bracket may contain more than one root. Use the graph, interval or context to identify the required one.
If a starting value is given, use it exactly. If none is given, use a sensible value from the located interval.
For a trigonometric iteration, use the angle unit shown in the question and check that each inverse-trigonometric output stays in the required interval.
A decimal without its meaning is unfinished. State the requested coordinate or quantity at the requested precision.

Your turnIndependent transfer [5 marks]

The curves $y=3e^{-2x}$ and $y=1+x$ meet at $P$ for $0<x<1$ . Show that the $x$ -coordinate satisfies $x=\tfrac12\ln\!\left(\tfrac3{1+x}\right)$ . Starting with $x_0=0.4$ , find the coordinates of $P$ to 3 significant figures. Show iterations to 5 significant figures.

Show worked answer

3e^{-2x}=1+x

e^{-2x}=\frac{1+x}{3}

x=\frac12\ln\left(\frac3{1+x}\right)

Value	5 significant figures
$x_0$	$0.40000$
$x_1$	$0.38107$
$x_2$	$0.38788$
$x_3$	$0.38542$
$x_4$	$0.38630$
$x_5$	$0.38599$
$x_6$	$0.38610$
$x_7$	$0.38606$
$x_8$	$0.38607$

Since $P$ also lies on $y=1+x$ , $y=1.38607\ldots$ .

Answer

P=(0.386,\ 1.39)