Numerical Methods (Edexcel A Level Maths: Pure): Exam Questions

The table below shows corresponding values of $x$ and $y$ for $y=\sqrt{\frac{x}{1+x}}$

The values of $y$ are given to 4 significant figures.

Use the trapezium rule, with all the values of $y$ in the table, to find an estimate for

${\int }_{0.5}^{2.5}\sqrt{\frac{x}{1+x}}\mathrm{d}x$

giving your answer to 3 significant figures.

Using your answer to part (a), deduce an estimate for ${\int }_{0.5}^{2.5}\sqrt{\frac{9x}{1+x}}\mathrm{d}x$

Given that

${\int }_{0.5}^{2.5}\sqrt{\frac{9x}{1+x}}\mathrm{d}x=4.535$ to 4 significant figures

comment on the accuracy of your answer to part (b).

The curve with equation $y=2\ln \left(8-x\right)$ meets the line $y=x$ at a single point, $x=\alpha$.

Show that $3<\alpha <4$.

Figure 2 shows the graph of $y=2\ln \left(8-x\right)$and the graph of $y=x$.

A student uses the iteration formula

$x_{n+1}=2\ln \left(8-x_n\right),n\in \mathrm{\mathbb{N}}$

in an attempt to find an approximation for $\alpha$.

Using the graph and starting with $x_1=4$, determine whether or not this iteration formula can be used to find an approximation for $\alpha$, justifying your answer.

The curve with equation $y=\mathrm{f}\left(x\right)$ where

$\mathrm{f}\left(x\right)=x^2+\ln \left(2x^2-4x+5\right)$

has a single turning point at $x=\alpha$.

Show that $\alpha$ is a solution of the equation $2x^3-4x^2+7x-2=0$.

The iterative formula

$x_{n+1}=\frac{1}{7}\left(2+4{x_n}^2-2{x_n}^3\right)$

is used to find an approximate value for $\alpha$.

Starting with $x_1=0.3$, calculate, giving each answer to 4 decimal places,

(i) the value of $x_2$

(ii) the value of $x_4$

Using a suitable interval and a suitable function that should be stated, show that $\alpha$ is $0.341$ to 3 decimal places.

A continuous curve has equation $y=\mathrm{f}\left(x\right)$.

The table shows corresponding values of $x$ and $y$ for this curve, where $a$ and $b$ are constants.

The trapezium rule is used, with all the $y$ values in the table, to find an approximate area under the curve between $x=3$ and $x=4$

Given that this area is $17.59$, show that $a+2b=51$

Given also that the sum of all the $y$ values in the table is $97.2$, find the value of $a$ and the value of $b$.

The equation $2x^3+x^2-1=0$ has exactly one real root.

Show that, for this equation, the Newton-Raphson formula can be written

$x_{n+1}=\frac{4{x_n}^3+{x_n}^2+1}{6{x_n}^2+2x_n}$

Using the formula given in part (a) with $x_1=1$, find the values of $x_2$ and $x_3$

Explain why, for this question, the Newton-Raphson method cannot be used with $x_1=0$

$\mathrm{f}\left(x\right)=x^3-4x+1$

Show that the equation $\mathrm{f}\left(x\right)=0$ has a root $\alpha$ in the interval $1<x<2$.

The iterative formula

$x_{n+1}=\sqrt{4-\frac{1}{x_n}}$

is used with $x_1=1.5$ to find an approximate value for $\alpha$.

Calculate the values of $x_2$ and $x_3$, giving your answers to 4 decimal places.

Show that if the iteration converges to a limit $\alpha$, then $\alpha$ must be a root of the equation $\mathrm{f}\left(x\right)=0$.

Figure 2 shows a sketch of part of the curve with equation $y=\mathrm{f}\left(x\right)$ where

$\mathrm{f}\left(x\right)=8\sin \left(\frac{1}{2}x\right)-3x+9x>0$

and $x$ is measured in radians.

The point $P$, shown in Figure 2, is a local maximum point on the curve.

Using calculus and the sketch in Figure 2, find the $x$ coordinate of $P$, giving your answer to 3 significant figures.

The curve crosses the $x$-axis at $x=\alpha$, as shown in Figure 2.

Given that, to 3 decimal places, $\mathrm{f}\left(4\right)=4.274$ and $\mathrm{f}\left(5\right)=-1.212$, explain why α must lie in the interval $\left[4,5\right]$

Taking $x_0=5$ as a first approximation to $\alpha$, apply the Newton-Raphson method once to $\mathrm{f}\left(x\right)$ to obtain a second approximation to $\alpha$.

Show your method and give your answer to 3 significant figures.

The diagram below shows part of the graph $y=\mathrm{f}\left(x\right)$ where

$\mathrm{f}\left(x\right)=2x\cos \left(3x\right)-1$

Show that a solution to the equation $\mathrm{f}\left(x\right)=0$ exists in the interval $1.6<x<1.7$

Using a suitable interval. show that $x=2.55$ is a solution to the equation $\mathrm{f}\left(x\right)=0$, correct to 3 significant figures.

The function $\mathrm{f}\left(x\right)$ is defined as

 $\mathrm{f}\left(x\right)=x^2-\ln (x+2)x>0$

Show that there is a solution to the equation $\mathrm{f}\left(x\right)=0$ in the interval $1<x<1.2$

Find an expression for $\mathrm{f}\text{'}\left(x\right)$

Taking $x_0=1$ as the first approximation, apply the Newton-Raphson method repeatedly to find a solution to the equation $\mathrm{f}\left(x\right)=0$ in the interval $\left[1,1.2\right]$.

Give your answer correct to 3 decimal places.

The diagram below shows part of the curve with the equation $y=2^{\ln x}$

The trapezium rule is used to estimate the area of the shaded region, given by

${\int }_5^{10}{2}^{\ln x}\mathrm{d}x$

Four equally spaced trapezia are used, each of width $h$.

Find $h$.

Complete the table of values below, correct to 3 significant figures.

Using the trapezium rule with all the values of $y$ in the table, find an estimate for

${\int }_5^{10}{2}^{\ln x}\mathrm{d}x$

State whether your answer to part (c) is an overestimate or an underestimate.

The figure below shows the line $y=x$ and the curve $y=\ln \left(x-1\right)+3$

The iteration formula

$x_{n+1}=\ln \left(x_n-1\right)+3$

with $x_0=2$ is used to find an estimate for a root, $\alpha$, of the equation$\mathrm{f}\left(x\right)=0$.

Write down an expression for $\mathrm{f}\left(x\right)$.

Draw a staircase diagram on the graph in part (a) to determine whether the iteration formula starting with $x_0=2$ finds an approximation for the $x$-coordinate of point $S$ or the $x$-coordinate of point $T$.

Find the values of the estimates $x_1,x_2,x_3$ and $x_4$, giving each answer to 3 decimal places.

Using a suitable interval, show that $\alpha =4.146$ to 3 decimal places.

The table below shows corresponding values of $x$ and $y$ for $y=\sqrt{\ln x}$.

Use the trapezium rule with all the values in the table to find an estimate for

${\int }_2^4\sqrt{\ln x}dx$

State, with a reason, whether your answer to part (a) is an overestimate or an underestimate of the value of the integral.

The graph of $y=\mathrm{f}\left(x\right)$ where

$\mathrm{f}\left(x\right)=2x-{\left(\ln x\right)}^3-3x>0$

is shown below, where $\alpha$ and $\beta$ are solutions to the equation $\mathrm{f}\left(x\right)=0$

The Newton-Raphson method is to be used to estimate the values of $\alpha$ and $\beta$.

Indicate on the diagram the starting value, $x_0$, that would lead to the Newton-Raphson method failing to find either solution, $\alpha$ or $\beta$.

[You do not need to calculate the value of $x_0$.]

Find an expression for $\mathrm{f}\text{'}\left(x\right)$.

Apply the Newton-Raphson method with $x_0=1$ to find $\beta$.

Give your answer to 5 significant figures.

The curve with equation $y=\sqrt{{\mathrm{e}}^{-x}+2x}$ is shown below.

The shaded area is represented by

${\int }_2^8\sqrt{{\mathrm{e}}^{-x}+2x}dx$

Use the trapezium rule with 6 equally spaced trapezia to find an estimate for the area of the shaded region.

Give your answer to 3 significant figures.

Explain how the accuracy of the estimate in part (a) can be improved.

Sketch two separate diagrams to show how the trapezium rule can lead to either an underestimate or an overestimate.

Use the trapezium rule with step size $h=0.25$ to find an estimate for the area bounded by the curve with equation $y=1+0.3x^2\sin x$ and the lines $x=1$, $x=2$ and the $x$-axis.

Give your answer to 3 significant figures.

The trapezium rule is to be used to approximate

${\int }_4^8\mathrm{f}\left(x\right)\mathrm{d}x$

The table below shows values of $x$ and $\mathrm{f}\left(x\right)$.

Using the values in the table, find an estimate for the integral using

(i) 2 trapezia,
(ii) 4 trapezia,
(iii) 8 trapezia.

Explain which estimate from part (a) is likely to be the most accurate approximation of

${\int }_4^8\mathrm{f}\left(x\right)\mathrm{d}x$

The diagram below shows part of the graph with equation $y=3x-{\mathrm{e}}^{x^2}.$

The area between the curve and the $x$-axis from $x=0.5$ to $x=1$ is shaded.

Use the trapezium rule with 5 equally spaced trapezia to find an estimate for the area of the shaded region.

Give your answer to 3 significant figures.

The table below shows corresponding values of $x$ and $y$ for $y={\log }_{3}2x$

The values of $y$ are given to 2 decimal places as appropriate.

Using the trapezium rule with all the values of $y$ in the table, find an estimate for

${\int }_3^9{\log }_{3}2x\mathrm{d}x$

Using your answer to part (a) and making your method clear, estimate

(i) ${\int }_3^9{\log }_3{\left(2x\right)}^{10}\mathrm{d}x$

(ii) ${\int }_3^9{\log }_{3}18x\mathrm{d}x$

Figure 1 shows a sketch of the curve $C$ with equation

$y=\frac{4x^2+x}{2\sqrt{x}}-4\ln xx>0$

Show that

$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{12x^2+x-16\sqrt{x}}{4x\sqrt{x}}$

The point $P$, shown in Figure 1, is the minimum turning point on $C$.

Show that the $x$ coordinate of $P$ is a solution of

$x={\left(\frac{4}{3}-\frac{\sqrt{x}}{12}\right)}^{\frac{2}{3}}$

Use the iteration formula

$x_{n+1}={\left(\frac{4}{3}-\frac{\sqrt{x_n}}{12}\right)}^{\frac{2}{3}}$with $x_1=2$

to find

(i) the value of $x_2$ to 5 decimal places,

(ii) the $x$ coordinate of $P$ to 5 decimal places.

Figure 8 shows a sketch of the curve $C$ with equation $y=x^x,x>0$.

The point $P\left(\alpha ,2\right)$ lies on $C$.

Show that $1.5<\alpha <1.6$.

A possible iteration formula that could be used in an attempt to find $\alpha$ is

$x_{n+1}=2{x_n}^{1-x_n}$

Using this formula with $x_1=1.5$, find $x_4$ to 3 decimal places.

Describe the long-term behaviour of $x_n$.

The diagram below shows part of the curve $y=\mathrm{f}\left(x\right)$ where

$\mathrm{f}\left(x\right)=3x^2{\sin }^{2}x-2-\frac{3\pi }{2}<x<\frac{3\pi }{2}$

You are given that $\mathrm{f}\left(0.9\right)=-0.509$ and$\mathrm{f}(3.4)=0.265$, to 3 significant figures,

A student wishes to find an estimate of the root of $\mathrm{f}\left(x\right)=0$ that is close to $x=0.98$

Explain why a change of sign in the interval $\left[0.9,3.4\right]$ is not necessarily helpful to the student.

Using a suitable interval, show that there is a root close to $x=0.98$.

Show that the root close to $x=0.98$ is $0.982$, correct to 3 significant figures.

The graph below shows a sketch of the line $y=x$ and the curve  $y=\sqrt[3]{3x^2+2x-1}$

An iteration formula is used to find the three roots of the equation

$x^3-3x^2-2x+1=0$

Draw a staircase diagram on the graph above to show that the iteration formula

$x_{n+1}=\sqrt[3]{3{x_n}^2+2x_n-1}$

with a starting value of $x_0=0.5$ converges to the largest positive root of the equation.

(i) Use the iteration formula from part (a) with $x_0=0.5$ to find $x_1,x_2$ and $x_3$, to 3 significant figures.

(ii) If the root is close to $x=3.5$, describe the speed of convergence to the root of the values $x_1,x_2$ and $x_3$.

Show that the root close to $x=3.5$ is $3.49$, correct to 3 significant figures.

The function $\mathrm{f}\left(x\right)$ is defined as

 $\mathrm{f}\left(x\right)=\sin 3x-\ln 2xx>0$

where $x$ is in radians.

Apply the Newton-Raphson method with a starting approximation of $x_0=0.8$ to find a solution to the equation

$\sin 3x=\ln 2x$

giving your answer correct to 4 decimal places.

The graph of $y=\mathrm{f}\left(x\right)$ has a local maximum point with coordinates $\left(\beta ,\mathrm{f}\left(\beta \right)\right)$.

Describe what happens when applying the Newton-Raphson method with a starting approximation of $x_0=\beta$.

Graphs of $y=\mathrm{f}\left(x\right)$ for four different functions are shown below.

Match each graph above with the correct statement below:

1. $\mathrm{f}\left(x\right)$ is not continuous and it is possible to have an interval showing no sign change that contains exactly one root

2. $\mathrm{f}\left(x\right)$ is not continuous and it is possible to have an interval showing no sign change that contains more than one root

3. $\mathrm{f}\left(x\right)$ is continuous and it is possible to have an interval showing no sign change that contains exact one root

4. $\mathrm{f}\left(x\right)$ is continuous and it is possible to have an interval showing no sign change that contains more than one root

The diagram below shows the line $y=x$ and the curve $y=\mathrm{g}\left(x\right)$.

Draw a cobweb diagram on the graph above, using the starting approximation $x_0$ indicated.

You must show

• the first two estimates, $x_1$ and $x_2$

• convergence to a root of the equation $x-\mathrm{g}\left(x\right)=0$

By forming a suitable iteration formula with $x_0=2$ and using repeated iteration, find a root of the equation

$x-\sin 0.8x=2.5$

correct to 2 significant figures.