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

The table below shows corresponding values of and for

The values of are given to 4 significant figures.

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

giving your answer to 3 significant figures.

Using your answer to part (a), deduce an estimate for

Given that

to 4 significant figures

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

The curve with equation meets the line at a single point, .

Show that .

Figure 2 shows the graph of and the graph of .

A student uses the iteration formula

in an attempt to find an approximation for .

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

The curve with equation where

has a single turning point at .

Show that is a solution of the equation .

The iterative formula

is used to find an approximate value for .

Starting with , calculate, giving each answer to 4 decimal places,

(i) the value of

(ii) the value of

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

A continuous curve has equation .

The table shows corresponding values of and for this curve, where and are constants.

The trapezium rule is used, with all the values in the table, to find an approximate area under the curve between and

Given that this area is , show that

Given also that the sum of all the values in the table is , find the value of and the value of .

The equation has exactly one real root.

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

Using the formula given in part (a) with , find the values of and

Explain why, for this question, the Newton-Raphson method cannot be used with

Figure 2 shows a sketch of part of the curve with equation where

and is measured in radians.

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

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

The curve crosses the -axis at , as shown in Figure 2.

Given that, to 3 decimal places, and , explain why α must lie in the interval

Taking as a first approximation to , apply the Newton-Raphson method once to to obtain a second approximation to .

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

The diagram below shows part of the graph where

Show that a solution to the equation exists in the interval

Using a suitable interval. show that is a solution to the equation , correct to 3 significant figures.

Show that the equation  can be written in the form

where and are integers to be found.

Taking  as the first approximation, use the iteration formula

with your values of and in part (a) to find, by repeated iteration, a solution to the equation

The function is defined as

Show that there is a solution to the equation in the interval

Find an expression for

Taking as the first approximation, apply the Newton-Raphson method repeatedly to find a solution to the equation in the interval .

Give your answer correct to 3 decimal places.

The diagram below shows part of the curve with the equation

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

Four equally spaced trapezia are used, each of width .

Find .

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

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

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

Part of the curve is shown below, where  is measured in radians.

A student uses a change of sign argument to show that the interval contains a solution to the equation

Explain whether, or not, this is a valid method.

The figure below shows the line and the curve

The iteration formula

with is used to find an estimate for a root, , of the equation.

Write down an expression for .

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

Find the values of the estimates and , giving each answer to 3 decimal places.

Using a suitable interval, show that to 3 decimal places.

The graph of where

is shown below, where and are solutions to the equation

The Newton-Raphson method is to be used to estimate the values of and .

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

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

Find an expression for .

Apply the Newton-Raphson method with to find .

Give your answer to 5 significant figures.

The curve with equation  is shown below.

The shaded area is represented by

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 to find an estimate for the area bounded by the curve with equation and the lines , and the -axis.

Give your answer to 3 significant figures.

The trapezium rule is to be used to approximate

The table below shows values of and .

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

The table below shows corresponding values of and for

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

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

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

(i)

(ii)

Figure 1 shows a sketch of the curve with equation

Show that

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

Show that the coordinate of is a solution of

Use the iteration formula

with

to find

(i) the value of to 5 decimal places,

(ii) the coordinate of to 5 decimal places.

Figure 8 shows a sketch of the curve with equation .

The point lies on .

Show that .

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

Using this formula with , find to 3 decimal places.

Describe the long-term behaviour of .

The diagram below shows part of the curve where

You are given that and, to 3 significant figures,

A student wishes to find an estimate of the root of that is close to

Explain why a change of sign in the interval is not necessarily helpful to the student.

Using a suitable interval, show that there is a root close to .

Show that the root close to is , correct to 3 significant figures.

The graph below shows a sketch of the line and the curve 

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

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

with a starting value of converges to the largest positive root of the equation.

(i) Use the iteration formula from part (a) with to find and , to 3 significant figures.

(ii) If the root is close to , describe the speed of convergence to the root of the values and .

Show that the root close to is , correct to 3 significant figures.

The function is defined as

where is in radians.

Apply the Newton-Raphson method with a starting approximation of to find a solution to the equation

giving your answer correct to 4 decimal places.

The graph of has a local maximum point with coordinates .

Describe what happens when applying the Newton-Raphson method with a starting approximation of .

The diagram below shows part of the graph with equation

The area between the curve and the -axis from to 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.

Graphs of for four different functions are shown below.

Match each graph above with the correct statement below:

1. is not continuous and it is possible to have an interval showing no sign change that contains exactly one root

2. is not continuous and it is possible to have an interval showing no sign change that contains more than one root

3. is continuous and it is possible to have an interval showing no sign change that contains exact one root

4. 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  and the curve .

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

You must show

• the first two estimates, and

• convergence to a root of the equation

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

correct to 2 significant figures.

Using a suitable interval and a suitable function that should be stated, show that your answer to part (b) is correct to 2 significant figures.

The diagram below shows part of the graph of where

Two roots of the equation are shown, and .

Write down the -coordinate of the maximum point, , on the graph.

The Newton-Raphson method is applied to find an estimate for the root .

The starting approximation, , is the smallest positive integer value greater than the -coordinate of the maximum point .

Find the first four estimates and use a suitable interval to find correct to 5 significant figures.

The diagram below shows the graph of where the function is defined by

The function has a positive root close to

An iteration formula is given by

where is an integer to be found.

Use repeated iteration to find an estimate of the positive root, to 6 decimal places, using a starting value of

Apply the Newton-Raphson method with to find an estimate of the positive root, to 6 decimal places.

Compare the rates at which the estimates converge between the different methods in part (a) and part (b).

The function, , is defined by

Show that the equation can be written in the form

where and are integers to be found.

On the same diagram, sketch the graphs of and , using your values of and from part (a).

The equation has a root, , close to .

Draw a cobweb diagram on the graph in part (b) to show how the iteration formula

with converges to .

(i) Find the values ofand , giving each answer correct to 3 significant figures.

(ii) How many iterations, , are required before and agree with each other to 2 decimal places?

The root lies in the interval .

Find the values of and that give the largest interval such that can be found to 2 decimal places.

Apply the Newton-Raphson method with to find a solution to equation

correct to 4 significant figures.

Show that there is a solution to the equation in the interval [0.605 , 0.615].

Without further calculation, state the value of this solution to the highest degree of accuracy possible.

The diagram below shows the graph of where .

Use the trapezium rule in step sizes of 0.2 to find an estimate of the integral

to 3 significant figures.

Figure 2 shows a sketch of part of the curve with equation

The finite region , shown shaded in Figure 2, is bounded by the curve, the line with equation , the -axis and the line with equation .

The table below shows corresponding values of and , with the values of given to 4 decimal places.

Use the trapezium rule, with all the values of in the table, to obtain an estimate for the area of , giving your answer to 3 significant figures.

Use algebraic integration to find the exact area of , giving your answer in the form

where , and are integers to be found.

A curve has equation , where

Show that

where and are constants to be found.

Hence show that the coordinates of the turning points of the curve are solutions of the equation

The equation has two positive roots and where

A student uses the iteration formula

in an attempt to find approximations for and

Diagram 1 shows a plot of part of the curve with equation and part of the line with equation

Using Diagram 1 draw a staircase diagram to show that the iteration formula starting with can be used to find an approximation for

Use the iteration formula with , to find, to decimal places,

(i) the value of

(ii) the value of

Using a suitable interval and a suitable function that should be stated show that to decimal places.

The diagram below shows part of the graph with equation where

A student searches for a root of the equation .

They find that and . They then conclude that there is a root in the interval .

Explain why the student’s conclusion is not correct.

Explain why a change of sign method would fail when searching for the root of the equation

The function is defined as

Show that the Newton-Raphson formula can be written as

where and are integers to be found.

Apply the Newton-Raphson method with to find, to 5 significant figures, a solution of the equation

A student wants to use the Newton-Raphson method to find a solution to the equation

in the range .

A teacher tells the student that the Newton-Raphson method is not necessary.

Explain why and find a solution to the equation, giving the highest accuracy possible.

The diagram below shows the graph of where is defined by

The equation has a solution close to

Use repeated iteration with to find this solution to 4 decimal places.

You must state your iteration formula clearly.

Apply the Newton-Raphson method to find the same root, using a different starting value of

Give your answer to 4 decimal places.

Compare the rates at which the estimates converge between the different methods in part (a) and part (b).

The figure below shows a sketch of the curve .

The coordinates of the local maximum point are .

Use the trapezium rule with 4 equally-spaced trapezia to find an approximation to

By replacing the trapezia with rectangles that fit above the curve, find an upper bound for the area shaded.

Use rectangles that fit below the curve to find a lower bound.