Numerical Solutions of Equations (CIE A Level Maths: Pure 3)

The diagram below shows part of the graph where

Find  and , giving your answers to three significant figures.

One of the solutions to the equation  is , correct to three significant figures.

Hence, use the sign change rule to confirm that this is a solution
(to three significant figures) to the equation .

Show that the equation   can be rewritten as

Starting with , use the iterative formula

to find a root of the equation , correct to two decimal places.

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

Explain why the change of sign rule would fail if attempting to locate a root of the function using the values of θ = 1.55 and θ = 1.65.

The diagram below shows the graphs of and .

The iterative formula

is to be used to find an estimate for a root, , of the function.

Write down an expression for .

Using an initial estimate, , show, by adding to the diagram above, which of the two points (S or T) the sequence of estimates will converge to.
Hence deduce whether   is the -coordinate of point S or point T.

Find the estimates and , giving each to three decimal places.

Confirm that  correct to three decimal places.

The village of Greendale lies on a straight road, as modelled by the line on the graph below.  To ease rush hour congestion, a bypass is to be built around Greendale.
The path of the bypass is modelled by the equation .

The bypass runs from the origin to the point .

On the diagram show how using the iterative formula  with will lead to convergence at the point  P.

Use the iterative method  with  to find the value of , correct to two decimal places.

Use the sign change rule with the function  to show your answer to part (b) is correct to two decimal places.

The game of Curveball is played on a flat table.

A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest in the winning zone.

 A particular player decides to roll the ball at an angle of 45°.
This is illustrated by the graph below with the ball being rolled from the origin and the shaded area being the winning zone.

The boundary of the winning zone is given by part of the curve with equation

Use the iterative formula with initial starting value , to show that the -coordinate of the point where this player’s ball should cross the winning zone boundary is 0.497 to three significant figures.

Use your answer to part (a) to find the minimum distance the ball should travel for this player to win Curveball.

According to legend, a unicorn can heal an injury almost instantly by touching it with its horn.

When a unicorn touches a cut in human skin of length , it will heal according to the model

where  is the length of the cut in millimeters, at time  seconds after the unicorn has touched the injury with its horn.

Write down the value  would be when the cut is completely healed.

Show that, for a cut in human skin of length  the equation  can be rearranged into the form

            

Use the iterative formula  , with initial value  to find how many seconds it takes a cut of size  to heal once a unicorn has touched it with its horn.

Write down the values of the estimates  to four decimal places.

Briefly explain why the model should also restrict the range of  to be greater than or equal to zero?

The diagram below shows part of the function where.

Correct to three significant figures, and.
Explain why using the sign change rule with these values would not necessarily be helpful in finding the root close to .

Using suitable values of x, show that there is a root close to .

Show that the root close to is 0.982, correct to three significant figures.

The diagram below shows a sketch of the graphs , and  .

An iterative formula is used to find roots to the equation . 

On the diagram above show that the iterative formula

would converge to the root close to when using a starting value of .

Confirm that the root close to is 3.49 correct to three significant figures.

The diagrams below show the graphs of four different functions.

Match each graph above with the correct statement below. 

1. The sign change rule with values of and would indicate a root but has failed due to the discontinuity (asymptote) at .
2. The sign change rule with values of and would indicate no root but has failed because there are two roots in the interval (1 , 5).
3. The sign change rule with values of and would indicate no root but fail as there are two roots in the interval (3 , 5).
4. The sign change rule with values of and would indicate no root but has failed to find the root as the graph has a turning point at .

The diagram below shows the graphs of and .

Show on the diagram, using the value of indicated, how an iterative process will lead to a sequence of estimates that converge to the x-coordinate of the point P.
Mark the estimates and on your diagram.

By finding a suitable iterative formula, use to estimate a root to the equation correct to two significant figures.

Confirm that your answer to part (b) is correct to two significant figures.

The village of Crinkley Bottom lies on a straight road, as modelled by the line  on the graph below.  Rush hour traffic causes much air pollution in the village so to improve the air quality around Crinkley Bottom a bypass is to be built.
The path of the bypass is modelled by part of the equation .

The bypass is to be built with a roundabout south of the village at the origin and a northern roundabout which re-joins the road through Crinkley Bottom at the point .

On the diagram show how using the iterative formula 

with  will lead to convergence at the southern roundabout

Use the alternative iterative method

with  to find the position of the roundabout at  to four significant figures.

Verify that your answer to part (b) is correct to four significant figures.

The game of Logball is played on a flat table.

A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest within a winning zone.

A particular player decides to roll the ball at an angle of 45°, as illustrated in the graph below, with the ball being rolled from the origin and the shaded area being the winning zone.

The lower boundary of the winning zone has equation  ,   

 The upper boundary of the winning zone has equation   ,  

Using an appropriate iterative formula with initial value , find the minimum distance this player’s ball needs to travel to stop within the winning zone. Give your answer to two significant figures.

Using another iterative formula with initial value , find the maximum distance this player’s ball can travel yet remain within the winning zone.
Give your answer to two significant figures.

According to legend, unicorn tears have magical healing powers.

When a unicorn tear is applied to a bruise of size  it will heal according to the model

where  is the area of the bruise, in square millimeters, at time  seconds after the unicorn tear is applied.

Show that, for a bruise of initial size , the equation  can be rearranged into the form

Using the equation from part (a) as an iterative formula and initial value  find how many seconds it takes a bruise of siae  to heal once a unicorn tear is applied. Give your answer to three significant figures. 

It is rumoured that a unicorn tear can heal bruises one-hundred-thousand times faster than they would heal naturally. Approximately how many days would it take a bruise of initial size  to heal without a unicorn tear?

The diagram below shows part of the graph with equation .

A student searches for a root of the equation .
They find that and that .
The student concludes that there is a root in the interval .
Explain why the student’s conclusion is incorrect.

Verify that is a solution to the equation

Explain why the sign change rule would fail if searching for the root of the equation .

The function, is defined by

Show that the equation can be written in the form

On the same diagram sketch the graphs of and .

The equation has a root, , close to .
The iterative formula  with  is to be used to find correct to three significant figures.

Show, using a diagram and your answer to part (b), that this formula and initial x value will converge to the root .

The root   lies in the interval .
Write down the values of p and q such that  can be deduced accurate to two decimal places from the interval.

Sketch three separate graphs with values of and , to show how the sign change rule would fail to find a root α in the interval (p , q) for the following reasons:.

On each diagram, clearly labelled p, q and the root α.

Sketch two separate diagrams to show how an iterative formula of the form

 can diverge in two different ways when being used to find an estimate for a root to the equation .

The village of Camberwick Green lies on a straight road, as modelled by the line 
 on the graph below.  Rush hour traffic through the village causes both congestion and air pollution.  To ease congestion and improve the air quality around Camberwick Green a bypass , modelled by part of the equation    is to be built.

The bypass is to be built with a roundabout south of the village at point and a northern roundabout which re-joins the road through Camberwick Green at the point .

Write down the coordinates of the southern roundabout at point .

Find the length of road through Camberwick Green that will benefit from the construction of the bypass. Decide on an appropriate unit of measurement.

The game of Funcball is played on a flat table. A player rolls a ball from a fixed point, at any angle, with the aim of it coming to rest within a winning zone.

The winning zone is modelled by the function

The lower boundary of the winning zone has equation

The upper boundary of the winning zone has equation ,

A particular player decides to roll the ball at an angle of 45°, as illustrated by the graph below, with the ball being rolled from the origin and the shaded area being the winning zone.

Using iterative formulas with initial values  and  as appropriate, find the exact distances between which the ball must stop for this player to win Funcball.
Give your answers to two significant figures.

According to legend, unicorn tears have magical healing powers.

Without unicorn tears, a bruise of initial size  will heal according to the model

where  is the area of the bruise, in square millimeters, at time  days since the bruise first appeared.

Use the iterative formula

with  to find how many days it takes a bruise to heal without unicorn tears.

Give your answer to three significant figures.

With unicorn tears, a bruise of the same initial size will heal according to the model

where time  is measured in seconds.

Using your answer from parts(a) and (b), work out approximately how many times quicker a bruise heals when unicorn tears are used.