The diagram below shows part of the graph where .
Write down an interval, in the form , such that , explain clearly your choice of values for a and b.
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The diagram below shows part of the graph where .
Write down an interval, in the form , such that , explain clearly your choice of values for a and b.
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A solution to the equation is , correct to two significant figures.
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Show that the equation can be rewritten as
Starting with , use the iterative formula
to find values for and , giving each to four decimal places where appropriate.
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The graph of where is shown below. θ is measured in radians and -
Given that .
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A student is trying to find a solution to the equation using an iterative formula.
The student rearranges into the form .
The diagram below shows a sketch of the graphs of and .
The student is trying to find the root , starting with an initial estimate .
Show on the diagram, how the iterative formula will converge and find the root .
Mark the -axis with the positions of and .
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A bypass is to be built around a village.
On the graph below the road through the village is modelled by the line The bypass is modelled by the equation .
The bypass runs from the origin to the point
Use the iterative formula with to find the value of , correct to three significant figures.
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The game of Tanball 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 radians.
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
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According to legend, unicorn tears can heal an injury almost instantly.
If a unicorn tear is applied to a burn of initialize size on human skin it will heal according to the model
where is the area of the burn, in square millimeters, at time seconds after the unicorn tear has been applied.
Show that the equation can be written as
Use the iterative formula with initial value , to find how many seconds it takes a burn of size to heal once a unicorn tear is applied .
Give your final answer to three significant figures.
An alternative iterative formula is ,
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The diagram below shows part of the graph where
One of the solutions to the equation is , correct to three significant figures.
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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.
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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.
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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.
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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.
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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.
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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.
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.
Briefly explain why the model should also restrict the range of to be greater than or equal to zero?
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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.
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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.
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The diagrams below show the graphs of four different functions.
Match each graph above with the correct statement below.
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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.
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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.
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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.
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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?
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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 .
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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.
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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:.
Sign change rule indicates a root but there isn’t one due to a discontinuity in the graph.
On each diagram, clearly labelled p, q and the root α.
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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 .
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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.
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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.
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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.
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