True or False? A sign change guarantees a root only if the function is continuous across the interval.

True. Continuity is part of the condition, not an optional extra. Without it a sign change is equally consistent with a jump across an asymptote , where nothing has been crossed and there is no root.

What must you state when concluding a change of sign argument?

That the function is continuous on the interval, that there is a sign change , and therefore that a root lies within it. The conclusion is not valid on the sign change alone, since the continuity condition is what rules out a jump.

How do you find an interval to test in the first place?

Sketch the function, or evaluate it at a few whole-number values until the sign flips. Even a rough sketch shows roughly where the curve crosses, which narrows the search down to one or two integers.

How can the change of sign method miss roots entirely?

If the interval contains an even number of roots, the signs at the two ends match and no change is detected. Two roots inside an interval look exactly the same from the endpoints as no roots at all.

There is a sign change. Why might there still be more roots than the one you have found?

Because any odd number of roots produces a single sign change, so three roots look like one. The method reports whether the number of crossings is odd or even, and nothing more than that.

How can a sign change appear where there is no root?

At a discontinuity , most often an asymptote, where the function jumps from large positive values to large negative ones. Nothing has been crossed, so the sign change is real but the root is not.

True or False? Making the interval smaller removes every one of these failures.

False. It does deal with the failures caused by several roots sitting in the interval at once, since a small enough interval will hold only one. It does nothing at all about a curve that merely touches the axis, which produces no sign change at any interval size.

The sequence converges, but extremely slowly. What is likely to be wrong?

The tangent gradient is close to zero , so each tangent meets the axis a long way from where it was drawn. The same thing can send the sequence away from the root altogether, or towards a different root.

When would you use the trapezium rule?

When the integral cannot be found analytically , or when all you have is a table of values rather than a function. It builds the area out of trapeziums, so it never needs the integration itself to be possible.

Why does the trapezium rule give only an approximation?

Because the top of each strip is a straight line joining two points on the curve, rather than the curve itself. The small gap between that chord and the arc it replaces is the error contributed by each strip.

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Numerical Methods (Edexcel A Level Maths: Pure): Flashcards

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FrontChange of Sign Method
What does a change of sign between $f (a)$ and $f (b)$ tell you?

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Cards in this collection (27)

What does a change of sign between $f (a)$ and $f (b)$ tell you?
That a root lies somewhere between $a$ and $b$ .
To get from a positive value to a negative one the curve has to cross the $x$ -axis, and crossing the axis is exactly what a root is.
True or False?
A sign change guarantees a root only if the function is continuous across the interval.
True.
Continuity is part of the condition, not an optional extra.
Without it a sign change is equally consistent with a jump across an asymptote, where nothing has been crossed and there is no root.
How do you show that a root is $2.35$ correct to 2 decimal places?
Test the two bounds of the values that round to it, by evaluating $f (2.345)$ and $f (2.355)$ .
A sign change between them puts the root inside the interval that rounds to $2.35$ .
What must you state when concluding a change of sign argument?
That the function is continuous on the interval, that there is a sign change, and therefore that a root lies within it.
The conclusion is not valid on the sign change alone, since the continuity condition is what rules out a jump.
How do you find an interval to test in the first place?
Sketch the function, or evaluate it at a few whole-number values until the sign flips.
Even a rough sketch shows roughly where the curve crosses, which narrows the search down to one or two integers.
How can the change of sign method miss roots entirely?
If the interval contains an even number of roots, the signs at the two ends match and no change is detected.
Two roots inside an interval look exactly the same from the endpoints as no roots at all.
There is a sign change. Why might there still be more roots than the one you have found?
Because any odd number of roots produces a single sign change, so three roots look like one.
The method reports whether the number of crossings is odd or even, and nothing more than that.
Why does a repeated root defeat the method?
Because the curve touches the $x$ -axis without passing through it, so the function never changes sign.
There is a genuine root present and no sign change anywhere near it.
How can a sign change appear where there is no root?
At a discontinuity, most often an asymptote, where the function jumps from large positive values to large negative ones.
Nothing has been crossed, so the sign change is real but the root is not.
True or False?
Making the interval smaller removes every one of these failures.
False.
It does deal with the failures caused by several roots sitting in the interval at once, since a small enough interval will hold only one.
It does nothing at all about a curve that merely touches the axis, which produces no sign change at any interval size.
Complete the notation for an iterative formula:
$x_{n + 1} =______$
The completed notation is:
$x_{n + 1} = g (x_{n})$
Each answer becomes the input for the next step, which makes an iterative formula a recurrence relation.
What has to be done to an equation before it can be iterated?
Rearrange it into the form $x = g (x)$ , with a single $x$ on its own on the left.
The same equation can usually be rearranged in several different ways, and they do not all behave alike.
How do you use an iterative formula?
Start from the given value $x_{0}$ , substitute it to get $x_{1}$ , then feed each answer straight back in.
Stop once successive values agree to the accuracy the question has asked for.
What are staircase and cobweb diagrams?
Pictures of an iteration, drawn by plotting $y = x$ against $y = g (x)$ and stepping between the two.
A staircase closes in from one side, while a cobweb spirals in from alternate sides.
Where is the root on such a diagram?
At the point where $y = x$ and $y = g (x)$ cross.
There the equation $x = g (x)$ is satisfied exactly, which is what being a root of the rearranged equation means.
True or False?
Every rearrangement into $x = g (x)$ converges to the root.
False.
Some rearrangements diverge, moving further from the root at every step.
That is why a question tells you which iterative formula to use rather than leaving the rearrangement to you.
Complete the Newton-Raphson formula:
$x_{n + 1} = x_{n} -______$
The completed formula is:
$x_{n + 1} = x_{n} - \frac{f (x_{n})}{f' (x_{n})}$
It is given in the formulae booklet, and it needs the derivative, so the function has to be one you can differentiate.
What is the Newton-Raphson method doing geometrically?
Drawing the tangent at the current estimate and taking the point where it crosses the $x$ -axis as the next estimate.
Each tangent lands nearer the root than the last, which is why repeating it closes in.
What happens if $f' (x_{n}) = 0$ ?
The method fails outright, since the formula would divide by zero.
Geometrically the tangent there is horizontal, so it never meets the $x$ -axis and there is no next estimate to take.
The sequence converges, but extremely slowly. What is likely to be wrong?
The tangent gradient is close to zero, so each tangent meets the axis a long way from where it was drawn.
The same thing can send the sequence away from the root altogether, or towards a different root.
Why does the starting value $x_{0}$ matter so much?
One too far from the root can produce a divergent sequence, or converge neatly to a different root from the one wanted.
A sketch, or a change of sign check, is what tells you where it is safe to start.
True or False?
Newton-Raphson can be used on any equation that has a root.
False.
The formula needs $f' (x)$ , so the function has to be differentiable, and one that is merely awkward to differentiate is a practical obstacle too.
An iteration of the form $x = g (x)$ needs no derivative at all, which is one reason both methods are on the course.
Complete the strip width used in the trapezium rule:
$h = \frac{______}{n}$
The completed formula is:
$h = \frac{b - a}{n}$
Here $a$ and $b$ are the limits of the integral and $n$ is the number of strips, so $h$ is the width of each one.
How many $y$ -values does the trapezium rule need for $n$ strips?
$n + 1$ , because every strip needs a value at each of its edges and neighbouring strips share one.
Using $n$ values instead of $n + 1$ is the commonest error in the whole method.
When would you use the trapezium rule?
When the integral cannot be found analytically, or when all you have is a table of values rather than a function.
It builds the area out of trapeziums, so it never needs the integration itself to be possible.
Why does the trapezium rule give only an approximation?
Because the top of each strip is a straight line joining two points on the curve, rather than the curve itself.
The small gap between that chord and the arc it replaces is the error contributed by each strip.
True or False?
Using more strips improves the estimate.
True.
Narrower strips leave less room between each chord and the curve, so the total error falls.
The cost is simply that more strips mean more $y$ -values to work out.

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