Why is continuity needed for the sign-change test?

A discontinuous function can jump over zero, for example across an asymptote.

Which function changes sign on [1,3] without a root: x^2-4 or \frac{1}{x-2}?

\frac{1}{x-2}: it has an asymptote at x=2.

How can sign change miss roots in a large interval?

The interval may hold an even number of roots, so the end values have the same sign.

How do you rearrange x^3-3x-5=0 for iteration?

x=\sqrt[3]{3x+5}, so x{n+1}=\sqrt[3]{3xn+5}.

What happens if |g'|1 at the root?

The iteration diverges away from the root.

How do you show a root is 2.15 to 2 d.p.?

Show a sign change of f on [2.145,\,2.155].

Why use numerical methods?

Many equations cannot be solved algebraically; numerical methods give a root to any required accuracy.

Locating roots and iterationEdexcel A-Level Maths: Flashcards

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State the change of sign test.

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State the change of sign test.: If $f$ is continuous on $[a,b]$ and $f(a)$ , $f(b)$ have opposite signs, there is a root in $[a,b]$ .
Why is continuity needed for the sign-change test?: A discontinuous function can jump over zero, for example across an asymptote.
Which function changes sign on $[1,3]$ without a root: $x^2-4$ or $\frac{1}{x-2}$ ?: $\frac{1}{x-2}$ : it has an asymptote at $x=2$ .
How can sign change miss roots in a large interval?: The interval may hold an even number of roots, so the end values have the same sign.
What is the form of an iterative formula?: $x_{n+1}=g(x_n)$
How do you rearrange $x^3-3x-5=0$ for iteration?: $x=\sqrt[3]{3x+5}$ , so $x_{n+1}=\sqrt[3]{3x_n+5}$ .
Convergence condition for $x_{n+1}=g(x_n)$ ?: $|g'(x)|<1$ near the root.
Staircase or cobweb: $0<g'<1$ ?: Staircase.
Staircase or cobweb: $-1<g'<0$ ?: Cobweb (spirals in).
What happens if $|g'|>1$ at the root?: The iteration diverges away from the root.
How do you show a root is $2.15$ to 2 d.p.?: Show a sign change of $f$ on $[2.145,\,2.155]$ .
Why use numerical methods?: Many equations cannot be solved algebraically; numerical methods give a root to any required accuracy.

Exam questions on Locating roots and iteration

Let $f(x)=x^3-3x-5$ , a continuous function. The equation $f(x)=0$ has a single real root $\alpha$ .
Show that $\alpha$ lies in the interval $[2.2,\,2.3]$ .2 marks
The root $\alpha$ of $x^3-3x-5=0$ is estimated using the iteration $x_{n+1}=\sqrt[3]{3x_n+5}$ with $x_0=2$ .
Find $x_3$ and $x_4$ , and hence write down the value of $\alpha$ correct to 2 decimal places.2 marks
Let $f(x)=\frac{1}{x-2}$ for $x\ne2$ , and $g(x)=x^2-5x+6$ .
Show that $f(1)$ and $f(3)$ have opposite signs, and explain why this does not show that $f(x)=0$ has a root in $[1,3]$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).