What does x{n+1} represent geometrically?

Where the tangent to y=f(x) at x=xn crosses the x-axis.

When can Newton-Raphson not be applied?

When f'(xn)=0, because the tangent is horizontal.

Why can a poor starting value give the wrong root?

The tangent at x0 may meet the axis nearer a different root.

Compare speed of Newton-Raphson and bisection.

Newton-Raphson usually converges much faster.

How do you choose x0?

Close to the root, from a sign change or sketch, and where f'(x0) is not near zero.

Differentiate f(x)=e^x-4 for the iteration.

f'(x)=e^x, so x{n+1}=xn-\frac{e^{xn}-4}{e^{xn}}

What is a stationary point and why does it matter here?

A point where f'(x)=0; an iterate there gives division by zero.

The Newton-Raphson processEdexcel International A Level Further Maths: Flashcards

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State the Newton-Raphson formula.

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State the Newton-Raphson formula.: $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$
What does $x_{n+1}$ represent geometrically?: Where the tangent to $y=f(x)$ at $x=x_n$ crosses the $x$ -axis.
When can Newton-Raphson not be applied?: When $f'(x_n)=0$ , because the tangent is horizontal.
Why can a poor starting value give the wrong root?: The tangent at $x_0$ may meet the axis nearer a different root.
Newton-Raphson for $f(x)=x^2-k$ ?: $x_{n+1}=\frac{x_n^2+k}{2x_n}$
$f(x)=x^3-2x-5$ , $x_0=2$ : find $x_1$ .: $2-\frac{-1}{10}=2.1$
$f(x)=\cos x-x$ : find $f'(x)$ .: $-\sin x-1$
What calculator mode for $\sin x$ and $\cos x$ in calculus?: Radians.
Compare speed of Newton-Raphson and bisection.: Newton-Raphson usually converges much faster.
How do you choose $x_0$ ?: Close to the root, from a sign change or sketch, and where $f'(x_0)$ is not near zero.
Differentiate $f(x)=e^x-4$ for the iteration.: $f'(x)=e^x$ , so $x_{n+1}=x_n-\frac{e^{x_n}-4}{e^{x_n}}$
What is a stationary point and why does it matter here?: A point where $f'(x)=0$ ; an iterate there gives division by zero.

Exam questions on The Newton-Raphson process

The equation $x^3-2x-5=0$ has a root $\alpha$ close to $2$ . Let $f(x)=x^3-2x-5$ and use the Newton-Raphson method with $x_0=2$ .
Find $x_2$ , giving your answer to 4 decimal places.2 marks
Let $f(x)=x^3-3x+1$ . The equation $f(x)=0$ has a root $\alpha$ between $0$ and $1$ .
Taking $x_0=\frac13$ , find $x_1$ to 4 decimal places.2 marks
Let $f(x)=\cos x-x$ , where $x$ is in radians. The equation $f(x)=0$ has a root $\alpha$ near $0.7$ .
Show that the Newton-Raphson iteration for this equation can be written as $x_{n+1}=x_n+\frac{\cos x_n-x_n}{1+\sin x_n}$ .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).