A-level · Numerical methods

The Newton–Raphson methodExam-style questions

A fast iteration that uses the tangent — x_(n+1) = x_n − f(x_n)/f'(x_n) — its geometric meaning, why it converges so quickly, and the cases where it fails (a horizontal tangent or a poor starting value).

Start practising Read the lesson

23 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
Using the Newton–Raphson method on $f(x) = x^3 - 5$ with $x_0 = 2$ , find $x_1$ and $x_2$ to 4 decimal places. (This estimates $\sqrt[3]{5}$ .)
Level 3 · 3 marksTry it
2
The equation $e^x + x = 4$ has a root near $x = 1$ . Use the Newton–Raphson method with $x_0 = 1$ to find $x_1$ and $x_2$ to 4 decimal places.
Level 3 · 3 marksTry it
3
$f(x) = x^3 + x - 1$ . Using Newton–Raphson with $x_0 = 1$ , find $x_1$ and $x_2$ to 4 decimal places.
Level 3 · 3 marksTry it
4
A student applies the Newton–Raphson method to $f(x) = x^3 - 2x - 5$ , whose only real root is near $2.09$ , but starts at $x_0 = 0$ . Find $x_1$ and explain why this starting value is a poor choice.
Level 3 · 3 marksTry it
5
(a) Show that applying the Newton–Raphson method to $f(x) = x^2 - N$ gives the iteration $x_{n+1} = \dfrac12\left(x_n + \dfrac{N}{x_n}\right)$ .
(b) Hence, with $N = 10$ and $x_0 = 3$ , estimate $\sqrt{10}$ to 4 decimal places.
Level 3 · 4 marksTry it

9 more questions on the newton–raphson method

Free with an account — just an email, no card. The whole bank, every mark scheme and worked solution.

Start free