The Newton-Raphson processEdexcel International A Level Further Maths: Mind map

Exam questions on The Newton-Raphson process

1. 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
2. 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
3. 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