Newton-Raphson methodEdexcel A-Level Maths: Mind map

Exam questions on Newton-Raphson method

1. The equation $f(x)=0$, where $f(x)=x^3-2x-5$, has a root $\alpha$ near $x=2$. The Newton-Raphson method is used with $x_0=2$.
   Find $x_2$, giving your answer to 4 decimal places.2 marks
2. Let $f(x)=\mathrm{e}^x-3x$. The equation $f(x)=0$ has a root $\alpha$ close to $0.6$, and the Newton-Raphson method is used with $x_0=0.5$.
   Given that $x_1=0.610060\ldots$, find $x_2$ to 4 decimal places.2 marks
3. The Newton-Raphson method is used to find $\sqrt7$, as the positive root of $x^2-7=0$, starting from $x_0=3$.
   Show that the Newton-Raphson formula for this equation simplifies to $x_{n+1}=\frac12\left(x_n+\frac{7}{x_n}\right)$, and find $x_1$ as an exact fraction.3 marks