Iteration and Newton-RaphsonAQA A-Level Maths: Mind map

Exam questions on Iteration and Newton-Raphson

1. The equation $x^3+x-3=0$ has a root $\alpha$ between 1 and 2. A student uses the iteration $x_{n+1}=\sqrt[3]{3-x_n}$ with $x_0=1$.
   Find $x_2$ and $x_3$, giving each to 4 decimal places.2 marks
2. The Newton-Raphson method is used to approximate $\sqrt7$ by solving $f(x)=0$, where $f(x)=x^2-7$, with starting value $x_0=3$.
   Find $x_2$, giving your answer to 4 decimal places.2 marks
3. The equation $f(x)=0$, where $f(x)=x^3-2x-5$, has a root $\alpha$ close to 2. The Newton-Raphson method is used with $x_0=2$.
   Show that the Newton-Raphson iteration is $x_{n+1}=x_n-\frac{x_n^3-2x_n-5}{3x_n^2-2}$, and hence show that $x_1=2.1$.3 marks