Numerical methods in contextAQA A-Level Maths: Mind map

Exam questions on Numerical methods in context

1. The depth $d$ metres of water in a tank satisfies the equation $d+\ln d=4$, where $d>0$. Let $f(d)=d+\ln d-4$.
   Use the iteration $d_{n+1}=4-\ln d_n$ with $d_0=3$ to find $d_1$ and $d_2$, giving your answers to 3 decimal places.2 marks
2. The angle $\theta$ (in radians) in a model of an orbit satisfies $\theta-0.5\sin\theta=1$. Let $f(\theta)=\theta-0.5\sin\theta-1$. The Newton-Raphson method is used with $\theta_0=1.5$.
   Find $\theta_2$ to 6 decimal places and hence state the value of $\theta$ to 3 decimal places, with a reason.2 marks
3. The integral $I=\int_0^1e^{-x^2}\,dx$ cannot be found by integrating $e^{-x^2}$ using the standard methods in the specification. It is estimated using the trapezium rule with 4 strips of equal width.
   Find the trapezium rule estimate of $I$, giving your answer to 3 significant figures.3 marks