Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
It is given that $\int_{1}^{a} x^{\frac{1}{2}} \ln x \, dx = 2$, with $a > 1$.
(i)[5]
Show, by algebraic manipulation, that $a^{\frac{3}{2}} = \dfrac{7 + 2a^{\frac{3}{2}}}{3 \ln a}$.
(ii)[2]
Show by calculation that $a$ lies between $2$ and $4$.
(iii)[3]
Use the iterative formula $a_{n+1} = \left( \dfrac{7 + 2a_n^{\frac{3}{2}}}{3 \ln a_n} \right)^{\frac{2}{3}}$ to find $a$ correct to $3$ decimal places. Show each iteration to $5$ decimal places.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Use integration by parts to reach $ax^2\ln x+b\int x^2\frac{1}{x}\,dx$” …