Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The diagram depicts the curve $y = x^2 + 3x + 1 + 5\cos\frac{1}{2}x$. This curve meets the $y$-axis at $P$, where its gradient is $m$. Point $Q$ lies on the curve, has $x$-coordinate $q$, and the gradient there is $-m$.
(i)[4]
Find the value of $m$ and then show that $q$ satisfies the equation $x = a\sin\frac{1}{2}x + b$, where the constants $a$ and $b$ are to be found.
(ii)[2]
Show by calculation that $-4.5 < q < -4.0$.
(iii)[3]
Use an iterative formula based on the equation in part (i) to determine the value of $q$ correct to 3 significant figures. Give each iteration to 5 significant figures.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Differentiate so that the answer is written in the form $k_1x + k_2 + k_3\sin\tfrac{1}{2}x$” …