Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The sketch depicts the curve $y = x^2 + 3x + 1 + 5\cos\frac{1}{2}x$. It cuts the $y$-axis at the point $P$, and the gradient of the curve there is $m$. The point $Q$ lies on the curve and has $x$-coordinate $q$, with gradient $-m$ at $Q$.
(i)[4]
Find the value of $m$ and so show that $q$ satisfies the equation $x = a\sin\frac{1}{2}x + b$, where the constants $a$ and $b$ are to be determined.
(ii)[2]
Show by calculation that $q$ lies in the interval $(-4.5,-4.0)$.
(iii)[3]
Use the iterative formula based on the equation in part (i) to determine $q$ 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 as to get an expression of the form $k_1x+k_2+k_3\sin\dfrac12 x$” …