It is given that
f(x)=x3+2x2+x+1f(x) = x^3 + 2x^2 + x + 1f(x)=x3+2x2+x+1
Show that the equation f(x)=0f(x) = 0f(x)=0 can be written in the form
x=−(x+1x2+2)\displaystyle x = -\left(\frac{x + 1}{x^2} + 2\right)x=−(x2x+1+2)
Use the iteration formula
xn+1=−(xn+1xn2+2)\displaystyle x_{n+1} = -\left(\frac{x_n + 1}{x_n^2} + 2\right)xn+1=−(xn2xn+1+2)
with x0=−2x_0 = -2x0=−2 to find, to three decimal places, the values of x1x_1x1, x2 x_2\,x2 and x3x_3x3.
The root of f(x)=0f(x) = 0f(x)=0 is α\alphaα.
By choosing a suitable interval, prove that α=−1.755\alpha = -1.755α=−1.755 correct to three decimal places.
137 exam-style questions on OCR (MEI) A Level Maths 1.10 Numerical Methods (A-level only), covering 1.10.1 Locate roots by change of sign (A-level only), 1.10.2 When change of sign methods fail (A-level only), 1.10.3 Fixed point iteration (A-level only), 1.10.4 Newton-Raphson method (A-level only), 1.10.5 Convergence of iterations (A-level only), 1.10.6 Trapezium rule (A-level only), 1.10.7 Upper and lower bounds using rectangles (A-level only), and 1.10.8 Numerical methods to solve problems (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.