f(x)=x3+2x2+x+1f(x) = x^3 + 2x^2 + x + 1f(x)=x3+2x2+x+1
Show that f(x)f(x)f(x) has a root between -1.5 and -2
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=kx_0 = kx0=k to show that x1=−2k2+k+1k2\displaystyle x_1 = -\frac{2k^2+k+1}{k^2}x1=−k22k2+k+1. Hence, taking k=−2k = -2k=−2, find, to 3 decimal places, the values of x2 x_2\,x2 and x3x_3x3.
By choosing a suitable interval, prove that the root between -1.5 and -2 is -1.755 to 3 decimal places
329 exam-style questions on Edexcel A Level Maths Numerical Methods, covering 10.1 Locating Roots, 10.2 Iteration, 10.3 The Newton-Raphson Method, and 10.4 Applications to Modelling. Each one has a worked solution and a mark scheme showing where the marks go.