f(x)=x3+2x2+x+34\displaystyle f(x) = x^3 + 2x^2 + x + \frac{3}{4}f(x)=x3+2x2+x+43 Let α \alpha\,α be the root of f(x)=0f(x)=0f(x)=0.
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+34x2+2)\displaystyle x = -\left(\frac{x+\frac{3}{4}}{x^2} + 2\right)x=−(x2x+43+2)
Use the iteration formula xn+1=−(xn+34xn2+2)\displaystyle x_{n+1} = -\left(\frac{x_n+\frac{3}{4}}{x_n^2} + 2\right)xn+1=−(xn2xn+43+2) with x0=−2x_0 = -2x0=−2 to find the value of x1x_1x1.
Hence, find, to 3 decimal places, the values of x2 x_2\,x2 and x3x_3x3.
By choosing a suitable interval, prove that α=−1.670\alpha = -1.670α=−1.670 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.