Figure 1 below shows the graph of y=ln(4x+3)y = \ln(4x + 3)y=ln(4x+3) and the graph of y=xy = xy=x.
Show that the equation ln(4x+3)=x\ln(4x + 3) = xln(4x+3)=x has a root between 2 and 3.
Use Figure 1, starting with x1=2x_1 = 2x1=2, to determine whether the iteration formula
xn+1=ln(4xn+3)x_{n+1} = \ln(4x_n + 3)xn+1=ln(4xn+3)
can be used to find an approximation for this root. Justify your answer.
125 exam-style questions on OCR A Level Maths 1.9 Numerical Methods (A-level only), covering 1.9.1 Locating roots by sign change (A-level only), 1.9.2 Failure of sign change methods (A-level only), 1.9.3 Simple iterative methods (A-level only), 1.9.4 Newton-Raphson method (A-level only), 1.9.5 Failure of iterative methods (A-level only), 1.9.6 Numerical integration (A-level only), and 1.9.7 Numerical methods in context (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.