The function f f\,f is defined by
f(x)=exx−1f(x) = e^x\sqrt{x} - 1f(x)=exx−1, x≥0x \geq 0x≥0
The equation f(x)=0f(x) = 0f(x)=0 has a single solution at x=αx = \alphax=α. By considering a suitable change of sign, show that α \alpha\,α lies between 0 and 1.
Show that f′(x)=ex(2x+1)2xf'(x) = \dfrac{e^x(2x + 1)}{2\sqrt{x}}f′(x)=2xex(2x+1).
Use the Newton-Raphson method with x1=1x_1 = 1x1=1 to find x3x_3x3, an approximation for α\alphaα. Give your answer to five decimal places.
Explain why the Newton-Raphson method cannot be used to find α \alpha\,α with x1=0x_1 = 0x1=0.
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.