A positive increasing function f f\,f is defined on [a,b][a,b][a,b]. The interval is divided into n n\,n strips, each of width h=b−an\displaystyle h = \frac{b-a}{n}h=nb−a.
The left- and right-endpoint rectangle sums are denoted by Ln L_n\,Ln and Un U_n\,Un respectively.
Prove that Un−Ln=h(f(b)−f(a))U_n - L_n = h\bigl(f(b)-f(a)\bigr)Un−Ln=h(f(b)−f(a)).
The result is applied to f(x)=exf(x)=e^xf(x)=ex on [0,1][0,1][0,1]. Determine the least value of n n\,n for which Un−Ln<0.01U_n-L_n<0.01Un−Ln<0.01.
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.