Water flows into a tank at a rate modelled by
q(t)=te−t/2q(t)=te^{-t/2}q(t)=te−t/2 litres per minute, 0≤t≤40\leq t\leq40≤t≤4,
where t t\,t is measured in minutes. Four rectangles, each of width 1 minute, are used to bound the volume of water entering the tank.
Show that q q\,q is increasing on [0,2][0,2][0,2] and decreasing on [2,4][2,4][2,4].
Write down lower and upper rectangle sums that take account of this change in monotonicity.
Hence calculate lower and upper bounds for the volume, giving each answer to three decimal places.
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.