AS/A2 Pure 1 Pure 2

P3 Topic 4: Differentiation
P3.2
Observe the function y = xe-x/n. (a) Try various values of n, like n = 1, 2, 3, 4, ... What is the gradient at x=0? Use the tangent line provided to identify the maximum point on each graph. Zoom in to check whether the maxima really are locally flat. Now prove your answers on paper. (b) Try the function y = x2e-x/n for values of n like 0.5, 1, 1.5, 2, ... Identify the maximum for each graph. What is its x-coordinate in terms of n? Prove your answer. (c) Now generalise the function to y = xke-x/n for various values of n and k. Identify the maxima. Prove your answer. Does it give a general rule which works for (a) and (b) above as well? (d) Can you identify the turning points for the function y = xe-2x2/n2 ?

y = xe-x/n

y = x2e-x/n

y = xke-x/n

y = xe-2x2/n2

Summary This problem is adapted from "Exploring Maths with Computers" by Aydin Önaç, The Chase, Malvern Based on free Java applets from JavaMath