MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations
Problem
Let y(x) satisfy dxdy=cosx−y with y(0)=0. Suppose y(1) is to be approximated numerically. It can be shown that ∣y′′(x)∣≤2 for all x in [0,1].
A researcher wishes to guarantee that the magnitude of the global error at x=1 does not exceed 1×10−4.
(a) Derive an inequality that relates the global error at x=1 to the step size h using the fact that ∣y′′(x)∣ is bounded by 2 on [0,1]. Clearly state any theorems or estimates you invoke.
(b) Use the inequality from part (a) to determine a theoretical upper bound for the step size h that will meet the 1×10−4 error tolerance. State the corresponding minimum number of uniform steps on [0,1]. Express your answer to at least three significant digits.
(c) To test the sharpness of the theoretical bound, the researcher computed numerical approximations with step sizes h=0.125 and h=0.0625, obtaining absolute errors 4.6×10−4 and 2.3×10−4, respectively. Analyze these results to comment on whether the bound from part (b) is overly conservative or appropriately tight. Support your conclusion with quantitative reasoning.