A numerical procedure generates successive points that satisfy

MATHEMATICS · Single correct · AP · Calculus AB · Differential Equations · Differential Equations

Problem

A numerical procedure generates successive points (xk,yk)(x_k,y_k) that satisfy y′=f(x,y)y'=f(x,y), using a variable increment hkh_k at each step. The program measures the change in slope between two consecutive points with Δfk=∣f(xk,yk)−f(xk−1,yk−1)∣\Delta f_k=|f(x_{k},y_{k})-f(x_{k-1},y_{k-1})|. Which of the following update rules for the next increment hk+1h_{k+1} is most likely to reduce local error in regions where the slope varies rapidly? (a) Set hk+1=hk (Δfk/tol)h_{k+1}=h_k\,\bigl(\Delta f_k/\text{tol}\bigr). (b) If Δfk\Delta f_k exceeds a chosen tolerance, set hk+1=2hkh_{k+1}=2h_k; otherwise leave hk+1h_{k+1} unchanged. (c) If Δfk\Delta f_k exceeds a chosen tolerance, set hk+1=12hkh_{k+1}=\tfrac12 h_k; otherwise leave hk+1h_{k+1} unchanged. (d) If Δfk\Delta f_k is *below* the tolerance, set hk+1=12hkh_{k+1}=\tfrac12 h_k; otherwise leave hk+1h_{k+1} unchanged.

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