Two students approximate the solution to with initial condition by drawing polygonal…
MATHEMATICS · Single correct · AP · Calculus BC · Differential Equations · Differential Equations
Problem
Two students approximate the solution to dxdy=−y with initial condition y(0)=1 by drawing polygonal paths on the slope field.
• Student A uses equal horizontal steps of length 1.
• Student B uses equal horizontal steps of length 0.25.
Which statement best describes how their approximations compare with the exact solution y=e−x on the interval 0≤x≤1?
(a) Both paths lie above the exact solution, and Student B’s path is closer to the exact curve.
(b) Both paths lie below the exact solution, and Student B’s path is closer to the exact curve.
(c) Student A’s path is closer to the exact solution than Student B’s because the larger step size reduces accumulated error.
(d) Student A’s path lies below while Student B’s path lies above the exact solution throughout the interval.