Consider the differential equation with initial condition
MATHEMATICS · Single correct · AP · Calculus AB · Differential Equations · Differential Equations
Problem
Consider the differential equation dxdy=y2 with initial condition y(0)=1. For a small positive step size h, a single tangent-line step is taken to approximate y(h). Which statement about the resulting approximation is most accurate?
(a) It will overestimate the exact value because a concave-up curve always lies below its tangent line near the point of tangency.
(b) It will coincide with the exact value for sufficiently small h since the tangent-line approximation and the function have the same value and slope at x=0.
(c) It will underestimate the exact value because the solution curve is concave upward at x=0.
(d) It will overestimate the exact value because the slope of the solution is increasing for x>0, so using the initial slope is too large.