A slope field for a first-order differential equation is constructed by drawing short…
MATHEMATICS · Single correct · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A slope field for a first-order differential equation is constructed by drawing short line segments whose slopes match the value of dxdy at selected points in the plane. Which of the following best explains why tracing a curve that follows these segments yields a reasonable approximation to a particular solution?
(a) Because the short line segments themselves lie on the actual solution curve, drawing a smooth curve through them reproduces the solution exactly.
(b) Because the solution’s tangent at every point must share the same slope given by the differential equation, small steps taken in the indicated directions locally trace the curve.
(c) Because the segments only indicate whether the function is increasing or decreasing, following them preserves monotonicity but is unrelated to the tangent direction.
(d) Because the segments represent average slopes over large intervals, connecting their endpoints approximates secant rather than tangent behavior of the curve.