A first-order differential equation is represented by a slope field on the rectangle and
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A first-order differential equation is represented by a slope field on the rectangle −3≤x≤3 and −3≤y≤3. Selected features of the field are described below.
• Along the line y=x the line segments are horizontal.
• In the region above the line y=x the segments slope downward from left to right and become steeper as the vertical distance above y=x increases.
• In the region below the line y=x the segments slope upward from left to right and become steeper as the vertical distance below y=x increases.
A particular solution passes through the point (0,2).
(1) Part 1: (a) Sketch a possible graph of the solution that passes through (0,2) on the axes provided. Label clearly the point (0,2).
(2) Part 2: (b) Describe, in interval notation, where the solution is increasing and where it is decreasing for −3<x<3. Justify your answer using properties of the slope field.
(3) Part 3: (c) Determine the intervals of concavity for the solution on −3<x<3. Explain how changes in the orientation of the line segments support your conclusion.
(4) Part 4: (d) Does the solution have a relative extremum on the interval −3<x<3? If so, state whether it is a maximum or minimum, give its approximate coordinates, and justify your answer.