MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A direction field for dxdy=x+y−2 is drawn on −4≤x≤4, −2≤y≤6. The field suggests that along the line y=2−x the arrows are horizontal, with the arrows pointing upward when y>2−x and downward when y<2−x.
(a) Sketch a reasonable approximation of the integral curve that passes through the point (0,2) on the axes provided. Your sketch should reflect the local orientation of arrows and any notable changes in steepness.
(b) Determine whether the integral curve from part (a) has a local maximum, a local minimum, or neither at x=0. Explain your conclusion by referring both to the differential equation and to the orientation of arrows near (0,2).
(c) Without solving the differential equation analytically, use qualitative reasoning to explain why the integral curve approaches the line y=2−x as x→−∞, but diverges away from that line as x→∞.
(d) Estimate the x–coordinate at which the integral curve you sketched crosses the horizontal line y=4. Describe how features of the slope field guided your estimate.