A slope field for the differential equation is shown on a square grid of integer…
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A slope field for the differential equation dxdy=x2−y is shown on a square grid of integer lattice points for −3≤x≤3 and −2≤y≤7.
• Along the parabola y=x2 the arrows are horizontal.
• Above the parabola the arrows tilt downward (negative slope); below the parabola they tilt upward (positive slope).
Consider the integral curve passing through the point (−1,1).
(a) For each of the three x–intervals (−∞,−1), (−1,1), and (1,∞), determine whether the integral curve is increasing or decreasing. Justify your conclusions by referencing features of the slope field.
(b) Using only information visible in the slope field, estimate the x–coordinate at which this integral curve attains its minimum value. Explain the reasoning that leads to your estimate.
(c) Explain whether this integral curve intersects the parabola y=x2 again for any x>−1. Support your answer with appropriate justification based on the qualitative behavior of the field.