A slope field displays the following characteristics
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A slope field displays the following characteristics.
• The field is symmetric with respect to the origin: rotating any segment 180∘ about (0,0) leaves the field unchanged.
• All segments along both coordinate axes are horizontal.
• In Quadrants I and III the segments slope upward; in Quadrants II and IV they slope downward.
• The farther a point is from the origin, the steeper the segment located there, with the steepness growing proportionally to the product of the distances to the two axes.
Using only these qualitative observations, complete the tasks below.
(a) Identify the loci of points where the slope is zero and explain how the slope changes sign across those loci.
(b) Construct the simplest polynomial expression F(x,y) such that dxdy=F(x,y) is consistent with every characteristic listed above.
(c) Verify algebraically that your expression from part (b) has the correct sign in each quadrant and vanishes exactly on both axes.