A slope field for the differential equation is drawn on a square grid that is not to…
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A slope field for the differential equation dxdy=y2−x is drawn on a square grid that is not to scale: one horizontal grid unit represents 0.20 on the x–axis, while one vertical grid unit represents 0.50 on the y–axis. Starting at Q0=(0,0), a path Q0,Q1,Q2,… is generated by repeatedly moving one horizontal grid unit to the right and then one vertical movement equal to the product of the horizontal change in x and the slope indicated at the current point.
(a) Using the grid scale information, express the horizontal and vertical displacements in coordinate units associated with a single step.
(b) Compute the coordinates of Q1 and Q2.
(c) Estimate the value of y when x=0.40 as given by this procedure, and clearly indicate the reasoning used.
(d) Suppose instead that the horizontal displacement is reduced to one–half of a grid unit while the vertical scale remains unchanged. Explain qualitatively how this modification is expected to influence the approximation obtained at x=0.40. Support your answer with reference to features of the slope field.