For the differential equation

MATHEMATICS · Single correct · AP · Calculus AB · Differential Equations · Differential Equations

Problem

\dfrac{dy}{dx}=cy

, two students compare slope fields generated with

c=2

and

c=-2

. Which of the following correctly describes how the sign of

c

affects the overall appearance of the slope field? (a) For

c>0

, every segment slopes upward to the right, whereas for

c<0

, every segment slopes downward to the right. (b) Reversing the sign of

c

reverses the sign of the slope at every point, so the slope field for

c=-2

is obtained from that for

c=2

by flipping the tilt of every segment. (c) Changing the sign of

c

reverses the orientation of segments above the

x

-axis only; segments below the

x

-axis are unaffected. (d) Changing

c

from positive to negative alters the steepness (length) of line segments but not their upward or downward direction.