Three separate slope-field panels correspond to differential equations of the form…

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

Three separate slope-field panels correspond to differential equations of the form

\frac{dy}{dx} = y(b-y)

with three different positive constants

b

. • In Panel I, every solution curve levels off near

y=4

. • In Panel II, every solution curve levels off near

y=7

. • In Panel III, every solution curve levels off near

y=9

. (a) For a general positive value of

b

, determine the

y

-value that solution curves with initial values

0 < y_0 < b

approach as

x \to \infty

. (b) Identify which panel corresponds to

b=7

and justify your choice. (c) A student claims that the panel with

b=4

has steeper line segments near

y=2

than the panel with

b=9

. Determine whether the claim is correct by computing the slopes at

y=2

for

b=4

and

b=9

, then comparing the values.