5 Steps to a 5 AP Calculus BC 2019

More Applications of Definite Integrals 325

(e.g., (2,−1), (2, 0), (2, 3), etc. ),

dy

dx

=1. Also, remember that

dy

dx

represents the

slopes of the tangent lines to the curve at various points. You are now ready to draw

these tangents.

Step 2: Draw short line segments with the given slopes at the various points. The slope

field for the differential equation

dy

dx

= 0. 5 xis shown in Figure 13.5-1.

Figure 13.5-1

Example 2

Figure 13.5-2 shows a slope field for one of the differential equations given below. Identify
the equation.

Figure 13.5-2

(a)

dy

dx

= 2 x (b)

dy

dx

=− 2 x (c)

dy

dx

=y

(d)

dy

dx

=−y (e)

dy

dx

=x+y

Solution:

If you look across horizontally at any row of tangents, you’ll notice that the tangents
have the same slope. (Points on the same row have the samey-coordinate but different

x-coordinates.) Therefore, the numerical value of
dy
dx
(which represents the slope of the
tangent) depends solely on they-coordinate of a point and it is independent of thex-
coordinate. Thus, only choice (c) and choice (d) satisfy this condition. Also notice that
the tangents have a negative slope wheny>0 and have a positive slope wheny<0.

Therefore, the correct choice is (d)

dy

dx

=−y.