5 Steps to a 5 AP Calculus BC 2019

324 STEP 4. Review the Knowledge You Need to Score High

Example 5

Write an equation for the curve that passes through the point (3, 4) and has a slope at any

point (x,y)as

dy

dx

=

x^2 + 1

2 y

.

Step 1. Separate variables: 2ydy=(x^2 +1)dx.

Step 2. Integrate both sides:

∫

2 ydy=

∫

(x^2 +1)dx;y^2 =

x^3

3

+x+C.

Step 3. Substitute given condition: 4^2 =

33

3

+ 3 +C⇒C= 4.

Thus, y^2 =

x^3

3

+x+ 4.

Step 4. Verify the result by differentiating:

2 y

dy

dx

=x^2 + 1

dy

dx

=

x^2 + 1

2 y

.

13.5 Slope Fields

Main Concepts:Slope Fields, Solution of Different Equations

Aslope field(or adirection field) for first-order differential equations is a graphic represen-

tation of the slopes of a family of curves. It consists of a set of short line segments drawn

on a pair of axes. These line segments are the tangents to a family of solution curves for the

differential equation at various points. The tangents show the direction the solution curves

will follow. Slope fields are useful in sketching solution curves without having to solve a

differential equation algebraically.

Example 1

If

dy

dx

= 0. 5 x, draw a slope field for the given differential equation.

Step 1: Set up a table of values for

dy

dx

for selected values ofx.

x − 4 − 3 − 2 − 101234

dy

dx

− 2 −1.5 − 1 −0.5 0 0.5 1 1.5 2

Note that since

dy

dx

= 0. 5 x, the numerical value of

dy

dx

is independent of the value

ofy. For example, at the points (1,−1), (1, 0), (1, 1), (1, 2), (1, 3), and at all

the points whosex-coordinates are 1, the numerical value of

dy

dx

is 0.5 regard-

less of theiry-coordinates. Similarly, for all the points whosex-coordinates are 2