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AP Calculus
Notes: Slope Fields
1. Consider the differential equation
dy 1

dx x
Name_________________________
a. Sketch a slope field for the given differential equation.
y
x
b. Let y = f(x) be the particular solution to the differential equation with initial condition f(2) = 3. Draw a
solution curve through this point.
c. Find the particular solution y = f(x) to the given differential equation with initial condition f(2) = 3.
2. Consider the differential equation
dy
 y +2
dx
a. Sketch a slope field for the given differential equation.
y
x
b. Let y = f(x) be the particular solution to the differential equation with initial condition f(1) = 2. Draw a
solution curve through this point.
c. Find the particular solution y = f(x) to the given differential equation with initial condition f(1) = 2.
3. Consider the differential equation
dy
x

dx
y
a. Sketch a slope field for the given differential equation.
y
x
b. Let y = f(x) be the particular solution to the differential equation with the initial condition f(-2) = -1. Draw a
solution curve through this point. Write an equation for the line tangent to the graph of f at (-2, -1) and use it
to approximate f(-1.9).
c. Find the particular solution y = f(x) to the given differential equation with initial condition f(-2) = -1.
Analyzing slope fields
dy
 ex
dx
dy x

dx y
dy
 y (2  y )
dx
dy 3x 2  xy
The slope field for the differential equation
will have vertical segments when:______

dx 4 x  8 y
a) x = 2y
b) y = 2x
horizontal segments when:______
c) x = 0
d) y = 3x
e) x = 0, y = 3x