04/19/23 Perkins

AP Calculus AB

Day 6Section 6.1

2

-2

Slope Field a visual representation of the derivative of a function at individual points

2dy

xdx

used to show the rate of change of a function at a specific set of given points

b. Sketch two possible solution curves for the differential equation.

The slope of each tangent line depends on the x-value.

Solution Curve a possible graph of the differential equation for a given slope field.

1a. Sketch the slope field for x and y in [-3,3] to represent the differential equation

2. This is the slope field for:

.dy

a xdx

. 2dy

b xdx

.dy

c ydx

. 2dy

d ydx

. 1dy

e ydx

2

-2

The slope of each tangent line depends on the y-value.

There are horizontal tangents when y = 0.

Slopes are negative when y’s are negative.

2

-2

3. Sketch a solution of the differential equation which must pass though the point (2,0). Then solve the differential equation analytically.

dy x

dx y

ydy xdx

y dy xdx 2 21 1

2 2y x C

2 2y x C 2 2x y C At (2,0)

4 0 C 2 2 4x y

Perkins

AP Calculus AB

Day 6Section 6.1

2

-2

Slope Field a visual representation of the derivative of a function at individual points

2dy

xdx

used to show the rate of change of a function at a specific set of given points

b. Sketch two possible solution curves for the differential equation.

Solution Curve a possible graph of the differential equation for a given slope field.

1a. Sketch the slope field for x and y in [-3,3] to represent the differential equation

2. This is the slope field for:

.dy

a xdx

. 2dy

b xdx

.dy

c ydx

. 2dy

d ydx

. 1dy

e ydx

2

-2

3. Sketch a solution curve for the differential equation which must pass though the point (2,0). Then solve the differential equation analytically.

dy x

dx y

2

-2