Slope Fields (College Board AP® Calculus AB): Revision Note

Slope fields

What is a slope field?

• A slope field for a differential equation is a diagram with short tangent lines drawn at a number of points

  • The slope of the tangent line drawn at any given point will be equal to the value of $\frac{\mathit{d}y}{\mathit{d}x}$ at that point

    • I.e. equal to the slope of the solution curve that goes through that point

  • Normally the tangent lines will be drawn for points that form a regularly-spaced grid of x and y values

How can I calculate the slope at a point on a slope field?

• Rewrite the differential equation in the form $\frac{\mathit{d}y}{\mathit{d}x}=g(x,y)$ (if it isn't in that form already)

  • I.e., the derivative $\frac{\mathit{d}y}{\mathit{d}x}$ is equal to some function of x and y

• Calculate the derivative $\frac{\mathit{d}y}{\mathit{d}x}$ at any point (x, y) by substituting the x and y values into $g(x,y)$

• This gives the slope of the solution curve at that point

Estimating a solution using slope fields

How can I use slope fields to estimate a solution for a differential equation?

• In some cases it may be possible to solve the differential equation analytically

  • But in other cases this is not possible

• The tangent lines in a slope field diagram give a general sense for what the solution curves will look like

  • Remember that the general solution to a differential equation is actually a family of solutions

  • At each point, the tangent line gives a sense of what one of those solutions is doing

• Think of the tangent lines in a slope diagram as ‘flow lines’

  • From a given point the solution curve through that point will ‘flow’ away from the point in the direction of the tangent line

To sketch a solution that goes through a particular point on a slope field

• The given point serves as a boundary condition,

  • letting you know which solution curve is the one you want to sketch

• The sketch should go through the given point,

  • and follow the general ‘flow’ of the tangent lines through the rest of the slope field diagram

• The sketched solution curve should not attempt to connect together different tangent lines in the diagram

  • You don't know that the solution curve goes through any exact point in the ‘grid’ of points at which tangent lines have been drawn

• The only tangent line that your solution curve should definitely go through

  • is the one at the given point

• The sketched solution curve may go along some of the tangent lines

  • but it should not cut across any of them

What else should I look out for on a slope field diagram?

• Look out for places where the tangent lines are horizontal

  • At such points $\frac{\mathit{d}y}{\mathit{d}x}=0$

  • Such points may indicate local minimum or maximum points for a solution curve

    • Be careful – not every point where $\frac{\mathit{d}y}{\mathit{d}x}=0$ is a local minimum or maximum

    • But every local minimum or maximum will be at a point where $\frac{\mathit{d}y}{\mathit{d}x}=0$

• You can solve the equation $\frac{\mathit{d}y}{\mathit{d}x}=g(x,y)=0$ directly to identify points where the slope is zero

  • This is another way to identify possible local minimum and maximum points

  • If such a point falls between the ‘grid points’ at which the tangent lines have been drawn

    • then this may be the only way to identify such a point exactly