**Resources** from OpenStax Calc II

College Board YouTube

& Khan Academy Calc AB

Slope Fields are a unit 7 topic that show the expected slope for any point in the x-y plane. You will not be asked to create these graphs but you may have to interpret them.

A **direction field (slope field)** is a mathematical object used to graphically represent solutions to a first-order differential equation. At each point in a direction field, a line segment appears whose slope is equal to the slope of a solution to the differential equation passing through that point.

## Example

An applied example of this type of differential equation appears in Newton’s law of cooling, which we will solve explicitly later in this chapter. First, though, let us create a direction field for the differential equation.

T′(𝑡) = −0.4(𝑇−72)

Here 𝑇(𝑡)T(t) represents the temperature (in degrees Fahrenheit) of an object at time 𝑡,t, and the ambient temperature is 72°F.72°F. Figure 4.6 shows the direction field for this equation.