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.