# Slope Fields

A *Slope Field* is one type of qualitative tool for studying first order ODEs. In the first order
equation,

**y' = f(t,y)**

the left hand side **y'** (or **dy/dt** as we sometimes denote it) represents the *rate of change* of **y** with respect to **t**. If we pick a point **(t,y)** in the ty-plane, the right hand side **f(t,y)** gives the value of the rate of change at that point. Note that this assumes that
we are analyzing the behavior of a solution to the ODE which actually passes through
the given point.

Here is an alternate way of thinking about it. Since **y(t)** represents the solution curve, then we can also interpret the left hand side of the
equation, **y'=y'(t)** , as the slope of the solution curve at any point on the curve. Again, if we pick
a point **(t,y)** in the ty-plane, the right hand side **f(t,y)** gives the value of the slope of the tangent to the curve at that point. By choosing
a point in the ty-plane and computing the value of **f**at that point, we are actually computing the slope of the tangent to the solution
curve at the point **(t,y)**. We can then use this slope value to draw a small piece of a tangent line at that
point. A slope field is nothing more than a graph of lots of mini-tangent lines at
various points in the ty-plane. The qualitative behavior of solution curves can then
be sketched by following the tangents.

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Updated on: 08/22/07.