∫ = ∫k dt
Then, integrate both sides (ln y = kt + C) and put them in exponential form.
y = ekt+c = CektNext, use the information about the population to solve for the constants. If you treat 1980 as t = 0 and
1990 as t = 10, then
10,000 = Cek(0) and 13,000 = Cek(10)So C = 10,000 and k = ln 1.3 ≈ 0.0262.
The equation for population growth is approximately y = 10,000e .0262t. We can estimate that the
population in 2000 will be
y = 10,000e 0.0262(20) = 16,900SLOPE FIELDS
The idea behind slope fields, also known as direction fields, is to make a graphical representation of the
slope of a function at various points in the plane. We are given a differential equation, but not the equation
itself. So how do we do this? Well, it’s always easiest to start with an example.
Example 1: Given = x, sketch the slope field of the function.
What does this mean? Look at the equation. It gives us the derivative of the function, which is the slope of
the tangent line to the curve at any point x. In other words, the equation tells us that the slope of the curve
at any point x is the x-value at that point.
For example, the slope of the curve at x = 1 is 1. The slope of the curve at x = 2 is 2. The slope of the
curve at the origin is 0. The slope of the curve at x = −1 is −1. We will now represent these different
slopes by drawing small segments of the tangent lines at those points. Let’s make a sketch.