Slopes Along Curves

The same method can be applied to other curves. The third degree curve y = x³ is shown in Figure 1.4.6. Let (x₀, y₀) be any point on the curve y = x³, and let Δx be a positive or a negative infinitesimal. Let Δy be the corresponding change in y along the curve.

In Figure 1.4.7, Δx and Δy are shown under a microscope. We again define the slope at (x₀, y₀) by

[slope at (x₀, y₀)] = [the real number infinitely close to ]

We now compute the hyperreal number .

and finally

= 3x₀² + 3x₀ Δx + (Δx)².

In the next section we shall develop some rules about infinitesimals which will enable us to show that since Δx is infinitesimal,

3x₀ Δx + (Δx)²

is infinitesimal as well. Therefore the hyperreal number

3x₀² + 3x₀ Δx + (Δx)²

is infinitely close to the real number 3x₀², whence

[slope at (x₀,y₀)] = 3x₀².

For example, at (0, 0) the slope is zero, at (1, 1) the slope is 3, and at (2, 8) the slope is 12.

We shall return to the study of the slope of a curve in Chapter 2 after we have learned more about hyperreal numbers. From the last example it is evident that we need to know how to show that two numbers are infinitely close to each other. This is our next topic.