Summary

The exponential function has a really interesting property: it is the only interesting function that has itself as its own derivative. (Do you know what the other function with this astonishing property is?)

This means that the function value at a given value of x simultaneously represents the slope of the tangent line.

Don't worry too much about other bases. You can always write an exponential function in base e. Convert, and then work with base e.

Another very useful function is introduced in this section, and it's worth paying attention to: a logistic function is of the form

Notice that this function contains some parameters: k and a. Each has its implications for the shape of the curve, but Figure 12 on page 251 shows the important facets of the logistic: they grow exponentially at the beginning, then taper off to a horizontal asymptote (given by the value of k). a determines how fast the transition occurs between low and high values.

Examples:

#1, p. 251

#8, p. 251

#16, p. 251

#36, p. 252