MATH 221 Rates Of Change Assignment

The two previous examples have much in common. If we ignore all the details about geometry, graphs, highways and motion, the following happened in both examples:

We had a function y = f(x), and we wanted to know how much f(x) changes if x changes. If you change x to x + ?x, then y will change from f(x) to f(x + ?x). The change in y is therefore
y = f(x + x)  f(x),
and the average rate of change is
y /x = f(x + x)  f(x) /x

This is the average rate of change of f over the interval from x to x + x. To define the rate of change of the function f at x we let the length x of the interval become smaller and smaller, in the hope that the average rate of change over the shorter and shorter time intervals will get closer and closer to some number.

If that happens then that “limiting number” is called the rate of change of f at x, or, the derivative of f at x. It is written as
f0 (x) = lim x0 f(x + x)  f(x) /x

Derivatives and what you can do with them are what the first half of this semester is about. The description we just went through shows that to understand what a derivative is you need to know what a limit is. In the next chapter we’ll study limits so that we get a less vague understanding of formulas like

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