Value, Change, Rate of Change

Many important questions about functions are local questions. We often wish to know what a function's behavior is like at or near specific choices for the independent variable. For example, we may ask:

This question asks: "For this particular value of  x, what will be the value of the associated output  y0 = f (x0) ?" In particular, we may wonder if this value is big or small.

If  y0 = f (x0 and  y1 = f (x1), this question asks for the change in value of the output  Dy = y1 – y0  as  x  changes from  x0  to  x1. In other words, it asks "How much has the value of the function gone up or gone down over this specific interval?"

Rate, as in everyday usage, refers to "how fast." To say how fast something – like the value of a function – is changing, we need to know not only how much it has changed, but also the length of the interval over which it has changed. If  y0 = f (x0 and  y1 = f (x1), we measure the rate of change of the function over the interval  Dx = x1 – x0  by:

Notice that the rate goes up if either  Dy  increases or if  Dx  decreases. That is, a certain change in value over a short interval will correspond to a greater rate of change than, say, the same change in value over a longer interval. Similarly, the rate will go down if either  Dy  decreases or if  Dx  increases.

The units for measuring rates are "change in output per change in input." Some familiar examples include miles per hour and cycles per second.

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