121616949-math.54 - 40 Chapter 2 Instantaneous Rate Of...

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40 Chapter 2 Instantaneous Rate Of Change: The Derivative Make sure you indicate any places where the derivative does not exist. 0 1 2 3 4 0 1 2 3 4 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Adjectives For Functions As we have defined it in Section 1.3 , a function is a very general object. At this point, it is useful to introduce a collection of adjectives to describe certain kinds of functions. Consider the graphs of the functions in Figure 2.3 . It would clearly be useful to have words to help us describe the distinct features of each of them. We will point out and define a few adjectives (there are many more) for the functions pictured here. For the sake of the discussion, we will assume that the graphs do not exhibit any unusual behavior off-stage (i.e., outside the view of the graphs).
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Unformatted text preview: Functions. Each graph in Figure 2.3 certainly represents a function—since each passes the vertical line test . In other words, as you sweep a vertical line across the graph of each function, the line never intersects the graph more than once. If it did, then the graph would not represent a function. Bounded. The graph in (c) appears to approach zero as x goes to both positive and negative infinity. It also never exceeds the value 1 or drops below the value 0. Because the graph never increases or decreases without bound, we say that the function represented by the graph in (c) is a bounded function. DEFINITION 2.17 Bounded A function f is bounded if there is a number M such that | f ( x ) | < M for every x in the domain of f ....
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