APEX Calculus

Our study of “nice” functions \(f\) in this chapter has so far focused on individual points: points where \(f\) is maximal/minimal, points where \(\fp(x) = 0\) or \(\fp\) does not exist, and points \(c\) where \(\fp(c)\) is the average rate of change of \(f\) on some interval.

In this section we begin to study how functions behave between special points; we begin studying in more detail the shape of their graphs.

We start with an intuitive concept. Given the graph in Figure 3.3.1, where would you say the function is increasing? Decreasing? Even though we have not defined these terms mathematically, one likely answered that \(f\) is increasing when \(x \gt 1\) and decreasing when \(x\lt 1\text{.}\) We formally define these terms here.

Informally, a function is increasing if as \(x\) gets larger (i.e., looking left to right) \(f(x)\) gets larger.

Our interest lies in finding intervals in the domain of \(f\) on which \(f\) is either increasing or decreasing. Such information should seem useful. For instance, if \(f\) describes the speed of an object, we might want to know when the speed was increasing or decreasing (i.e., when the object was accelerating vs. decelerating). If \(f\) describes the population of a city, we should be interested in when the population is growing or declining.

To find such intervals, we again consider secant lines. Let \(f\) be an increasing, differentiable function on an open interval \(I\text{,}\) such as the one shown in Figure 3.3.4, and let \(a\lt b\) be given in \(I\text{.}\) The secant line on the graph of \(f\) from \(x=a\) to \(x=b\) is drawn; it has a slope of \((f(b)-f(a))/(b-a)\text{.}\)

But note, since \(b \gt a\) and \(f\) is increasing, \(f(b) \gt f(a)\text{.}\) And these facts imply \(b-a \gt 0\) and \(f(b)-f(a) \gt 0\text{.}\) Therefore:

By considering all such secant lines in \(I\text{,}\) we strongly imply that \(\fp(x) \gt 0\) on \(I\text{.}\) A similar statement can be made for decreasing functions.

Our above logic can be summarized as “If \(f\) is increasing, then \(\fp\) is probably positive.” Theorem 3.3.5 below turns this around by stating “If \(\fp\) is positive, then \(f\) is increasing.” This leads us to a method for finding when functions are increasing and decreasing.

Let \(f\) be differentiable on an interval \(I\) and let \(a\) and \(b\) be in \(I\) where \(\fp(a) \gt 0\) and \(\fp(b)\lt 0\text{.}\) If \(\fp\) is continuous on \([a,b]\text{,}\) it follows from the Intermediate Value Theorem that there must be some value \(c\) between \(a\) and \(b\) where \(\fp(c) = 0\text{.}\) (It turns out that this is still true even if \(\fp\) is not continuous on \([a,b]\text{.}\)) This leads us to the following method for finding intervals on which a function is increasing or decreasing.

We demonstrate using this process in the following example.

One is justified in wondering why so much work is done when the graph seems to make the intervals very clear. We give three reasons why the above work is worthwhile.

First, the points at which \(f\) switches from increasing to decreasing are not precisely known given a graph. The graph shows us something significant happens near \(x=-1\) and \(x=0.3\text{,}\) but we cannot determine exactly where from the graph.

One could argue that just finding critical values is important; once we know the significant points are \(x=-1\) and \(x=1/3\text{,}\) the graph shows the increasing/decreasing traits just fine. That is true. However, the technique prescribed here helps reinforce the relationship between increasing/decreasing and the sign of \(\fp\text{.}\) Once mastery of this concept (and several others) is obtained, one finds that either (a) just the critical points are computed and the graph shows all else that is desired, or (b) a graph is never produced, because determining increasing/decreasing using \(\fp\) is straightforward and the graph is unnecessary. So our second reason why the above work is worthwhile is this: once mastery of a subject is gained, one has options for finding needed information. We are working to develop mastery.

Finally, our third reason: many problems we face “in the real world” are very complex. Solutions are tractable only through the use of computers to do many calculations for us. Computers do not solve problems “on their own,” however; they need to be taught (i.e., programmed) to do the right things. It would be beneficial to give a function to a computer and have it return maximum and minimum values, intervals on which the function is increasing and decreasing, the locations of relative maxima, etc. The work that we are doing here is easily programmable. It is hard to teach a computer to “look at the graph and see if it is going up or down.” It is easy to teach a computer to “determine if a number is greater than or less than \(0\text{.}\)”

In Section 3.1 we learned the definition of relative maxima and minima and found that they occur at critical points. We are now learning that functions can switch from increasing to decreasing (and vice-versa) at critical points. This new understanding of increasing and decreasing creates a great method of determining whether a critical point corresponds to a maximum, minimum, or neither. Imagine a function increasing until a critical point at \(x=c\text{,}\) after which it decreases. A quick sketch helps confirm that \(f(c)\) must be a relative maximum. A similar statement can be made for relative minimums. We formalize this concept in a theorem.

One is often tempted to think that functions always alternate “increasing, decreasing, increasing, decreasing,…” around critical values. Our previous example demonstrated that this is not always the case. While \(x=1\) was not technically a critical value, it was an important value we needed to consider. We found that \(f\) was decreasing on “both sides of \(x=1\text{.}\)”

We examine one more example.

We have seen how the first derivative of a function helps determine when the graph of a function is going “up” or “down.” In the next section, we will see how the second derivative helps determine how the graph of a function curves.