Active Calculus

So far, we have used the words increasing and decreasing intuitively to describe a function’s graph. Here we define these terms more formally.

Simply put, an increasing function is one that is rising as we move from left to right along the graph, and a decreasing function is one that falls as the value of the input increases. If the function has a derivative, the sign of the derivative tells us whether the function is increasing or decreasing.

Let \(f\) be a function that is differentiable on an interval \((a,b)\text{.}\) It is possible to show that if \(f'(x) > 0\) for every \(x\) such that \(a \lt x \lt b\text{,}\) then \(f\) is increasing on \((a,b)\text{;}\) similarly, if \(f'(x) \lt 0\) on \((a,b)\text{,}\) then \(f\) is decreasing on \((a,b)\text{.}\)

For example, the function pictured in Figure 1.6.3 is increasing on the entire interval \(-2 \lt x \lt 0\text{,}\) and decreasing on the interval \(0 \lt x \lt 2\text{.}\) Note that the value \(x = 0\) is not included in either interval since at this location, the function is changing from increasing to decreasing.

Because \(f'\) is itself a function, it is perfectly feasible for us to consider the derivative of the derivative, which is the new function \(y = [f'(x)]'\text{.}\) We call this resulting function the second derivative of \(y = f(x)\text{,}\) and denote the second derivative by \(y = f''(x)\text{.}\) Consequently, we will sometimes call \(f'\) “the first derivative” of \(f\text{,}\) rather than simply “the derivative” of \(f\text{.}\)

The meaning of the derivative function still holds, so when we compute \(y = f''(x)\text{,}\) this new function measures slopes of tangent lines to the curve \(y = f'(x)\text{,}\) as well as the instantaneous rate of change of \(y = f'(x)\text{.}\) In other words, just as the first derivative measures the rate at which the original function changes, the second derivative measures the rate at which the first derivative changes. The second derivative will help us understand how the rate of change of the original function is itself changing.

In addition to asking whether a function is increasing or decreasing, it is also natural to inquire how a function is increasing or decreasing. There are three basic behaviors that an increasing function can demonstrate on an interval, as pictured in Figure 1.6.5: the function can increase more and more rapidly, it can increase at the same rate, or it can increase in a way that is slowing down. Fundamentally, we are beginning to think about how a particular curve bends, with the natural comparison being made to lines, which don’t bend at all. More than this, we want to understand how the bend in a function’s graph is tied to behavior characterized by the first derivative of the function.

On the leftmost curve in Figure 1.6.5, draw a sequence of tangent lines to the curve. As we move from left to right, the slopes of those tangent lines will increase. Therefore, the rate of change of the pictured function is increasing, and this explains why we say this function is increasing at an increasing rate. For the rightmost graph in Figure 1.6.5, observe that as \(x\) increases, the function increases, but the slopes of the tangent lines decrease. This function is increasing at a decreasing rate.

Similar options hold for how a function can decrease. Here we must be extra careful with our language, because decreasing functions involve negative slopes. Negative numbers present an interesting tension between common language and mathematical language. For example, it can be tempting to say that “\(-100\) is bigger than \(-2\text{.}\)” But we must remember that “greater than” describes how numbers lie on a number line: \(x \gt y\) provided that \(x\) lies to the right of \(y\text{.}\) So of course, \(-100\) is less than \(-2\text{.}\) Informally, it might be helpful to say that “\(-100\) is more negative than \(-2\text{.}\)” When a function’s values are negative, and those values get more negative as the input increases, the function must be decreasing.

Now consider the three graphs shown in Figure 1.6.6. Clearly the middle graph depicts a function decreasing at a constant rate. Now, on the first curve, draw a sequence of tangent lines. We see that the slopes of these lines get less and less negative as we move from left to right. That means that the values of the first derivative, while all negative, are increasing, and thus we say that the leftmost curve is decreasing at an increasing rate.

This leaves only the rightmost curve in Figure 1.6.6 to consider. For that function, the slopes of the tangent lines are negative throughout the pictured interval, but as we move from left to right, the slopes get more and more negative. Hence the slope of the curve is decreasing, and we say that the function is decreasing at a decreasing rate.

We now introduce the notion of concavity which provides simpler language to describe these behaviors. When a curve opens upward on a given interval, like the parabola \(y = x^2\) or the exponential growth function \(y = e^x\text{,}\) we say that the curve is concave up on that interval. Likewise, when a curve opens down, like the parabola \(y = -x^2\) or the opposite of the exponential function \(y = -e^{x}\text{,}\) we say that the function is concave down. Concavity is linked to both the first and second derivatives of the function.

In Figure 1.6.7, we see two functions and a sequence of tangent lines to each. On the lefthand plot, where the function is concave up, observe that the tangent lines always lie below the curve itself, and the slopes of the tangent lines are increasing as we move from left to right. In other words, the function \(f\) is concave up on the interval shown because its derivative, \(f'\text{,}\) is increasing on that interval. Similarly, on the righthand plot in Figure 1.6.7, where the function shown is concave down, we see that the tangent lines always lie above the curve, and the slopes of the tangent lines are decreasing as we move from left to right. The fact that its derivative, \(f'\text{,}\) is decreasing makes \(f\) concave down on the interval.

We state these most recent observations formally as the definitions of the terms concave up and concave down.

Exploring the context of position, velocity, and acceleration is an excellent way to understand how a function, its first derivative, and its second derivative are related to one another. In Activity 1.6.2, we can replace \(s\text{,}\) \(v\text{,}\) and \(a\) with an arbitrary function \(f\) and its derivatives \(f'\) and \(f''\text{,}\) and essentially all the same observations hold. In particular, note that the following are equivalent: on an interval where the graph of \(f\) is concave up, \(f'\) is increasing and \(f''\) is positive. Likewise, on an interval where the graph of \(f\) is concave down, \(f'\) is decreasing and \(f''\) is negative.