APEX Calculus

A common amusement park ride lifts riders to a height then allows them to freefall a certain distance before safely stopping them. Suppose such a ride drops riders from a height of $150$ feet. Students of physics may recall that the height (in feet) of the riders, $t$ seconds after freefall (and ignoring air resistance, etc.) can be accurately modeled by $f(t)=-16t^2+150\text{.}$

Using this formula, it is easy to verify that, without intervention, the riders will hit the ground when $f(t)=0$ so at $t=2.5\sqrt{1.5}\approx 3.06$ seconds. Suppose the designers of the ride decide to begin slowing the riders’ fall after $2$ seconds (corresponding to a height of $f(2)=$ 86 ft). How fast will the riders be traveling at that time?

We have been given a position function, but what we want to compute is a velocity at a specific point in time, i.e., we want an instantaneous velocity. We do not currently know how to calculate this.

We can approximate the instantaneous velocity at $t=2$ by considering the average velocity over some time period containing $t=2\text{.}$ If we make the time interval small, we will get a good approximation. (This fact is commonly used. For instance, high speed cameras are used to track fast moving objects. Distances are measured over a fixed number of frames to generate an accurate approximation of the velocity.)

We really want to use $h=0\text{,}$ but this, of course, returns the familiar “$0/0$” indeterminate form. So we employ a limit, as we did in Section 1.1.

We can approximate the value of this limit numerically with small values of $h$ as seen in Figure 2.1.2. It looks as though the velocity is approaching -64 ^ft⁄_s.

Graphically, we can view the average velocities we computed numerically as the slopes of secant lines on the graph of $f$ going through the points $(2,f(2))$ and $(2+h,f(2+h))\text{.}$ In Figures 2.1.3–2.1.5, the secant line corresponding to $h=1$ is shown in three contexts. Figure 2.1.3 shows a “zoomed out” version of $f$ with its secant line. In Figure 2.1.4, we zoom in around the points of intersection between $f$ and the secant line. Notice how well this secant line approximates $f$ between those two points — it is a common practice to approximate functions with straight lines.

As $h\to 0\text{,}$ these secant lines approach the tangent line, a line that goes through the point $(2,f(2))$ with the special slope of $-64\text{.}$ In Figure 2.1.5 and Figure 2.1.6, we zoom in around the point $(2,86)\text{.}$ We see the secant line, which approximates $f$ well, but not as well the tangent line shown in Figure 2.1.6.

We have just introduced a number of important concepts that we will flesh out more within this section. First, we formally define two of them.

Some examples will help us understand these definitions.

In Definition 2.1.7, we assumed that the function is continuous, but this is actually not necessary. One can in fact prove that a function has to be continuous at any point where it is differentiable. Or, in other words, a function cannot be differentiable at a point of discontinuity. This is explained in the video in Figure 2.1.12.

Another important line that can be created using information from the derivative is the normal line. It is perpendicular to the tangent line, hence its slope is the negative-reciprocal of the tangent line’s slope.

Linear functions are easy to work with; many functions that arise in the course of solving real problems are not easy to work with. A common practice in mathematical problem solving is to approximate difficult functions with not-so-difficult functions. Lines are a common choice. It turns out that at any given point on the graph of a differentiable function $f\text{,}$ the best linear approximation to $f$ is its tangent line. That is one reason we’ll spend considerable time finding tangent lines to functions.

One type of function that does not benefit from a tangent line approximation is a line; it is rather simple to recognize that the tangent line to a line is the line itself. We look at this in the following example.

We often desire to find the tangent line to the graph of a function without knowing the actual derivative of the function. While we will eventually be able to find derivatives of many common functions, the algebra and limit calculations on some functions are complex. Until we develop further techniques, the best we may be able to do is approximate the tangent line. We demonstrate this in the next example.

Recall from Section 1.3 that $\underset{x\to 0}{lim}\frac{\sin (x)}{x}=1\text{,}$ meaning for values of $x$ near $0\text{,}$ $\sin (x)\approx x\text{.}$ Since the slope of the line $y=x$ is $1$ at $x=0\text{,}$ it should seem reasonable that “the slope of $f(x)=\sin (x)$” is near $1$ at $x=0\text{.}$ In fact, since we approximated the value of the slope to be $0.9983\text{,}$ we might guess the actual value is 1. We’ll come back to this later.

This process describes a function; given one input (the value of $c$), we return exactly one output (the value of $f^{\prime }(c)$). The “do something” box is where the tedious work (taking limits) of this function occurs.

The output is the derivative function, $f^{\prime }(x)\text{.}$ The $f^{\prime }(x)$ function will take a number $c$ as input and return the derivative of $f$ at $c\text{.}$ This calls for a definition.

Important: The notation $\frac{dy}{dx}$ is one symbol; it is not the fraction “$dy/dx$”. The notation, while somewhat confusing at first, was chosen with care. A fraction-looking symbol was chosen because the derivative has many fraction-like properties. Among other places, we see these properties at work when we talk about the units of the derivative, when we discuss the Chain Rule, and when we learn about integration (topics that appear in later sections and chapters).

Examples will help us understand this definition.

The point of non-differentiability came where the piecewise defined function switched from one piece to the other. Our next example shows that this does not always cause trouble.

For one more example involving piecewise-defined functions, we turn to the video in Figure 2.1.32.

Recall we pseudo-defined a continuous function as one in which we could sketch its graph without lifting our pencil. We can give a pseudo-definition for differentiability as well: it is a continuous function that does not have any “sharp corners” or a vertical tangent line. One such sharp corner is shown in Figure 2.1.27. Even though the function $f$ in Example 2.1.29 is piecewise-defined, the transition is “smooth” hence it is differentiable. Note how in the graph of $f$ in Figure 2.1.30 it is difficult to tell when $f$ switches from one piece to the other; there is no “corner.”

When we defined the derivative at a point in Definition 2.1.7, we specified that the interval $I$ over which a function $f$ was defined needed to be an open interval. Open intervals are required so that we can take a limit at any point $c$ in $I\text{,}$ meaning we want to approach $c$ from both the left and right.

Recall we also required open intervals in Definition 1.5.1 when we defined what it meant for a function to be continuous. Later, we used one-sided limits to extend continuity to closed intervals. We now extend differentiability to closed intervals by again considering one-sided limits.

Our motivation is three-fold. First, we consider “common sense.” In Example 2.1.22 we found that when $f(x)=3x^2+5x-7\text{,}$ $f^{\prime }(x)=6x+5\text{,}$ and this derivative is defined for all real numbers, hence $f$ is differentiable everywhere. It seems appropriate to also conclude that $f$ is differentiable on closed intervals, like $[0,1]\text{,}$ as well. After all, $f^{\prime }(x)$ is defined at both $x=0$ and $x=1\text{.}$

Secondly, consider $f(x)=\sqrt{x}\text{.}$ The domain of $f$ is $[0,\mathrm{\infty })\text{.}$ Is $f$ differentiable on its domain — specifically, is $f$ differentiable at $0\text{?}$ (We’ll consider this in the next example.)

Finally, in later sections, having the derivative defined on closed intervals will prove useful. One such place is Section 7.4 where the derivative plays a role in measuring the length of a curve.

After a formal definition of differentiability on a closed interval, we explore the concept in an example.

For all the functions $f$ in this text, we can determine differentiability on $[a,b]$ by considering the limits $\underset{x\to a^{+}}{lim}{f}^{\prime }(x)$ and $\underset{x\to b^{-}}{lim}{f}^{\prime }(x)\text{.}$ This is often easier to evaluate than the limit of the difference quotient.

Most calculus textbooks omit this topic and simply avoid specific cases where it could be applied. We choose in this text to not make use of the topic unless it is “needed.” Many theorems in later sections require a function $f$ to be differentiable on an open interval $I\text{;}$ we could remove the word “open” and just use “$\dots$ on an interval $I\text{,}$” but choose to not do so in keeping with the current mathematical tradition. Our first use of differentiability on closed intervals comes in Chapter 7, where we measure the lengths of curves.

This section defined the derivative; in some sense, it answers the question of “What is the derivative?” The next section addresses the question “What does the derivative mean?”