Active Calculus

Subsection 1.3.1 The Derivative of a Function at a Point

Just as we defined instantaneous velocity in terms of average velocity, we now define the instantaneous rate of change of a function at a point in terms of the average rate of change of the function \(f\) over related intervals. This instantaneous rate of change of \(f\) at \(a\) is called “the derivative of \(f\) at \(a\text{,}\)” and is denoted by \(f'(a)\text{.}\)

Aloud, we read the symbol \(f'(a)\) as either “\(f\)-prime at \(a\)” or “the derivative of \(f\) evaluated at \(x = a\text{.}\)” Much of the next several chapters will be devoted to understanding, computing, applying, and interpreting derivatives. For now, we observe the following important things.

We first consider the derivative at a given value as the slope of a certain line.

When we compute an instantaneous rate of change, we allow the interval \([a,a+h]\) to shrink as \(h \to 0\text{.}\) We can think of one endpoint of the interval as “sliding towards” the other. In particular, provided that \(f\) has a derivative at \((a,f(a))\text{,}\) the point \((a+h,f(a+h))\) will approach \((a,f(a))\) as \(h \to 0\text{.}\) Because the process of taking a limit is a dynamic one, it can be helpful to use computing technology to visualize it. One option is a java applet in which the user is able to control the point that is moving. For a helpful collection of examples, consider the work of David Austin¹ of Grand Valley State University, and this particularly relevant example². For applets that have been built in Geogebra³, see Marc Renault's library⁵ via Shippensburg University, with this example⁶ being especially fitting for our work in this section.

Figure 1.3.5 shows a sequence of figures with several different lines through the points \((a, f(a))\) and \((a+h,f(a+h))\text{,}\) generated by different values of \(h\text{.}\) These lines (shown in the first three figures in magenta), are often called secant lines to the curve \(y = f(x)\text{.}\) A secant line to a curve is simply a line that passes through two points on the curve. For each such line, the slope of the secant line is \(m = \frac{f(a+h) - f(a)}{h}\text{,}\) where the value of \(h\) depends on the location of the point we choose. We can see in the diagram how, as \(h \to 0\text{,}\) the secant lines start to approach a single line that passes through the point \((a,f(a))\text{.}\) If the limit of the slopes of the secant lines exists, we say that the resulting value is the slope of the tangent line to the curve. This tangent line (shown in the right-most figure in green) to the graph of \(y = f(x)\) at the point \((a,f(a))\) has slope \(m = f'(a)\text{.}\)

If the tangent line at \(x = a\) exists, the graph of \(f\) looks like a straight line when viewed up close at \((a,f(a))\text{.}\) In Figure 1.3.6 we combine the four graphs in Figure 1.3.5 into the single one on the left, and zoom in on the box centered at \((a,f(a))\) on the right. Note how the tangent line sits relative to the curve \(y = f(x)\) at \((a,f(a))\) and how closely it resembles the curve near \(x = a\text{.}\)

The following example demonstrates several key ideas involving the derivative of a function.

Example 1.3.8. Using the limit definition of the derivative.

For the function \(f(x) = x - x^2\text{,}\) use the limit definition of the derivative to compute \(f'(2)\text{.}\) In addition, discuss the meaning of this value and draw a labeled graph that supports your explanation.

Solution.

From the limit definition, we know that

\begin{equation*} f'(2) = \lim_{h \to 0} \frac{f(2+h)-f(2)}{h}\text{.} \end{equation*}

Now we use the rule for \(f\text{,}\) and observe that \(f(2) = 2 - 2^2 = -2\) and \(f(2+h) = (2+h) - (2+h)^2\text{.}\) Substituting these values into the limit definition, we have that

\begin{equation*} f'(2) = \lim_{h \to 0} \frac{(2+h) - (2+h)^2 - (-2)}{h}\text{.} \end{equation*}

In order to let \(h \to 0\text{,}\) we must simplify the quotient. Expanding and distributing in the numerator,

\begin{equation*} f'(2) = \lim_{h \to 0} \frac{2+h - 4 - 4h - h^2 + 2}{h}\text{.} \end{equation*}

Combining like terms, we have

\begin{equation*} f'(2) = \lim_{h \to 0} \frac{ -3h - h^2}{h}\text{.} \end{equation*}

Next, we remove a common factor of \(h\) in both the numerator and denominator and find that

\begin{equation*} f'(2) = \lim_{h \to 0} (-3-h)\text{.} \end{equation*}

Finally, we are able to take the limit as \(h \to 0\text{,}\) and thus conclude that \(f'(2) = -3\text{.}\) We note that \(f'(2)\) is the instantaneous rate of change of \(f\) at the point \((2,-2)\text{.}\) It is also the slope of the tangent line to the graph of \(y = x - x^2\) at the point \((2,-2)\text{.}\) Figure 1.3.9 shows both the function and the line through \((2,-2)\) with slope \(m = f'(2) = -3\text{.}\)

The following activities will help you explore a variety of key ideas related to derivatives.

Subsection 1.3.2 Summary

• The average rate of change of a function \(f\) on the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b-a}\text{.}\) The units on the average rate of change are units of \(f(x)\) per unit of \(x\text{,}\) and the numerical value of the average rate of change represents the slope of the secant line between the points \((a,f(a))\) and \((b,f(b))\) on the graph of \(y = f(x)\text{.}\) If we view the interval as being \([a,a+h]\) instead of \([a,b]\text{,}\) the meaning is still the same, but the average rate of change is now computed by \(\frac{f(a+h)-f(a)}{h}\text{.}\)

• The instantaneous rate of change with respect to \(x\) of a function \(f\) at a value \(x = a\) is denoted \(f'(a)\) (read “the derivative of \(f\) evaluated at \(a\)” or “\(f\)-prime at \(a\)”) and is defined by the formula

  \begin{equation*} f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}\text{,} \end{equation*}
  provided the limit exists. Note particularly that the instantaneous rate of change at \(x = a\) is the limit of the average rate of change on \([a,a+h]\) as \(h \to 0\text{.}\)

• Provided the derivative \(f'(a)\) exists, its value tells us the instantaneous rate of change of \(f\) with respect to \(x\) at \(x = a\text{,}\) which geometrically is the slope of the tangent line to the curve \(y = f(x)\) at the point \((a,f(a))\text{.}\) We even say that \(f'(a)\) is the “slope of the curve” at the point \((a,f(a))\text{.}\)

• Limits allow us to move from the rate of change over an interval to the rate of change at a single point.