Introduction to Derivatives

Read this section to lay the groundwork for introducing the concept of a derivative. Work through practice problems 1-5.

Fig. 1 is the graph of a function $y = f(x)$ . We can use the information in the graph to fill in the table:


$x$	$y = f(x)$	$m(x)$ = the estimated SLOPE of the tangent line to $y=f(x)$ at the point $(x,y)$
0	0	1
1	1	0
2	0	– 1
3	– 1	0
4	1	1
5	2	1/2

We can estimate the values of $m(x)$ at some non-integer values of $x$ , $m(.5) ≈ 0.5$ and $m(1.3) ≈ –0.3$ , and even over entire intervals, if $0 < x < 1$ , then $m(x)$ is positive.

The values of

$m(x)$ definitely depend on the values of

$x$ , and

$m(x)$ is a function of

$x$ . We can use the results in the table to help sketch the graph of

$m(x)$ in Fig. 2.

Practice 1: The graph of $y = f(x)$ is given in Fig. 3. Set up a table of values for $x$ and $m(x)$ (the slope of the line tangent to the graph of $y=f(x)$ at the point $(x,y)$ and then graph the function $m(x)$ .

In some applications, we need to know where the graph of a function $f(x)$ has horizontal tangent lines $(slopes = 0)$ . In Fig. 3, the slopes of the tangent lines to graph of $y = f(x)$ are 0 when $x = 2$ or $x ≈ 4.5$ .

Practice 2: At what values of $x$ does the graph of $y = g(x)$ in Fig. 4 have horizontal tangent lines?

Example 1: Fig. 5 is the graph of the height of a rocket at time $t$ . Sketch the graph of the velocity of the rocket at time $t$ . (Velocity is the slope of the tangent to the graph of position or height.)

Solution: The lower graph in Fig. 6 shows the velocity of the rocket.

Practice 3: Fig. 7 shows the temperature during a summer day in Chicago. Sketch the graph of the rate at which the temperature is changing. (This is just the graph of the slopes of the lines which are tangent to the temperature graph.)

The function $m(x)$ , the slope of the line tangent to the graph of $f(x)$ , is called the derivative of $f(x)$ . We have used the idea of the slope of the tangent line throughout Chapter 1. In the Section 2.1, we will formally define the derivative of a function and begin to examine some of the properties of the derivative function, but first lets see what we can do when we have a formula for $f(x)$ .