The following exercises can be implemented using a graphing calculator, computer algebra system, or spreadsheet. The exercises can be used by instructors as demonstrations or as projects for students.

Graph and tabulate values for a function. Calculate the average rate of change between successive points:  (f(x₂)-f(x₁))/(x₂-x₁).  Each average rate of change is the slope of a secant line to the function. The value of the slope of the tangent line, which is also the instantaneous rate of change, is given by the derivative of the function. If the function is not too steep and/or the x-interval not too large, the average and instantaneous rates of change should be close.

Example 1: y = x². The limit definition of the derivative is relatively easy to apply to this function. Some tabular and graphical evidence may also help to build intuition.

Table 1.
 

x	y=x^2	avg rate of change	y=2x
-3.00	9.00
-2.80	7.84	-5.80	-5.60
-2.60	6.76	-5.40	-5.20
-2.40	5.76	-5.00	-4.80
-2.20	4.84	-4.60	-4.40
-2.00	4.00	-4.20	-4.00
-1.80	3.24	-3.80	-3.60
-1.60	2.56	-3.40	-3.20
-1.40	1.96	-3.00	-2.80
-1.20	1.44	-2.60	-2.40
-1.00	1.00	-2.20	-2.00
-0.80	0.64	-1.80	-1.60
-0.60	0.36	-1.40	-1.20
-0.40	0.16	-1.00	-0.80
-0.20	0.04	-0.60	-0.40
0.00	0.00	-0.20	0.00
0.20	0.04	0.20	0.40
0.40	0.16	0.60	0.80
0.60	0.36	1.00	1.20
0.80	0.64	1.40	1.60
1.00	1.00	1.80	2.00
1.20	1.44	2.20	2.40
1.40	1.96	2.60	2.80
1.60	2.56	3.00	3.20
1.80	3.24	3.40	3.60
2.00	4.00	3.80	4.00
2.20	4.84	4.20	4.40

 

As you can see in Figure 1 and in Table 1, the average rate of change is almost identical to the instantaneous rate of change for y = x².

Note that where the function has a minimum, the rates of change functions are approximately zero. Where the function is decreasing, the rates of change are negative. Where the function is increasing, the rates of change are positive.

Also notice that the function is concave up and its derivative is increasing.

Example 2: y = e^x.

For this function, notice that the average rate of change function has values that almost identically match the function values. Thus, it makes sense that the derivative of the exponential function is the function itself.

Notice that the function increases and its derivative is positive. Notice further that the function is concave up and its derivative is increasing.

As the function becomes steeper, the average rates of change do not approximate the derivative as well. Better approximations are obtained by taking smaller increments of x, as can be seen in Table 2 and Figure 3.

Table 2:
 

x	exp(x)	avg rate of change
1	2.72
1.1	3.00	2.86
1.2	3.32	3.16
1.3	3.67	3.49
1.4	4.06	3.86
1.5	4.48	4.26
1.6	4.95	4.71
1.7	5.47	5.21
1.8	6.05	5.76
1.9	6.69	6.36
2	7.39	7.03
2.01	7.46	7.43
2.02	7.54	7.50
2.03	7.61	7.58
2.04	7.69	7.65
2.05	7.77	7.73
2.06	7.85	7.81
2.07	7.92	7.89
2.08	8.00	7.96
2.09	8.08	8.04

 

Example 3: y= lnx

This is perhaps a more elusive function for most students. In Table 3 and Figure 4, the average rate of change function and the derivative are very close where the curve is not too steep, that is for x>0.6. The derivative is positive, and the function increases. The derivative increases, and the function is concave up.

Table 3
 

	y=ln(x)	avg rate of change	y=1/x
0.01	-4.61		100.00
0.21	-1.56	15.22	4.76
0.41	-0.89	3.35	2.44
0.61	-0.49	1.99	1.64
0.81	-0.21	1.42	1.23
1.01	0.01	1.10	0.99
1.21	0.19	0.90	0.83
1.41	0.34	0.76	0.71
1.61	0.48	0.66	0.62
1.81	0.59	0.59	0.55
2.01	0.70	0.52	0.50
2.21	0.79	0.47	0.45
2.41	0.88	0.43	0.41
2.61	0.96	0.40	0.38
2.81	1.03	0.37	0.36
3.01	1.10	0.34	0.33
3.21	1.17	0.32	0.31
3.41	1.23	0.30	0.29
3.61	1.28	0.28	0.28
3.81	1.34	0.27	0.26
4.01	1.39	0.26	0.25
4.21	1.44	0.24	0.24
4.41	1.48	0.23	0.23
4.61	1.53	0.22	0.22
4.81	1.57	0.21	0.21
5.01	1.61	0.20	0.20
5.21	1.65	0.20	0.19