Table view of the Derivative

We have looked at the derivative function from a graphical point of view. Let's now look at approximating the derivative function using a table of values.

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See About the calculus applets for operating instructions.

1. A parabola

The applet shows a table that represents selected values of a function. The first row shows a selection of x values, while the second row shows the corresponding values for f (x). The third row shows estimated values for the derivative. These are computed from the pair of x values and the pair of y = f (x) values above each estimate from the difference quotient (y2-y1)/(x2-x1) Note that this is only an estimate for the derivative, as this is calculating the average rate of change on the interval, not the instantaneous rate of change. But, since we only have selected values for x and f (x), that's the best we can do just using the data in the table. For example, the first derivative estimate is computed as (1-0)/(1-0)=1 and the next value is (4-1)/(2-1)=3 Can you calculate some of the other values for the estimated derivative, using the data in the table?

2. Parabola with wider intervals

Select the second example from the drop down menu. This is the same function, but now the x intervals are farther apart. The derivative is still estimated using the difference quotient, but in this example the denominator is 2 instead of 1.

3. A linear function

Select the third example, a linear function. Compute one or two derivative estimates and check that your calculations match those in the table. What do you notice about the derivative?

4. A constant function

Select the fourth example, a constant function. Why is the derivative always 0?

5. An exponential function

Select the fifth example, an exponential function. Does there seem to be some relationship between the derivative and the value of the function? We will return to this in the future.

6. A hyperbola

Select the sixth example, a hyperbola. Why are the derivative estimates negative?

7. A sine curve

Select the seventh example, a sine curve. Does it make sense for the values of the derivative to change around like they seem to do? Is there an upper and lower bound on the values of the derivative function?

Explore

You can make your own example by typing a function definition into the "f(x)=" box. You can also select the starting x value and the step size between x values.