# Calculus Applets

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## A Tabular View of the Second Derivative

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

Try the following:

1. 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, and the fourth row shows estimates for the second derivative. The estimates for the second derivative are computed from the pair of derivative estimates above and from a pair of x values that enclose both x-intervals used to compute the derivative estimates. For example, the first second derivative estimate shown is computed as , where the 3 and the 1 are the two derivative estimates from the table immediately above the second derivative estimate, while the 2 and the 0 in the denominator come from the x-values used to compute the first derivative estimates. Note that because each derivative estimate spanned an x-interval, the second derivative estimate will span two x-intervals. Can you calculate some of the other values for the estimated second derivative, using the data in the table?

2. Select the second example from the drop down menu. This is the same function, but now the x intervals are farther apart.

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

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

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

6. Select the sixth example, a hyperbola. The derivative estimates are all negative, but the second derivative estimates are all positive. Why?

7. 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.