Estimating Derivatives from Tables Examples

Use the following table of values to estimate f  ' (1.2).

Since we're trying to estimate f ' (1.2), that means we're using a = 1.2:

The table gives us f(a + h) for h = -0.2, -0.1, 0.1, 0.2. That's not great, but we've got to use what we've got. We can rearrange the information we're given into a table that looks like the ones we've been using:

Then all that remains is to calculate the slopes:

As h approaches 0 from the right, we find slopes of 5 and 5. As h approaches 0 from the right, we find slopes of 3.5 and 4. It looks like f ' (1.2) is somewhere in between 4 and 5.