For each problem, we rearrange the information given into a more familiar table, then calculate the slopes.
As h approaches 0 from the right we find the slopes 6 and 10. As h approaches 0 from the left we find the slopes 5 and 9. We estimate that f'(2.5) is in between 9 and 10.
Estimate g'(4) given the following table of values:
We estimate that g'(4) is in between 0.2 and 0.4. Notice that we used g instead of f for the function name; this changes the labels on the tables but nothing else.
Estimate f'(2) given the following table of values:
It looks like f'(2) is between 1 and 2. We had more values of a + h to the left of 2 than we did to the right of 2, but again, that didn't change the way we did things.
Make it rain.