Suppose we have the following data for a function
| x |
0.0 |
0.5 |
1.0 |
1.5 |
2.0 |
2.5 |
3.0 |
| f(x) |
0.0 |
4.8 |
16.8 |
29.9 |
26.4 |
29.9 |
8.5 |
How can we generate values for the derivative? We know the derivative
is slope, and can calculate a bunch of slopes from these---e.g., the
slope between the
x-values 0 and 0.5, or between 0 and 2.5,
etc. Let's just find the slopes between pairs of points that are next
to each other in the table above. This gives us
| x-values: |
b/t x=0 & x=0.5 |
0.5 & 1.0 |
1.0 & 1.5 |
1.5 & 2.0 |
2.0 & 2.5 |
2.5 & 3.0 |
| average slope: |
9.6 |
24.1 |
26.2 |
12.9 |
-12.9 |
-42.9 |
(Check these!) Because these are slopes, we can say they are also
approximations to the derivative. In particular, the first value,
9.6, is an approximation to the derivative for
x-values between
x=0 and
x=0.5. Let's use it to approximate the value of
the derivative at
x=0.5. Applying similar logic to the rest of
the values, we find
| x-values: |
0.25 |
0.75 |
1.25 |
1.75 |
2.25 |
2.75 |
| f'(x) (approx.) |
9.6 |
24.1 |
26.2 |
12.9 |
-12.9 |
-42.9 |
Sense?