= f(x). Remember that functions defined numerically lack information about what happens in between the data points. So the operations of calculus are necessarily approximate. The processes you see here are not really differentiation and integration, but their approximate, discrete versions. Shorthand analytic statements of the theorems are given for reference.
Part 1: = f(x).
Take a sequence of numbers y1, y2, y3, ..., yn and let Y0 be a "starting number." Form a new sequence Yi by successively adding the old sequence to Y0. Thus, Y1 = Y0 + y1, Y2 = Y1 + y2, ..., etc. Using the new sequence, find consecutive differences. You get the original sequence back.
| yi | Yi | formula | differences | formula |
|---|---|---|---|---|
| - | 3 | (start) | - | |
| 1 | 4 | = 3+1 | 1 | = 4-3 |
| 6 | 10 | = 4+6 | 6 | = 10-4 |
| 5 | 15 | = 10+5 | 5 | = 15-10 |
| -2 | 13 | = 15-2 | -2 | = 13-15 |
| 7 | 20 | = 13+7 | 7 | = 20-13 |
| 8 | 28 | = 20+8 | 8 | = 28-20 |
| 4 | 32 | = 28+4 | 4 | = 32-28 |
Part 2: 
= F(b) - F(a).
Take a sequence of numbers y1, y2, y3, ..., yn and a starting number Y0, and form the new sequence Yi as above. Then the sum of the first sequence is the net change Yn - Y0 in the new sequence.
In the above example, the sum of the yi is 1 + 6 + 5 - 2 + 7 + 8 + 4 = 29. The net change in the Yi is 32 - 3 = 29. The result is the same.
In symbols,= Yn - Y0. Compare this with the analytic statement.
back to FTOC
back to Calculus I
back to Calculus II
back to Calculus for Life Sciences
back to Business Calculus
back to Margie's home page