1. Derive the relationship between the backward difference operator,
~∆
f ≡
f(
x) −
f(
x −
h), and the
differential operator,
D f ≡
f′(
x), and numerically obtain
f′(1.0) from the table below using all the
available information:
| x | f(x) |
| 0.20 | 1.22140276 |
| 0.25 | 1.28402542 |
| 0.30 | 1.34985881 |
| 0.35 | 1.41906755 |
| 0.40 | 1.49182470 |
| 0.45 | 1.56831219 |
| 0.50 | 1.64872127 |
| 0.55 | 1.73325302 |
| 0.60 | 1.82211880 |
| 0.65 | 1.91554083 |
| 0.70 | 2.01375271 |
| 0.75 | 2.11700002 |
| 0.80 | 2.22554093 |
| 0.85 | 2.33964685 |
| 0.90 | 2.45960311 |
| 0.95 | 2.58570966 |
| 1.00 | 2.71828183 |
2.
(a) Is f(x) continuous at x=0 ? State your rationale.
(b) Is f(x) differentiable at x=0 ? State your rationale.
