Question

Over which interval are the exponential and linear function approximately the same?

mc003-1.jpg
from 0.25 to 0.5
from 0.5 to 0.75
from 0.75 to 1.0
from 1.25 to 1.5

Answer 1

From 0.25 to 0.5.

Reason: near 0 the exponential and its linear tangent (≈ 1 + x) are close. For example,
- e^0.25 ≈ 1.284 vs 1 + 0.25 = 1.25 (difference ≈ 0.034),
- e^0.5 ≈ 1.649 vs 1 + 0.5 = 1.5 (difference ≈ 0.149).

The discrepancy grows for larger x, so the functions are approximately the same on 0.25–0.5.