Final answer:
By using linear approximation with the function f(x) = e^x, the estimated value of e^(-0.01) is approximately 0.99. This method is quick and suitable for small changes, but for higher precision, more accurate methods should be used.
Step-by-step explanation:
Using Linear Approximation to Estimate e(-0.01)
When attempting to estimate the value of e(-0.01) using linear approximation, one begins by considering the function f(x) = ex at a point close to x = 0, since e0 = 1 is an easily calculable value. The derivative, f'(x) = ex, at x = 0 is also 1. Therefore, the differential df can be used to estimate the change in f for a small dx, here dx = -0.01. Thus, the linear approximation can be written as:
df = f'(0) \* dx
Which leads to:
df = 1 \* (-0.01) = -0.01
Therefore, the estimated value of e(-0.01) is:
1 + df = 1 - 0.01 = 0.99
The approximation method avoids the need to solve for the roots of a quadratic equation and is suitable for small changes in levels. However, for more accurate measures, using the first formula or more precise techniques would be necessary. This is important to keep in mind for applications in scientific calculations and various systems where precision is crucial.