Let f(x) = ex. For the value of a, fill in the table and use the resulting values to estimate f ' (a).
a = 0
Answer
a = 0
The slope of the secant line from x = 0 to x = .1 is
The slope of the secant line from x = 0 tox = 0.01 is
The slope of the secant line from x = 0 tox = 0.001 is
The numbers 1.052, 1.01, 1 appear to be approaching 1. We can estimate that f ' (0) = 1.
Example 2
Let f(x) = ex. For the value of a, fill in the table and use the resulting values to estimate f ' (a).
a = 1
Answer
a = 1
The slope of the secant line from x = 1 to x = 1.1 is
The slope of the secant line from x = 1 to x = 1.01 is
The slope of the secant line from x = 1 to x = 1.001 is
The numbers 2.8589, 2.732, 2.72 appear to be approaching something in the vicinity of 2.72 or 2.71 - in other words, something in the vicinity of e. It's reasonable to guess
f ' (1) = e.
Example 3
Let f(x) = ex. For the value of a, fill in the table and use the resulting values to estimate f ' (a).
a = 2
Answer
a = 2
The slope of the secant line from x = 2 to x = 2.1 is
The slope of the secant line from x = 2 to x = 2.01 is
The slope of the secant line from x = 2 to x = 2.001 is
The numbers 7.7711, 7.426, 7.39 appear to be approaching something in the vicinity of 7.39 - in other words, something in the vicinity of e2. We'll guessf ' (2) = e2.
Example 4
Use the fact that f(x) = ex is its own derivative to find the following value.
f ' (5)
Answer
f ' (5) = e5
Example 5
Use the fact that f(x) = ex is its own derivative to find the following value.
f ' (-2)
Answer
f ' (-2) = e-2
Example 6
Use the fact that f(x) = ex is its own derivative to find the following value.
f ' (ln 5)
Answer
f ' (ln 5) = eln 5 = 5. Don't forget the exponent laws.
Example 7
Use the fact that f(x) = ex is its own derivative to find the following value.
f ' (0)
Answer
f ' (0) = e0 = 1
Example 8
Use the fact that f(x) = ex is its own derivative to find the following value.
