Chapter 4: Inverse Functions
Section 4.1: Exponential Functions and their Derivatives
Definition: An exponential function is a function of the form f (x) = ax , where a 6= 1 is
a positive constant called the base of f and x is a real variable called the exponent of f .
The domain of an exponential function is R = (−∞, ∞) and the range is (0, ∞).
Note: If 0 < a < 1, then f (x) = ax is decreasing and if a > 1, then f (x) = ax is increasing.
Theorem: (Limits of Exponential Functions)
1. If 0 < a < 1, then
lim ax = 0
x→∞
and
lim ax = ∞.
x→−∞
2. If a > 1, then
lim ax = ∞
x→∞
and
Theorem: (Properties of Exponential Functions)
Suppose that a, b > 0 and x, y ∈ R. Then
1. ax ay = ax+y
2. (ab)x = ax bx
ax
= ax−y
ay
a x ax
4.
= x
b
b
3.
5. (ax )y = axy
1
lim ax = 0.
x→−∞
Note: The most common base of an exponential function is the natural exponential base
e ≈ 2.7182818284590452.
Example: Evaluate each limit.
2e3x − 3e−3x
x→∞ 3e3x + 4e−3x
(a) lim
2e3x − 3e−3x
x→−∞ 3e3x + 4e−3x
(b) lim
(c) lim+ e2/(x−1)
x→1
2
Theorem: (Derivative of the Exponential Function)
d x
e = ex
dx
By the Chain Rule, it follows that
d f (x)
e
= f 0 (x)ef (x) .
dx
Example: Differentiate each function.
(a) f (x) = ex
3 +2x2 +3x−4
(b) f (x) = e−3x cos(5x)
(c) f (x) = ex tan x
3
(d) f (x) =
e3x
1 + ex
(e) f (x) = sin
1 − e2x
1 + e2x
2
(f) f (x) = sec(etan x )
4
Example: Find an equation of the tangent line to the graph of y = x2 e−x at (1, 1/e).
Example: If f (x) = xe−x , find f (100) (x).
Example: For what values of λ does y = eλx satisfy the equation y 00 − 5y 0 − 6y = 0?
5
Example: Find an equation of the tangent line to the curve xey + yex = 4 at (0, 4).
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