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Notes 5.4: Exponential Functions: Differentiation and Integration
Definition of the Natural Exponential Function
The inverse function of the natural logarithmic function f(x) = ln x is called the natural exponential
function and is denoted by
f –1 (x) = 𝑒 𝑥 .
That is, y = 𝑒 𝑥 if and only if x = ln y.
*Inverse relationship between the two → ln (𝒆𝒙 )= x and 𝒆𝐥𝐧 𝒙 = x
Ex 1: Solve 7 = 𝑒 𝑥+1
Ex 2: Solve ln(2x – 3) = 5
Theorem 5.10 Operations with Exponential Functions
Let a and b be any real numbers.
1. 𝑒 𝑎 ·𝑒 𝑏 = 𝑒 𝑎+𝑏
2.
𝑒𝑎
𝑒𝑏
= 𝑒 𝑎−𝑏
Properties of the Natural Exponential Function
1. The domain of f(x) = 𝑒 𝑥 is (−∞, ∞) and the range is (0, ∞).
2. The function f(x) = 𝑒 𝑥 is continuous, increasing, and one-to-one on its entire domain.
3. The graph of f(x) = 𝑒 𝑥 is concave upward on its entire domain.
4. 𝐥𝐢𝐦 𝒆𝒙 = 0 and 𝐥𝐢𝐦 𝒆𝒙 = ∞
𝒙 → −∞
𝒙→∞
Theorem 5.11 The Derivative of the Natural Exponential Function
Let u be a differentiable function of x.
1.
𝑑
𝑑𝑥
[𝑒 𝑥 ] = 𝑒 𝑥
2.
𝑑
𝑑𝑥
[𝑒 𝑢 ] = 𝑒 𝑢
Ex 3: Differentiate the following:
a. 𝑒 2𝑥−1
b. 𝑒
−3
𝑥
Ex 4: Find the relative extrema of f(x) = x𝑒 𝑥
𝑑𝑢
𝑑𝑥
Ex 5: Show that the normal probability density function f(x) =
1
√2𝜋
𝑒
−𝑥2
2
has points of inflection when
x = ± 1.
Ex 6: The number y of medical doctors (in thousands) in the U.S. from 1980 through 1997, can be
modeled by
y = 475,520𝑒 0.0271𝑡
where t = 0 represents 1980. At what rate was the number of M.D.’s changing in 1992?
Theorem 5.12 Integration Rules for Exponential Functions
Let u be a differentiable function of x.
1. ∫ 𝑒 𝑥 dx = 𝑒 𝑥 + C
2. ∫ 𝑒 𝑢 du = 𝑒 𝑢 + C
Ex 7: Find ∫ 𝑒 3𝑥+1 dx
2
Ex 8: Find ∫ 5𝑥𝑒 −𝑥 dx
Ex 9: Find the following:
1
a.
𝑒𝑥
∫ 𝑥 2 dx
b. ∫ sin 𝑥 𝑒 cos 𝑥 dx
Ex 10: Evaluate each definite integral:
1
a. ∫0 𝑒 − 𝑥 dx
1 𝑒𝑥
b. ∫0
c.
0
1+ 𝑒 𝑥
dx
∫– 1 [𝑒 𝑥 cos(𝑒 𝑥 )] dx