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Exponential Functions

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.

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Exponential Functions

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  1. Exponential Functions Differentiation and Integration

  2. 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) = ex. • That is, • y = ex if and only if x = ln y

  3. Inverse Relationship • ln (ex) = x and eln x = x

  4. Solving Exponential Equations • eln 2x = 12 • ln (x – 2)2 = 12 • 200e-4x = 15

  5. Operations with Exponential Functions • Let a and b be any real numbers. • eaeb = ea + b

  6. Properties of the Natural Exponential Function • 1. The domain of f(x) = ex is (-∞, ∞), and the range is (0, ∞). • 2. The function f(x) = ex is continuous, increasing, and one-to-one on its entire domain. • 3. The graph of f(x) = ex is concave upward on its entire domain. • 4.

  7. Properties of the Natural Exponential Function

  8. Properties of the Natural Exponential Function • Sketch the graph of y = e-x using transformations. • p. 347 problems 25 – 26

  9. The Derivative of the Natural Exponential Function • Let u be a differentiable function of x.

  10. Integrals of Exponential Functions • Let u be a differentiable function of x.

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