Exponential functions: differentiation & integration (5.4) March 1st, 2013.

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5.4: Exponential Functions: Differentiation and Integration

Presentation transcript:

Exponential functions: differentiation & integration (5.4) March 1st, 2013

I. the natural exponential function DEF. OF THE NATURAL EXPONENTIAL FUNTION: The inverse of is. THus, if and only if. It follows that and.

THM. 5.10: OPERATIONS WITH EXPONENTIAL FUNCTIONS: Let and be any real numbers

PROPERTIES OF THE NATURAL EXPONENTIAL FUNCTION: 1. the domain of is and the range is 2. is continuous, increasing, and one-to-one 3. is concave upward 4. and

II. derivatives of exponential functions THM. 5.11: DERIVATIVE OF THE NATURAL EXPONENTIAL FUNCTION: Let u be a differentiable function of x

EX. 1: Find the derivative of each function. a. b.

EX. 3: Find the extrema and points of inflection of.

III. integrals of exponential functions THM. 5.12: INTEGRATION RULES FOR EXPONENTIAL FUNCTIONS: Let u be a differentiable function of x

EX. 3: Find each indefinite integral. a. b. c.

EX. 4: Evaluate.