1. Exponential Functions
4.4
OBJECTIVES
Differentiate exponential functions.
Solve application problems with exponential functions.
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2. DEFINITION: An exponential function f is given by
where x is any real number, a > 0, and a ≠ 1. The
number a is called the base.
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3. Example 1: Look at the graph First, we find some function values.
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4. DEFINITION: e is a number, named for the Swiss mathematician
Leonhard Euler.
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5. THEOREM 1
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6. Example 1: Find dy/dx:
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8. Example 1 (concluded):
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9. THEOREM 2 OR The derivative of e to some power is the product of e
to that power and the derivative of the power.
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10. Example 2: Differentiate each of the following with
respect to x:
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12. Example 2 (concluded):
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13. The sales of a new computer ( in thousands) are given by:
t represents time in years.
Find the rate of change of sales at each time.
after 1 year b) after 5 years
c) What is happening to the rate of change of sales?
Answers: a) 20 b) 6 c) decreasing
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14. Write an equation of the tangent line to at x = 0.
Answer: y = -6x+2
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