1. Mrs. McConaughyHonors Algebra 21 Objectives:  To use the natural base e as a base of an exponential function  To use the natural base e in real-life problems The Natural Base e

2. Mrs. McConaughyHonors Algebra 22 VOCABULARY The Natural Base e Natural base e __________________________ __________________________ ≈ __________ The number is_______________ (non-terminating, non-repeating decimal). e = 1/0! + 1/1! + ½! + 1/3! + ¼! + … = 1 + 1 + ½ + 1/6 + 1/24 + …. 2.718281828459. irrational.

3. Mrs. McConaughyHonors Algebra 23 Most scientific calculators have a special key for evaluating natural base e functions of the form f(x) = Ce bx. For example, to evaluate e 2 on the TI- calculators, you can use the key strokes: 2 = ______ 2 nd exex enter Evaluating Functions of the Form f(x) = Ce bx 7.38905

4. Mrs. McConaughyHonors Algebra 24 SIMPLIFYING NATURAL BASE e EXPRESSIONS EXAMPLE 1 SIMPLIFYING NATURAL BASE e EXPRESSIONS A. e 2 ∙ e 3 = _____ B. 6 e 2 /2 e = _____ C. (e -1 ) 2 = _____ The graph of the function f(x) = Ce bx represents exponential ______ if b is positive and exponential _______ if b is negative. e5e5 3e 1/e 2 growth decay SIMPLIFYING NATURAL BASE e EXPRESSIONS

5. Mrs. McConaughyHonors Algebra 25 EXAMPLE 2 GRAPHING THE NATURAL BASE e FUNCTION In general, the graph of f(x) = Ce bx + d has the line _____as its horizontal asymptote. GRAPHING NATURAL BASE e FUNCTIONS f(x) = e x g(x) = 0.5 e 0.6x h(x) = 0.5 e -0.6x y = d

6. Mrs. McConaughyHonors Algebra 26 FORMULA: CONTINUOUSLY COMPOUNDED INTEREST Formula for Compounding Interest n times per year: ________________________________ Formula for Compounding Interest Continuously: ________________________________ A = p (1 + r/n) nt A = P e r t

7. Mrs. McConaughyHonors Algebra 27 EXAMPLE 3Finding Continuously Compounded Interest You deposit $10,000 in an account that pays 6% interest. Find the balance after 10 years if the interest is compounded a.quarterly b.continuously Finding Continuously Compounded Interest $18,140.18 $18,221.19 A = P e r t

8. Mrs. McConaughyHonors Algebra 28 Final Checks for Understanding 1.Simplify 5e 3 /20e -2 2.Sketch the graph of f(x) = ½ e 3x. Is this function an example of exponential growth or decay? Explain. 3.State the horizontal asymptote of the graph of f(x) = 2e x – 2.

9. Exit Ticket 1.Condense: 2/3log a 27 - 2 log a 3 2.Expand: log (x + 3 ) 2 3.Use properties of logarithms to solve for x: 5log 5 2 + log 5 x = log 5 64 Mrs. McConaughyHonors Algebra 29

10. Mrs. McConaughyHonors Algebra 210 Homework Assignment: Natural Base e WS 1-32 all; 33, 37, 41 (calculator); 47, 50.