Lesson 20 – Introducing and Applying Base e. Pre-Calculus 2/22/20161Pre-Calculus

Lesson Objectives (1) Investigate a new base to use in exponential expressions, graphs, and applications (2) Apply exponent laws and graph using base e (3) Understand WHEN the use of base e is appropriate 2/22/20162 Pre-Calculus

Irrational number Called the “natural base” is the “natural exponential function” Represented by the series 2/22/20163 Pre-Calculus (A) What is e?

Use your GDC: Graph and look at the table for What happens as ? 2/22/2016 Pre-Calculus 4

(A) What is e? Use your GDC: Graph and What do you notice? 2/22/2016 Pre-Calculus 5

2/22/20166 (B) Graphing Using Base e Graph Domain  Range  Intercept  Increase/decrease on  Asymptote  As x →-∞, y → As x → ∞, y → List 3 key points (in base e) Pre-Calculus

(C) Exponent Laws & Base e Simplify the following expressions using the exponent laws 2/22/2016 Math 2 Honors - Santowski 7

(C) Working With Exponential Equations in Base e (i) Using your GDC, approximate the value of x to 3 decimal places. a)b) c)d) 2/22/20168 Pre-Calculus

(C) Working With Exponential Equations in Base e (i) Solve the following equations (No Calc): a)b) c)d) 2/22/20169 Pre-Calculus

2/22/ (D) Graphing Using Base e Pre-Calculus Graph List the transformations applied to f(x) List 3 key points on the new function Solve C(x) = 0 and evaluate C(0) Draw the asymptote and label the intercept(s) Graph List the transformations applied to f(x) List 3 key points on the new function Solve D(x) = 0 and evaluate D(0) Draw the asymptote and label the intercept(s)

2/22/ (D) Graphing Using Base e Pre-Calculus Graph List the transformations applied to f(x) List 3 key points on the new function Solve E(x) = 0 and evaluate E(0) Draw the asymptote and label the intercept(s) Graph List the transformations applied to f(x) List 3 key points on the new function Solve F(x) = 0 and evaluate F(0) Draw the asymptote and label the intercept(s)

2/22/ (D) Graphing Using Base e Pre-Calculus Graph List the transformations applied to f(x) List 3 key points on the new function Solve A(x) = 0 and evaluate A(0) Draw the asymptote and label the intercept(s) Graph List the transformations applied to f(x) List 3 key points on the new function Solve B(x) = 0 and evaluate B(0) Draw the asymptote and label the intercept(s)

Lesson Objective #3 (3) Apply base e in word problems and equations 2/22/ Pre-Calculus

2/22/2016 Pre-Calculus 14 (E) Where is used in applications Our formula for situations featuring continuous change is A = Pe rt  P represents an initial amount, r the annual growth/decay rate and t the number of years In the formula, if r > 0, we have exponential growth and if r < 0, we have exponential decay Seen in population growth, radioactive decay, etc.

Homework Holt Textbook:  5.2: #53 Larson Textbook:  3.1: No GDC #34 – 38, 43, 45, 46 With GDC #47, 49, 51, 63 2/22/ Pre-Calculus