Lesson 20 Introducing and Applying Base e PreCalculus

Lesson 20 – Introducing and Applying Base e. Pre-Calculus 1/30/2022 Pre-Calculus 1

Lesson Objectives n (1) Investigate a new base to use in exponential expressions, graphs, and applications n (2) Apply exponent laws and graph using base e n (3) Understand WHEN the use of base e is appropriate 1/30/2022 Pre-Calculus 2

(A) What is e? n n Irrational number Called the “natural base” is the “natural exponential function” Represented by the series 1/30/2022 Pre-Calculus 3

(A) What is e? n Use your GDC: Graph and look at the table for What happens as 1/30/2022 ? Pre-Calculus 4

(A) What is e? n Use your GDC: Graph and What do you notice? 1/30/2022 Pre-Calculus 5

(B) Graphing Using Base e n Graph n Domain Range Intercept Increase/decrease on Asymptote As x →-∞, y → As x → ∞, y → List 3 key points (in base e) n n n n 1/30/2022 Pre-Calculus 6

(C) Exponent Laws & Base e n Simplify the following expressions using the exponent laws 1/30/2022 Math 2 Honors - Santowski 7

(C) Working With Exponential Equations in Base e n (i) Using your GDC, approximate the value of x to 3 decimal places. a) b) c) d) 1/30/2022 Pre-Calculus 8

(C) Working With Exponential Equations in Base e n (i) Solve the following equations (No Calc): a) b) c) d) 1/30/2022 Pre-Calculus 9

(D) Graphing Using Base e n Graph n 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) n 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) n n n 1/30/2022 n n n Pre-Calculus 10

(D) Graphing Using Base e n Graph n 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) n 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) n n n 1/30/2022 n n n Pre-Calculus 11

(D) Graphing Using Base e n Graph n 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) n 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) n n n 1/30/2022 n n n Pre-Calculus 12

Lesson Objective #3 n (3) Apply base e in word problems and equations 1/30/2022 Pre-Calculus 13

(E) Where is used in applications n Our formula for situations featuring continuous change is A = Pert P represents an initial amount, r the annual growth/decay rate and t the number of years n In the formula, if r > 0, we have exponential growth and if r < 0, we have exponential decay n Seen in population growth, radioactive decay, etc. 1/30/2022 Pre-Calculus 14

Homework n Holt Textbook: q n 5. 2: #53 Larson Textbook: q 3. 1: No GDC #34 – 38, 43, 45, 46 With GDC #47, 49, 51, 63 1/30/2022 Pre-Calculus 15