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1 COMMON CORE Learning Standards HSF-IF.C.7e HSF-LE.B.5. USING TOOLS STRATEGICALLY To be proficient in math, ou need to use technological tools to eplore and deepen our understanding of concepts. The Natural Base e Essential Question What is the natural base e? So far in our stud of mathematics, ou have worked with special numbers such as π and i. Another special number is called the natural base and is denoted b e. The natural base e is irrational, so ou cannot find its eact value. Approimating the Natural Base e Work with a partner. One wa to approimate the natural base e is to approimate the sum Use a spreadsheet or a graphing calculator to approimate this sum. Eplain the steps ou used. How man decimal places did ou use in our approimation? Approimating the Natural Base e Work with a partner. Another wa to approimate the natural base e is to consider the epression ( ). As increases, the value of this epression approaches the value of e. Cop and complete the table. Then use the results in the table to approimate e. Compare this approimation to the one ou obtained in Eploration ( ) Graphing a Natural Base Function Work with a partner. Use our approimate value of e in Eploration 1 or to complete the table. Then sketch the graph of the natural base eponential function = e. You can use a graphing calculator and the e ke to check our graph. What are the domain and range of = e? Justif our answers. 1 1 = e Communicate Your Answer. What is the natural base e? 5. Repeat Eploration for the natural base eponential function = e. Then compare the graph of = e to the graph of = e.. The natural base e is used in a wide variet of real-life applications. Use the Internet or some other reference to research some of the real-life applications of e. Section. The Natural Base e

2 . Lesson What You Will Learn Core Vocabular natural base e, p. Previous irrational number properties of eponents percent increase percent decrease compound interest Define and use the natural base e. Graph natural base functions. Solve real-life problems. The Natural Base e The histor of mathematics is marked b the discover of special numbers, such as π and i. Another special number is denoted b the letter e. The number is called the natural base e, or the Euler number, after its discoverer, Leonhard Euler (177 17). The epression ( ) approaches e as increases, as shown in the graph and table. 1 = e 1 = (1 + 1 ( ( ) Core Concept The Natural Base e The natural base e is irrational. It is defined as follows: As approaches +, ( ) approaches e.711. Simplifing Natural Base Epressions Check You can use a calculator to check the equivalence of numerical epressions involving e. e^()*e^() 1.9 e^(9) 1.9 Simplif each epression. a. e e b. 1e5 e c. (e ) SOLUTION a. e e = e + b. 1e5 e = e5 c. (e ) = (e ) = e 9 = e = 9e = 9 e Monitoring Progress Simplif the epression. Help in English and Spanish at BigIdeasMath.com 1. e 7 e. e e 5. (1e ) Chapter Eponential and Logarithmic Functions

3 Graphing Natural Base Functions Core Concept Natural Base Functions A function of the form = ae r is called a natural base eponential function. When a > and r >, the function is an eponential growth function. When a > and r <, the function is an eponential deca function. The graphs of the basic functions = e and = e are shown. eponential growth 7 5 (, 1) = e (1,.71) 7 5 = e (, 1) eponential deca (1,.) Graphing Natural Base Functions Tell whether each function represents eponential growth or eponential deca. Then graph the function. a. = e b. f () = e.5 LOOKING FOR STRUCTURE You can rewrite natural base eponential functions to find percent rates of change. In Eample (b), f () = e.5 = (e.5 ) (.5) = (1.95). So, the percent decrease is about 9.5%. SOLUTION a. Because a = is positive and b. Because a = 1 is positive and r = 1 is positive, the function is r =.5 is negative, the function an eponential growth function. is an eponential deca function. Use a table to graph the function. Use a table to graph the function (, 7.9) 1 (1,.15) ( 1, 1.1) (,.7) (,.1) (, ) (,.7) (, 1) Monitoring Progress Help in English and Spanish at BigIdeasMath.com Tell whether the function represents eponential growth or eponential deca. Then graph the function.. = 1 e 5. = e. f () = e Section. The Natural Base e 5

4 Solving Real-Life Problems You have learned that the balance of an account earning compound interest is given b A = P ( 1 + n) r nt. As the frequenc n of compounding approaches positive infinit, the compound interest formula approimates the following formula. Core Concept Continuousl Compounded Interest When interest is compounded continuousl, the amount A in an account after t ears is given b the formula A = Pe rt where P is the principal and r is the annual interest rate epressed as a decimal. Modeling with Mathematics Balance (dollars) Your Friend s Account A 1, 1,,,,, (, ) 1 1 t Year You and our friend each have accounts that earn annual interest compounded continuousl. The balance A (in dollars) of our account after t ears can be modeled b A = 5e.t. The graph shows the balance of our friend s account over time. Which account has a greater principal? Which has a greater balance after 1 ears? SOLUTION 1. Understand the Problem You are given a graph and an equation that represent account balances. You are asked to identif the account with the greater principal and the account with the greater balance after 1 ears.. Make a Plan Use the equation to find our principal and account balance after 1 ears. Then compare these values to the graph of our friend s account.. Solve the Problem The equation A = 5e.t is of the form A = Pe rt, where P = 5. So, our principal is $5. Your balance A when t = 1 is A = 5e.(1) = $71.1. Because the graph passes through (, ), our friend s principal is $. The graph also shows that the balance is about $75 when t = 1. MAKING CONJECTURES You can also use this reasoning to conclude that our friend s account has a greater annual interest rate than our account. So, our account has a greater principal, but our friend s account has a greater balance after 1 ears.. Look Back Because our friend s account has a lesser principal but a greater balance after 1 ears, the average rate of change from t = to t = 1 should be greater for our friend s account than for our account. Your account: Your friend s account: Monitoring Progress A(1) A() = = A(1) A() 75 = Help in English and Spanish at BigIdeasMath.com 7. You deposit $5 in an account that earns 5% annual interest compounded continuousl. Compare the balance after 1 ears with the accounts in Eample. Chapter Eponential and Logarithmic Functions

5 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. VOCABULARY What is the Euler number?. WRITING Tell whether the function f () = 1 e represents eponential growth or eponential deca. Eplain. Monitoring Progress and Modeling with Mathematics In Eercises 1, simplif the epression. (See Eample 1.). e e5. e e ANALYZING EQUATIONS In Eercises, match the function with its graph. Eplain our reasoning.. = e. = e 5. 11e 9 e 1. 7e 7 e 7. (5e 7 ). (e ) 9. 9e 1. e e e e 1. e e e + ERROR ANALYSIS In Eercises 1 and 1, describe and correct the error in simplifing the epression (e ) = e ()() e 5 = e = e 5 e = e 5. = e.5. =.75e A. B. ( 1, 7.9) ( 1,.59) 1 (, 1) C. D. (,.75) (1,.) (, ) (1, 7.9) (, 1) In Eercises 15, tell whether the function represents eponential growth or eponential deca. Then graph the function. (See Eample.) 15. = e 1. = e 17. = e 1. = e 19. =.5e. =.5e 1. =.e.5. =.e.5 USING STRUCTURE In Eercises 7, use the properties of eponents to rewrite the function in the form = a(1 + r) t or = a(1 r) t. Then find the percent rate of change. 7. = e.5t. = e.75t 9. = e.t. =.5e.t USING TOOLS In Eercises 1, use a table of values or a graphing calculator to graph the function. Then identif the domain and range. 1. = e. = e + 1. = e + 1. = e 5 Section. The Natural Base e 7

6 5. MODELING WITH MATHEMATICS Investment accounts for a house and education earn annual interest compounded continuousl. The balance H (in dollars) of the house fund after t ears can be modeled b H = e.5t. The graph shows the balance in the education fund over time. Which account has the greater principal? Which account has a greater balance after 1 ears? (See Eample.) H 1, Education Account. THOUGHT PROVOKING Eplain wh A = P ( 1 + r n ) nt approimates A = Pe rt as n approaches positive infinit. 9. WRITING Can the natural base e be written as a ratio of two integers? Eplain.. MAKING AN ARGUMENT Your friend evaluates f () = e when = 1 and concludes that the graph of = f () has an -intercept at (1, ). Is our friend correct? Eplain our reasoning. Balance (dollars),,,, (, 5) 1. DRAWING CONCLUSIONS You invest $5 in an account to save for college. Account 1 pas % annual interest compounded quarterl. Account pas % annual interest compounded continuousl. Which account should ou choose to obtain the greater amount in 1 ears? Justif our answer. 1 1 t Year. MODELING WITH MATHEMATICS Tritium and sodium- deca over time. In a sample of tritium, the amount (in milligrams) remaining after t ears is given b = 1e.5t. The graph shows the amount of sodium- in a sample over time. Which sample started with a greater amount? Which has a greater amount after 1 ears? Sodium- Deca Amount (milligrams) 1 1 t Year 7. OPEN-ENDED Find values of a, b, r, and q such that f () = ae r and g() = be q are eponential deca functions, but f () g() represents eponential growth.. HOW DO YOU SEE IT? Use the graph to complete each statement. a. f () approaches as approaches +. b. f () approaches as approaches. f. PROBLEM SOLVING The growth of Mcobacterium tuberculosis bacteria can be modeled b the function N(t) = ae.1t, where N is the number of cells after t hours and a is the number of cells when t =. a. At 1: p.m., there are M. tuberculosis bacteria in a sample. Write a function that gives the number of bacteria after 1: p.m. b. Use a graphing calculator to graph the function in part (a). c. Describe how to find the number of cells in the sample at :5 p.m. Maintaining Mathematical Proficienc Write the number in scientific notation. (Skills Review Handbook) ,, 7..7 Find the inverse of the function. Then graph the function and its inverse. (Section 5.). = = 1, 5. = = Reviewing what ou learned in previous grades and lessons Chapter Eponential and Logarithmic Functions