TEKS 7. 2A.4.B, 2A..B, 2A..C, 2A..F Before Use Functions Involving e You studied eponential growth and deca functions. Now You will stud functions involving the natural base e. Wh? So ou can model visibilit underwater, as in E. 59. Ke Vocabular 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 (707 78). The epression n 2 n approaches e as n increases. n 0 0 2 0 0 4 0 5 0 6 n 2 n 2.5974 2.7048 2.7692 2.785 2.7827 2.7828 KEY CONCEPT For Your Notebook The Natural Base e The natural base e is irrational. It is defined as follows: As n approaches `, n 2 n approaches e ø 2.7828828. E XAMPLE Simplif natural base epressions REVIEW EXPONENTS For help with properties of eponents, see p. 0. Simplif the epression. a. e 2 p e 5 5 e 2 5 b. 2e 4 e 5 4e4 2 c. (5e 2 ) 2 5 5 2 (e 2 ) 2 5 e 7 5 4e 5 25e 26 5 25 e 6 E XAMPLE 2 Evaluate natural base epressions Use a calculator to evaluate the epression. Epression Kestrokes Displa a. e 4 [e ] 4 54.598500 b. e 20.09 [e ].09 0.9985 492 Chapter 7 Eponential and Logarithmic Functions
GUIDED PRACTICE for Eamples and 2 Simplif the epression.. e 7 p e 4 2. 2e 2 p 6e 5. 24e 8 4e 5 4. (0e 24 ) 5. Use a calculator to evaluate e /4. KEY CONCEPT For Your Notebook Natural Base Functions A function of the form 5 ae r is called a natural base eponential function. If a > 0 and r > 0, the function is an eponential growth function. If a > 0 and r < 0, the function is an eponential deca function. The graphs of the basic functions 5 e and 5 e 2 are shown below. Eponential growth 5 e 5 e 2 Eponential deca (0, ) (, 2.78) (0, ) (, 0.68) E XAMPLE Graph natural base functions Graph the function. State the domain and range. a. 5 e 0.25 b. 5 e 20.75( 2 2) ANOTHER WAY You can also write the function from part (a) in the form 5 ab in order to graph it: 5 e 0.25 5 (e 0.25 ) ø (.28) Solution a. Because a 5 is positive and b. a 5 is positive and r 520.75 r 5 0.25 is positive, the function is is negative, so the function is an eponential growth function. an eponential deca function. Plot the points (0, ) and (,.85) Translate the graph of 5 e 20.75 and draw the curve. right 2 units and up unit. (,.85) (0, ) 5 e 0.25 (22, 4.48) 5 e 20.75 (0, ) 5 e 20.75( 2 2) (0, 5.48) (2, 2) The domain is all real numbers, The domain is all real numbers, and the range is > 0. and the range is >. 7. Use Functions Involving e 49
E XAMPLE 4 TAKS Solve a REASONING: multi-step problem Multi-Step Problem BIOLOGY The length l (in centimeters) of a tiger shark can be modeled b the function Adult shark l 5 7 2 276e 20.78t where t is the shark s age (in ears). Graph the model. Use the graph to estimate the length of a tiger shark that is ears old. Newborn shark INTERPRET VARIABLES On a graphing calculator, enter the function l 5 7 2 276e 20.78t using the variables and, as shown below: 5 7 2 276e 20.78 Solution STEP STEP 2 Graph the model, as shown. Use the trace feature to determine that l ø 75 when t 5. c The length of a -ear-old tiger shark is about 75 centimeters. X= Y=75.95 GUIDED PRACTICE for Eamples and 4 Graph the function. State the domain and range. 6. 5 2e 0.5 7. f() 5 2 e 2 8. 5.5e 0.25( 2 ) 2 2 9. WHAT IF? In Eample 4, use the given function to estimate the length of a tiger shark that is 5 ears old. CONTINUOUSLY COMPOUNDED INTEREST In Lesson 7., ou learned that the balance of an account earning compound interest is given b this formula: A 5 P r n 2 nt As the frequenc n of compounding approaches positive infinit, the compound interest formula approimates the following formula. KEY CONCEPT For Your Notebook Continuousl Compounded Interest When interest is compounded continuousl, the amount A in an account after t ears is given b the formula A 5 Pe rt where P is the principal and r is the annual interest rate epressed as a decimal. 494 Chapter 7 Eponential and Logarithmic Functions
E XAMPLE 5 Model continuousl compounded interest FINANCE You deposit $4000 in an account that pas 6% annual interest compounded continuousl. What is the balance after ear? Solution Use the formula for continuousl compounded interest. A 5 Pe rt Write formula. 5 4000e 0.06() Substitute 4000 for P, 0.06 for r, and for t. ø 4247.5 Use a calculator. c The balance at the end of ear is $4247.5. GUIDED PRACTICE for Eample 5 0. FINANCE You deposit $2500 in an account that pas 5% annual interest compounded continuousl. Find the balance after each amount of time. a. 2 ears b. 5 ears c. 7.5 ears. FINANCE Find the amount of interest earned in parts (a) (c) of Eercise 0. 7. EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 5, 5, and 57 5 TAKS PRACTICE AND REASONING Es. 5, 6, 52, 5, 60, 6, and 64. VOCABULARY Cop and complete: The number? is an irrational number approimatel equal to 2.7828. 2. WRITING Tell whether the function f() 5 e 4 is an eample of eponential growth or eponential deca. Eplain. EXAMPLE on p. 492 for Es. 8 SIMPLIFYING EXPRESSIONS Simplif the epression.. e p e 4 4. e 22 p e 6 5. (2e ) 6. (2e 22 ) 24 7. (e 5 ) 2 8. e p e 2 p e 4 9. Ï 9e 6 0. e p 5e. e e 2. 4e e 4. Ï 8e 9 4. 6e 4 8e 5. MULTIPLE TAKS REASONING CHOICE What is the simplified form of (4e 2 )? A 4e 6 B 4e 8 C 64e 6 D 64e 8 4(27e 6. MULTIPLE CHOICE What is the simplified form of Î ) TAKS REASONING e 7? 2 A 6e 0 B 6e 6 4 C 6e 2 D 6e 2 7. Use Functions Involving e 495
ERROR ANALYSIS Describe and correct the error in simplifing the epression. 7. (e 5 ) 2 5 e (5)(2) 5 e 0 8. e 6 2 2 5 e6 e 22 5 e 4 EXAMPLE 2 on p. 492 for Es. 9 0 EVALUATING EXPRESSIONS Use a calculator to evaluate the epression. 9. e 20. e 2/4 2. e 2.2 22. e /2 2. e 22/5 24. e 4. 25. e 7 26. e 24 27. 2e 20. 28. 5e 2/ 29. 26e 2.4 0. 0.4e 4. GROWTH OR DECAY Tell whether the function is an eample of eponential growth or eponential deca.. f() 5 e 2 2. f() 5 e 4. f() 5 e 24 4. f() 5 5 e 5. f() 5 4 e 25 6. f() 5 e 7. f() 5 2e 4 8. f() 5 4e 22 EXAMPLE on p. 49 for Es. 9 50 MATCHING GRAPHS Match the function with its graph. 9. 5 0.5e 0.5 40. 5 2e 0.5 4. 5 e 0.5 2 A. B. C. (22, 2.7) (0, ) (0, 0.5) (, 0.82) (0, 2) (,.0) GRAPHING FUNCTIONS Graph the function. State the domain and range. 42. 5 e 22 4. 5 e 44. 5 0.5e 45. 5 2e 2 2 46. 5 2.5e 20.5 2 47. 5 0.6e 2 2 48. f() 5 e 2 2 2 49. g() 5 4 e 2 50. h() 5 e 22( ) 2 5. GRAPHING CALCULATOR Use the table feature of a graphing calculator to find the value of n for which n 2 n gives the value of e correct to 9 decimal places. Eplain the process ou used to find our answer. 52. SHORT TAKS REASONING RESPONSE Can e be epressed as a ratio of two integers? Eplain our reasoning. 5. OPEN-ENDED TAKS REASONING MATH Find values of a, b, r, and q such that f() 5 ae r and g() 5 be q are eponential deca functions and f() is an eponential g() growth function. 54. CHALLENGE Eplain wh A 5 P r n 2 nt approimates A 5 Pe rt as n approaches positive infinit. Hint: Let m 5 n r. 2 5 WORKED-OUT SOLUTIONS 496 Chapter 7 Eponential p. WS and Logarithmic Functions 5 TAKS PRACTICE AND REASONING
PROBLEM SOLVING EXAMPLE 4 on p. 494 for Es. 55 56 55. CAMERA PHONES The number of camera phones shipped globall can be modeled b the function 5.28e. where is the number of ears since 997 and is the number of camera phones shipped (in millions). How man camera phones were shipped in 2002? 56. BIOLOGY Scientists used traps to stud the Formosan subterranean termite population in New Orleans. The mean number of termites collected annuall can be modeled b 5 78e 0.45t where t is the number of ears since 989. What was the mean number of termites collected in 999? EXAMPLE 5 on p. 495 for Es. 57 58 57. FINANCE You deposit $2000 in an account that pas 4% annual interest compounded continuousl. What is the balance after 5 ears? 58. FINANCE You deposit $800 in an account that pas 2.65% annual interest compounded continuousl. What is the balance after 2.5 ears? 59. MULTI-STEP PROBLEM The percent L of surface light that filters down through bodies of water can be modeled b the eponential function L() 5 00e k where k is a measure of the murkiness of the water and is the depth below the surface (in meters). a. A recreational submersible is traveling in clear water with a k-value of about 20.02. Write and graph an equation giving the percent of surface light that filters down through clear water as a function of depth. b. Use our graph to estimate the percent of surface light available at a depth of 40 meters. c. Use our graph to estimate how deep the submersible can descend in clear water before onl 50% of surface light is available. 60. EXTENDED TAKS REASONING RESPONSE The growth of the bacteria mcobacterium tuberculosis can be modeled b the function P(t) 5 P 0 e 0.6t where P(t) is the population after t hours and P 0 is the population when t 5 0. a. Model At :00 P.M., there are 0 mcobacterium tuberculosis bacteria in a sample. Write a function for the number of bacteria after :00 P.M. b. Graph Graph the function from part (a). c. Estimate What is the population at 5:00 P.M.? d. Reasoning Describe how to find the population at :45 P.M. 7. Use Functions Involving e 497
6. RATE OF HEALING The area of a wound decreases eponentiall with time. The area A of a wound after t das can be modeled b A 5 A 0 e 20.05t where A 0 is the initial wound area. If the initial wound area is 4 square centimeters, what is the area after 4 das? 62. CHALLENGE The height (in feet) of the Gatewa Arch in St. Louis, Missouri, can be modeled b the function 5 757.7 2 6.85(e /27.7 e 2/27.7 ) where is the horizontal distance (in feet) from the center of the arch. a. Use a graphing calculator to graph the function. How tall is the arch at its highest point? b. About how far apart are the ends of the arch? MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Skills Review Handbook p. 002; TAKS Workbook REVIEW TAKS Preparation p. 408; TAKS Workbook 6. TAKS PRACTICE Which of the following shows that the conjecture is false? The square root of a number is alwas less than. TAKS Obj. 0 A 5 4 B 5 4 C 5 48 D 5 900 64. TAKS PRACTICE Quadrilateral MNPQ is a rhombus. P measures 558. What are the measures of M, N, and Q? TAKS Obj. 6 F 558, 58, and 58 G 558, 558, and 558 H 558, 08, and 08 J 558, 258, and 258 QUIZ for Lessons 7. 7. Graph the function. State the domain and range.. 5 2 p 2 2 (p. 478) 2. 5 2 5 2 (p. 486). f() 5 8 2 2 (p. 486) Simplif the epression. (p. 492) 4. e 4 p e 5. (25e ) 6. e 4 5e 7. 8e 5 6e 2 Graph the function. State the domain and range. (p. 492) 8. 5 2e 9. 5 e 22 0. 5 e 2 2. g() 5 4e 2 2. TV SALES From 997 to 200, the number n (in millions) of black-and-white TVs sold in the United States can be modeled b n 5 26.8(0.85) t where t is the number of ears since 997. Identif the deca factor and the percent decrease. Graph the model and state the domain and range. Estimate the number of black-and-white TVs sold in 999. (p. 478). FINANCE You deposit $200 in an account that pas 4.5% annual interest compounded continuousl. What is the balance after 5 ears? (p. 492) 498 Chapter 7 EXTRA Eponential PRACTICE and Logarithmic for Lesson Functions 7., p. 06 ONLINE QUIZ at classzone.com