4.3 Assess Your Understanding

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1 CHAPTER Exponential and Logarithmic Functions. Assess Your Understanding Are You Prepared? Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.. = ; > = ; - =. (pp and. Find the average rate of change of fx = x - 5 from pp. 0 06) x = 0 to x = c. (pp. 5 ). Solve: x + 5x - = 0. (pp ). True or False: To graph y = x - 5. True or False: The function fx = has y = as a, shift the graph of x - y = x to the left units. (pp. 0) horizontal asymptote. (pp. 9 95) Concepts and Vocabulary 6. The graph of every exponential function fx = a x, a 7 0, a Z, passes through three points:,, and. 7. If the graph of the exponential function fx = a x, a 7 0, a Z, is decreasing, then a must be less than.. If x =, then x =. 9. True or False: The graphs of and y = a x y = x are b identical. 0. True or False: The range of the exponential function is the set of all real numbers. fx = a x, a 7 0, a Z,

2 Skill Building SECTION. Exponential Functions In Problems 0, approximate each number using a calculator. Express your answer rounded to three decimal places.. (a). (b). (c).6 (d) 5. (a) 5.7 (b) 5.7 (c) 5.7 (d) 5. (a). (b). (c).5 (d) p. (a).7 (b).7 (c).7 (d) e 5. (a)..7 (b)..7 (c)..7 (d) p e 6. (a).7. (b).7. (c).7. (d) e p e. e -. In Problems, determine whether the given function is exponential or not. For those that are exponential functions, identify the value of the base a. [Hint: Look at the ratio of consecutive values.] e -0.5 e.. x f(x) x g(x). x H(x). x F(x) x f(x) 6. x g(x) 7. x H(x). x F(x) 6 In Problems 9 6, the graph of an exponential function is given. Match each graph to one of the following functions. A. y = x B. y = -x C. y = - x D. y = - -x E. y = x - F. y = x - G. y = - x H. y = - x 9. y 0. y. y. y y. y. y 5. y 6. y y

3 CHAPTER Exponential and Logarithmic Functions In Problems 7, use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. 7. fx = x +. fx = x + 9. fx = -x - 0. fx = - x +. fx = + x. fx = - x. fx = + x>. fx = - -x> In Problems 5 5, begin with the graph of y = e x (Figure 0 (a)) and use transformations to graph each function. Determine the domain, range, and horizontal asymptote of each function. 5. fx = e -x 6. fx = -e x 7. fx = e x +. fx = e x - 9. fx = 5 - e -x 50. fx = 9 - e -x 5. fx = - e -x> 5. fx = 7 - e In Problems 5 66, solve each equation = = 55. x = 9 x 56. x = x a - x a - x 9 -x = x # -x = x 6. 5 b = 5 b = x - = x - x = e x # e x e x = e x # x = 9 = 7 = e 67. If x = 7, what does equal? 6. If x =, what does equal? If -x =, what does equal? 70. If 5 -x =, what does equal? e -x 5 x In Problems 7 7, determine the exponential function whose graph is given. 7. y y 0 6 (, 9) (0, ) (, ) (, ) x (, ) 5 (, 5) (0, ) x 7. y 7. (0, ) (, ) 6 x (, 6) (, 6) 0 (, e ) y (0, ) x (, e) (, e ) Applications and Extensions 75. Optics If a single pane of glass obliterates % of the light passing through it, then the percent p of light that passes through n successive panes is given approximately by the function pn = n (a) What percent of light will pass through 0 panes? (b) What percent of light will pass through 5 panes? 76. Atmospheric Pressure The atmospheric pressure p on a balloon or plane decreases with increasing height. This pressure, measured in millimeters of mercury, is related to the height h (in kilometers) above sea level by the function ph = 760e -0.5h (a) Find the atmospheric pressure at a height of kilometers (over a mile). (b) What is it at a height of 0 kilometers (over 0,000 feet)?

4 SECTION. Exponential Functions Depreciation The price p of a Honda Civic DX Sedan that is x years old is given by px = 6, x (a) How much does a -year-old Civic DX Sedan cost? (b) How much does a 9-year-old Civic DX Sedan cost? 7. Healing of Wounds The normal healing of wounds can be modeled by an exponential function. If A 0 represents the original area of the wound and if A equals the area of the wound, then the function An = A 0 e -0.5n describes the area of a wound after n days following an injury when no infection is present to retard the healing. Suppose that a wound initially had an area of 00 square millimeters. (a) If healing is taking place, how large will the area of the wound be after days? (b) How large will it be after 0 days? 79. Drug Medication The function Dh = 5e -0.h can be used to find the number of milligrams D of a certain drug that is in a patient s bloodstream h hours after the drug has been administered. How many milligrams will be present after hour? After 6 hours? 0. Spreading of Rumors A model for the number N of people in a college community who have heard a certain rumor is N = P - e -0.5d where P is the total population of the community and d is the number of days that have elapsed since the rumor began. In a community of 000 students, how many students will have heard the rumor after days?. Exponential Probability Between :00 PM and :00 PM, cars arrive at Citibank s drive-thru at the rate of 6 cars per hour (0. car per minute). The following formula from probability can be used to determine the probability that a car will arrive within t minutes of :00 PM: Ft = - e -0.t (a) Determine the probability that a car will arrive within 0 minutes of :00 PM (that is, before :0 PM). (b) Determine the probability that a car will arrive within 0 minutes of :00 PM (before :0 PM). (c) What value does F approach as t becomes unbounded in the positive direction? (d) Graph F using a graphing utility. (e) Using TRACE, determine how many minutes are needed for the probability to reach 50%.. Exponential Probability Between 5:00 PM and 6:00 PM, cars arrive at Jiffy Lube at the rate of 9 cars per hour (0.5 car per minute). The following formula from probability can be used to determine the probability that a car will arrive within t minutes of 5:00 PM: Ft = - e -0.5t (a) Determine the probability that a car will arrive within 5 minutes of 5:00 PM (that is, before 5:5 PM). (b) Determine the probability that a car will arrive within 0 minutes of 5:00 PM (before 5:0 PM). (c) What value does F approach as t becomes unbounded in the positive direction? (d) Graph F using a graphing utility. (e) Using TRACE, determine how many minutes are needed for the probability to reach 60%.. Poisson Probability Between 5:00 PM and 6:00 PM, cars arrive at McDonald s drive-thru at the rate of 0 cars per hour. The following formula from probability can be used to determine the probability that x cars will arrive between 5:00 PM and 6:00 PM. where (a) Determine the probability that x = 5 cars will arrive between 5:00 PM and 6:00 PM. (b) Determine the probability that x = 0 cars will arrive between 5:00 PM and 6:00 PM.. Poisson Probability People enter a line for the Demon Roller Coaster at the rate of per minute. The following formula from probability can be used to determine the probability that x people will arrive within the next minute. where Px = 0x e -0 x! x! = x # x - # x - # Á # # # Px = x e - x! x! = x # x - # x - # Á # # # (a) Determine the probability that x = 5 people will arrive within the next minute. (b) Determine the probability that x = people will arrive within the next minute. 5. Relative Humidity The relative humidity is the ratio (expressed as a percent) of the amount of water vapor in the air to the maximum amount that it can hold at a specific temperature. The relative humidity, R, is found using the following formula: R = 0 a T D b where T is the air temperature (in F) and D is the dew point temperature (in F). (a) Determine the relative humidity if the air temperature is 50 Fahrenheit and the dew point temperature is Fahrenheit. (b) Determine the relative humidity if the air temperature is 6 Fahrenheit and the dew point temperature is 59 Fahrenheit. (c) What is the relative humidity if the air temperature and the dew point temperature are the same?

5 6 CHAPTER Exponential and Logarithmic Functions 6. Learning Curve Suppose that a student has 500 vocabulary words to learn. If the student learns 5 words after 5 minutes, the function Lt = e t approximates the number of words L that the student will learn after t minutes. (a) How many words will the student learn after 0 minutes? (b) How many words will the student learn after 60 minutes? 7. Current in a RL Circuit The equation governing the amount of current I (in amperes) after time t (in seconds) in a single RL circuit consisting of a resistance R (in ohms), an inductance L (in henrys), and an electromotive force E (in volts) is E I = E R - e-r>lt (a) If E = 0 volts, R = 0 ohms, and L = 5 henrys, how much current I is flowing after 0. second? After 0.5 second? After second? (b) What is the maximum current? (c) Graph this function I = I t, measuring I along the y- axis and t along the x-axis. (d) If E = 0 volts, R = 5 ohms, and L = 0 henrys, how much current I is flowing after 0. second? After 0.5 second? After second? (e) What is the maximum current? (f) Graph this function I = I t on the same coordinate axes as I t.. Current in a RC Circuit The equation governing the amount of current I (in amperes) after time t (in microseconds) in a single RC circuit consisting of a resistance R (in ohms), a capacitance C (in microfarads), and an electromotive force E (in volts) is I I = E R e-t>rc I R R L (a) If E = 0 volts, R = 000 ohms, and C =.0 microfarad, how much current I is flowing initially t = 0? After 000 microseconds? After 000 microseconds? (b) What is the maximum current? (c) Graph this function I = I t, measuring I along the y-axis and t along the x-axis. (d) If E = 0 volts, R = 000 ohms, and C =.0 microfarads, how much current I is flowing initially? After 000 microseconds? After 000 microseconds? (e) What is the maximum current? (f) Graph this function I = I t on the same coordinate axes as I t. 9. Another Formula for e Use a calculator to compute the values of for n =, 6,, and 0. Compare each result with e. [Hint: +! +! + Á + n!! =,! = #,! = # #, n! = nn - # Á #. ] 90. Another Formula for e Use a calculator to compute the various values of the expression. Compare the values to e etc. 9. Difference Quotient If fx = a x, show that fx + h - fx 9. If fx = a x, show that fa + B = fa # fb. 9. If fx = ax, show that f-x = fx. 9. If fx = a x, show that fax = fx a. h = a x # ah - h Problems 95 and 96 provide definitions for two other transcendental functions. 95. The hyperbolic sine function, designated by sinh x, is defined as sinh x = ex - e -x (a) Show that fx = sinh x is an odd function. (b) Graph fx = sinh x using a graphing utility. 96. The hyperbolic cosine function, designated by cosh x, is defined as E C cosh x = ex + e -x (a) Show that fx = cosh x is an even function. (b) Graph fx = cosh x using a graphing utility.

6 SECTION. Logarithmic Functions 7 (c) Refer to Problem 95. Show that, for every x, cosh x - sinh x = 97. Historical Problem Pierre de Fermat (60 665) conjectured that the function Discussion and Writing fx = + 9. The bacteria in a -liter container double every minute. After 60 minutes the container is full. How long did it take to fill half the container? 99. Explain in your own words what the number e is. Provide at least two applications that require the use of this number. 00. Do you think that there is a power function that increases more rapidly than an exponential function whose base is greater than? Explain. for x =,,, Á, would always have a value equal to a prime number. But Leonhard Euler (707 7) showed that this formula fails for x = 5. Use a calculator to determine the prime numbers produced by f for x =,,,. Then show that f5 = 6 * 6,700,7, which is not prime. 0. As the base a of an exponential function fx = a x, a 7, increases, what happens to the behavior of its graph for x 7 0? What happens to the behavior of its graph for x 6 0? 0. The graphs of and y = a x y = a -x are identical. Why? a b Are You Prepared? Answers. 6; ;. e -,. False. 5. True 9 f