ch.4 Exponenital Functions

ch.4 Exponenital Functions - Skills Refresher for Chapter 4...

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Unformatted text preview: Skills Refresher for Chapter 4 Exponents Example Simplify each expression by hand. a) 82/3 b) (–32)–4/5 Pg. 177 36,44,48,50,52,54 Chapter 4 Exponential Functions 4.1 Introduction to The Family of Exponential Functions Key Points • Growth factors and growth rates • Decay factors and decay rates • The definition of an exponential function Salary example • New Salary = 100% of Old Salary + Percent Growth Rate Annual Growth Rate Factor Growing at a Constant Percent Rate Example 2 During the 2000s, the population of Mexico increased at a constant annual percent rate of 1.2%. Since the population grew by the same percent each year, it can be modeled by an exponential function. Let’s calculate the population of Mexico for the years after 2000. In 2000, the population was 100 million. The population grew by 1.2%, so Pop. in 2001 = Pop. in 2000 + 1.2% of Pop. in 2000 = 100 + 0.012(100) = 100 + 1.2 = 101.2 million. On the next slide, we extend this reasoning to estimate the Functions Modeling population of Mexico through 2007. Change: A Preparation for Calculus, 4th Growing at a Constant Percent Rate Example 2 continued Population of Mexico The population of Mexico increased by slightly more each year than it did the year before, because each year the increase is 1.2% of a larger number. Year 2000 2001 2002 2003 2004 2005 2006 2007 ΔP, % increase in population — 1.2 1.21 1.23 1.25 1.26 1.27 1.29 P, population (millions) 100 101.2 102.41 103.64 104.89 106.15 107.42 108.71 The projected population of Mexico, assuming 1.2% annual growth P, population (millions) 100 year Functions Modeling Change: A Preparation for Calculus, 4th Growth Factors and Percent Growth Rates The Growth Factor of an Increasing Exponential Function In Example 2, the population grew by 1.2%, so New Population = Old Population + 1.2% of Old Population = (1 + .012) ˑ Old Population = 1.012 ˑ Old Population We call 1.012 the growth factor. Functions Modeling Change: A Preparation for Calculus, 4th Growth Factors and Percent Growth Rates The Growth Factor of a Decreasing Exponential Function In Example 3, the carbon-14 changes by −11.4% every 1000 yrs. New Amount = Old Amount −11.4% of Old Amount = (1 − .114) ˑ Old Amount = 0.886 ˑ Old Amount Although 0.886 represents a decay factor, we use the term “growth factor” to describe both increasing and decreasing quantities. Functions Modeling Change: A Preparation for Calculus, 4th A General Formula for the Family of Exponential Functions An An exponential function Q = f(t) has the formula f(t) = a bt, a ≠ 0, b > 0, where a is the initial value of Q (at t = 0) and b, the base, is the growth factor. The growth factor is given by b=1+r where r is the decimal representation of the percent rate of change. • If there is exponential growth, then r > 0 and b > 1. • If there is exponential decay, then r < 0 and 0 < b < 1. Functions Modeling Change: A Preparation for Calculus, 4th Applying the General Formula for the Family of Exponential Functions Example 6 Using Example 2, find a formula for P, the population of Mexico (in millions), in year t where t = 0 represents the year 2000. Because the growth factor may change eventually, this formula may not give accurate results for large values of t. Functions Modeling Change: A Preparation for Calculus, 4th Definition of Exponential Function Function • A function represented by f(t) = abt, b > 0, 0, is an exponential function where a is the intial value of f (at t = 0) and base b being the growth factor. growth • • • If b > 1, then f is an exponential growth function If then If 0 < b < 1, then f is an exponential decay function If then b = 1 + r where r is the decimal representation of the percent change. percent • f(t) = abt or f(t) = a(1 + r)t The graph of f(t) = abt, b > 1 (exponential growth function) Range: (0, ∞ ) Domain: (–∞ , ∞ ) Horizontal Asymptote y=0 The graph of f(t) = abt, 0 < b < 1 (exponential decay) Range: (0, ∞ ) Horizontal Asymptote y=0 x 4 Domain: (–∞ , ) 2,6,8,30,40 Chapter 4 Exponential Functions 4.2 Comparing Functions and Linear Functions Key Points • How to determine when a function is linear • How to determine when a function is exponential • Finding formulas for exponential functions Population Growth by a Constant Number vs by a Constant Number Percentage Percentage Suppose a population is 10,000 in January 2004. Suppose the population increases by… • 500 people per year • 5% per year • What is the population in • What is the population in What What Jan 2005? Jan 2005? Jan – 10,000 + 500 = 10,500 – 10,000 + .05(10,000) = 10,000 10,500 10,000 10,000 + 500 = 10,500 10,500 • What is the population in What • What is the population in Jan 2006? Jan What Jan 2006? Jan – 10,500 + 500 = 11,000 10,500 11,000 – 10,500 + .05(10,500) = 10,500 10,500 + 525 = 11,025 11,025 Suppose a population is 10,000 in Jan 2004. Suppose the population increases by 500 per year. What is the by population in …. population • Jan 2005? – 10,000 + 500 = 10,500 • Jan 2006? – 10,000 + 2(500) = 11,000 • Jan 2007? – 10,000 + 3(500) = 11,500 • Jan 2008? – 10,000 + 4(500) = 12,000 Suppose a population is 10,000 in Suppose a population is 10,000 in Jan 2004 and increases by 500 per year. • Let t be the number of years after 2004. Let P(t) Let be the population in year t. What is the symbolic representation for P(t)? We know… • Population in 2004 = P(0) = 10,000 + 0(500) Population 10,000 • Population in 2005 = P(1) = 10,000 + 1(500) Population 10,000 • Population in 2006 = P(2) = 10,000 + 2(500) Population 10,000 • Population in 2007 = P(3) = 10,000 + 3(500) Population 10,000 • Population t years after 2004 = Population P(t) = 10,000 + t(500) 10,000 Population is 10,000 in 2004; increases Population is 10,000 in 2004; increases by 500 per yr P(t) = 10,000 + t(500) • P is a linear function of t. linear • What is the slope? – 500 people/year • What is the y-intercept? What – number of people at time 0 (the year 2004) = 10,000 When P increases When by a constant number of people per year, P is a linear function of t. Suppose a population is 10,000 in Jan 2004. Suppose a population is 10,000 in Jan 2004. More realistically, suppose the population increases by 5% per year. What is the population in …. • Jan 2005? – 10,000 + .05(10,000) = 10,000 10,000 + 500 = 10,500 10,000 • Jan 2006? – 10,500 + .05(10,500) = 10,500 10,500 + 525 = 11,025 10,500 • Jan 2007? – 11,025 + .05(11,025) = 11,025 11,025 + 551.25 = 11,576.25 11,576.25 Suppose a population is 10,000 in Jan Suppose a population is 10,000 in Jan 2004 and increases by 5% per year. • Let t be the number of years after 2004. Let P(t) be the Let be population in year t. What is the symbolic representation for P(t)? We know… • Population in 2004 = P(0) = 10,000 Population (0) • Population in 2005 = P(1) = 10,000 + .05 (10,000) = Population 10,000 1.05(10,000) = 1.051(10,000) =10,500 1.05 • Population in 2006 = P(2) = 10,500 + .05 (10,500) = Population 10,500 1.05 (10,500) = 1.05 (1.05)(10,000) = 1.052(10,000) = 1.05 1.05 (1.05)(10,000) 1.05 11,025 11,025 • Population t years after 2004 = Population P(t) = 10,000(1.05)t P(t) Population is 10,000 in 2004; increases Population is 10,000 in 2004; increases by 5% per yr P(t) = 10,000 (1.05)t • P iis an EXPONENTIAL function of t. More specifically, s EXPONENTIAL an exponential growth function. an • What is the base of the exponential function? – 1.05 • What is the y-intercept? – number of people at time 0 (the year 2004) = 10,000 When P increases by a When constant percentage per year, P is an exponential function of t. function Linear vs. Exponential Growth Linear • A Linear Function Linear adds a fixed amount to the previous value of y for each unit increase in x • For example, in For f(x) = 10,000 + 500x 500 is added to y for each increase of 1 in x. x. • An Exponential An Function multiplies a fixed amount to the previous value of y for each unit increase in x. • For example, in For f(x) = 10,000 (1.05)x y is multiplied by 1.05 for each increase of 1 in x. Comparison of Exponential and Linear Functions Linear y = 10000(1.05) x y = 10000 + 500 x Linear Function Linear y = 10000 + 500x x 0 1 2 3 4 5 6 y 10000 10500 11000 11500 12000 12500 13000 ∆x ∆y ∆y ∆x 1 1 1 1 1 1 500 500 500 500 500 500 500/1=500 500/1=500 500/1=500 500/1=500 500/1=500 500/1=500 Linear Function Slope is constant. Exponential Function Exponential Y = 10000 (1.05)x x 0 1 2 3 4 5 6 y 10,000 10,500 11,025 11,576 12,155 12,763 13,401 Ratios of consecutive y-values (corresponding to unit increases in x) 10500/10000 = 1.05 11025/10500 = 1.05 11576/11025 = 1.05 12155/11576 = 1.05 12763/12155 = 1.05 13401/12763 = 1.05 Note that this constant is the Note base of the exponential function. function. Exponential Exponential Function Ratios of consecutive y-values -values (corresponding to unit increases in x) are constant, in this case 1.05. Which function is linear and which is exponential? which x y -3 3/8 -2 3/4 -1 3/2 0 3 1 6 2 12 3 24 xy -3 9 -2 7 -1 5 03 11 2 -1 3 -3 For the linear function, tell the slope and y-intercept. For the -intercept. exponential function, tell the base and the y-intercept. Write the -intercept. equation of each. equation Identifying Linear and Exponential Functions From a Table For a table of data that gives y as a function of x and in which x is constant: • If the difference of consecutive y-values is constant, the table could represent a linear function. • If the ratio of consecutive y-values is constant, the table could represent an exponential function. Functions Modeling Change: A Preparation for Calculus, 4th 2,12, 36 Chapter 4 Exponential Functions 4.3 GRAPHS OF EXPONENTIAL FUNCTIONS Key Points • The possible appearances of the graphs of exponential functions • The effect of the initial value on the appearance of the graph of an exponential function • The effect of the growth factor on the appearance of the graph of an exponential function • Why exponential functions have horizontal asymptotes • Understanding limit notation and limits to infinity Graphs of the Exponential Family: The Effect of the Parameter a In the formula Q = abt, the value of a tells us where the graph crosses the Q-axis, since a is the value of Q when t = 0. Q Q Q=150 (1.2)t Q=50 (1.4)t Q=50 (1.2)t Q=100 (1.2)t 150 100 50 50 Q=50 (1.2)t 0 5 10 t Q=50 (0.8)t 0 Q=50 (0.6)t 5 t Functions Modeling Change: A Preparation for Calculus, 4th Graphs of the Exponential Family: The Effect of the Parameter b The growth factor, b, is called the base of an exponential function. Provided a is positive, if b > 1, the graph climbs when read from left to right, and if 0 < b < 1, the graph falls when read from left to right. Q Q=50 (1.4)t 50 Q=50 (1.2)t Q=50 (0.8)t Q=50 (0.6)t 0 5 t Functions Modeling Change: A Preparation for Calculus, 4th Horizontal Asymptotes The horizontal line y = k is a horizontal asymptote of The a function, f, if the function values get arbitrarily close to k as x gets large (either positively or negatively or both). We describe this behavior using the notation f(x) → k as x → ∞ or f(x) → k as x → −∞. Alternatively, using limit notation, we write lim f ( x) = k or lim f ( x) = k x →∞ x → −∞ Functions Modeling Change: A Preparation for Calculus, 4th Interpretation of a Horizontal Asymptote Example 1 A capacitor is the part of an electrical circuit that stores electric charge. The quantity of charge stored decreases exponentially with time. Stereo amplifiers provide a familiar example: When an amplifier is turned off, the display lights fade slowly because it takes time for the capacitors to discharge. If t is the number of seconds after the circuit is switched off, suppose that the quantity of stored charge (in micro-coulombs) is given by Q = 200(0.9)t, t ≥0. Q, charge (micro-coulombs) 200 The charge stored by a capacitor over one minute. 100 0 15 30 45 60 t (seconds) Functions Modeling Change: A Preparation for Calculus, 4th Solving Exponential Equations Graphically Exercise 42 The population of a colony of rabbits grows exponentially. The colony begins with 10 rabbits; five years later there are 340 rabbits. (a) Give a formula for the population of the colony of rabbits as a function of the time. (b) Use a graph to estimate how long it takes for the population of the colony to reach 1000 rabbits. R, # of rabbits Solution 1500 (6.5+, R = 10 (34)t/5 ≈ 10 (2.0244)t 1000 Based on the graph, one would estimate that the population of rabbits would reach 1000 in a little more than 6 ½ years. 1000) 500 (5,34) (0,10) 0 1 2 3 4 5 6 7 t, years Functions Modeling Change: A Preparation for Calculus, 4th Finding an Exponential Function for Data Example: Population data for the Houston Metro Area Since 1900 Table showing population Graph showing population data (in thousands) since 1900 t 0 10 20 30 40 50 N 184 236 332 528 737 1070 t 60 70 80 90 100 110 N 1583 2183 3122 3733 4672 5937 with an exponential model P (thousands) P = 190 (1.034)t t (years since 1900) Using an exponential regression feature on a calculator or computer the exponential function was found to be P = 190 Functions Modeling (1.034)t Change: A Preparation for Calculus, 4th • 18, Like 24, like 26, like 28 Chapter 4 Exponential Functions 4.5 The Number e Key Points • Basic facts about the number e • Continuous growth rates The Natural Number e An irrational number, introduced by Euler in 1727, is so important that it is given a special name, e. Its value is approximately e ≈ 2.71828 . . .. It is often used for the base, b, of the exponential function. Base e is called the natural base. This may seem mysterious, as what could possibly be natural about using an irrational base such as e? The answer is that the formulas of calculus are much simpler if e is used as the base for exponentials. Functions Modeling Change: A Preparation for Calculus, 4th Graphs of exponential functions with various bases Exponential Functions with Base e For the exponential function Q = a bt, the continuous growth rate, k, is given by solving ek = b. Then Q = a ekt. If a is positive, • If k > 0, then Q is increasing. • If k < 0, then Q is decreasing. Functions Modeling Change: A Preparation for Calculus, 4th Exponential Functions with Base e Example 1 Give the continuous growth rate of each of the following functions and graph each function: P = 5e0.2t, Q = 5e0.3t, and R = 5e−0.2t. 20 15 R = 5e−0.2t 10 P = 5e0.2t Q = 5e0.3t 5 5 0 5 t Functions Modeling Change: A Preparation for Calculus, 4th Exponential Functions with Base e Example 1 Give the continuous growth rate of each of the following functions and graph each function: P = 5e0.2t, Q = 5e0.3t, and R = 5e−0.2t. Solution The function P = 5e0.2t has a continuous growth rate of 20%, Q = 5e0.3t has a continuous 30% growth rate, and R = 5e−0.2t has a continuous growth rate of −20%. The negative sign in the exponent tells us that R is decreasing instead of increasing. Because a = 5 in all three Q = 5e0.3t functions, they each pass P = 5e0.2t through the point (0,5). They are all concave up and have R = 5e−0.2t horizontal asymptote y = 0. 20 15 10 5 5 0 5 t Functions Modeling Change: A Preparation for Calculus, 4th Exponential Functions with Base e Like Example 3 Caffeine leaves the body at a continuous rate of 17% per hour. How much caffeine is left in the body 4 hours after drinking a Monster Energy Drink containing 160 mg of caffeine? Functions Modeling Change: A Preparation for Calculus, 4th b = ek Exponential Functions with Base e Represent Continuous Growth • Any positive base b can be written as a power of e: • b = ek • The function Q = abt = a(ek)t = aekt S12, like10, like12, 18, 22 ...
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This note was uploaded on 04/04/2012 for the course MATH 2412 taught by Professor Staff during the Spring '08 term at Austin CC.

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