1.3 Exponential Functions

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1 Section. Eponential Functions. Eponential Functions You will be to model eponential growth and decay with functions of the form y = k a and recognize eponential growth and decay in algebraic, numerical, and graphical representations. Eponential functions Rules of eponents Applications of eponential functions (growth and decay) Compound interest The number e Eponential Growth Table. shows the growth of $00 invested in 996 at an interest rate of 5.5%, compounded annually. Year TABLE. Savings Account Growth Amount (dollars) Increase (dollars) 00 00(.055) 00( (.055 = =.0 = 7. 00(.055) = y=x After the first year, the value of the account is always.055 times its value in the previous year. After n years, the value is y = 00 (.055) n. Compound interest provides an eample of eponential growth and is modeled by a function of the form y P a, where P is the initial investment (called the principal) and a is equal to plus the interest rate epressed as a decimal. The equation y P a, a > 0, a #, identifies a family of functions called eponential functions. Notice that the ratio of consecutive amounts in Table. is always the same:.0/05.0 = 7./.0 =.88/ This fact is an important feature of eponential curves that has widespread application, as we will see. EXPLORATION Eponential Functions 6] by [-, 6] (a) Graph the function y a for a =,, 5, in a [ 5, 5] by [, 5] viewing window. For what values of is it true that < < For what values of is it true that > > For what values of is it true that Graph the function y = (l/a) = a- for a Repeat parts for the functions in part 5. DEFINITION Eponential Function Let a be a positive real number other than. The function f() = a is the eponential function with base a. 6] by [-, 6] (b) Figure.0 A graph of (a) y = and The domain of f() = a is ( 00, 00) and the range is (0, 00). If a >, the graph of f looks like the graph of y = in Figure.0a. If 0 < a < l, the graph of f looks like the graph of y = - in Figure.0b.

2 Chapter Prerequisites for Calculus EXAMPLE Graph the function y = Graphing an Eponential Function ). State its domain and range. Figure. shows the graph of the function y. It appears that the domain is ( 00, 00) The range is (, 00) because () > 0 for all. Now Try Eercise. [-5, 5] by [-5, 5] Figure. The graph of y (Eample l) Y = (X ) -. EXAMPLE Find the zeros of f() Finding Zeros 5.5 graphically. Figure.a suggests that f has a zero between = and =, closer to. We can use our grapher to find that the zero is approimately.756 (Figure.b). Now Try Eercise 9. Eponential functions obey the rules for eponents. Rules for Eponents y = 5-.5X 5] by [-8, 8] (a) If a > 0 and b > 0, the following hold for all real numbers and y. = a +y a.. (a)y. a. b = (ab) 5. a b a Zero X=l.7S6U708 [-5, 5] by [-8, 8] (b) Figure (a) Agraphoff() 5.5. (b) Showing the use of the ZERO feature to approimate the zero of f. (Eample ) t/0 Eponential Decay Eponential functions can also model phenomena that produce a decrease over time, such as happens with radioactive decay. The half-life of a radioactive substance is the amount of time it takes for half of the substance to change from its original radioactive state to a nonradioactive state by emitting energy in the form of radiation. EXAMPLE Modeling Radioactive Decay Suppose the half-life of a certain radioactive substance is 0 days and that there are 5 grams present initially. When will there be only gram of the substance remaining? The number of grams remaining after 0 days is 5 5 The number of grams remaining after 0 days is 5 5 Intersection =V6.k856 [0, 80 by [-, 5] Figure. (Eample ) The function y = 5( /)t/0 models the mass in grams of the radioactive substance after t days. Figure. shows that the graphs of Yl = 5( /) t/0 and h = I (for I gram) intersect when t is approimately 6.. There will be gram of the radioactive substance left after approimately 6. days, or about 6 days 0.5 hours. Now Try Eercise.

3 Section. Eponential Functions 5 Compound interest investments, population growth, and radioactive decay are all eamples of eponential growth and decay. DEFINITIONS Eponential Growth, Eponential Decay The function y = k a k > 0 is a model for eponential growth if a > l, and a model for eponential decay if 0 < a <. Compound Interest One common application of eponential growth in the financial world is the compounding effect of accrued interest in a savings account. We saw at the beginning of this section that an account paying 5.5% annual interest would multiply by a factor of I.055 every year, so an account starting with principal P would be worth.055 after t years. Some accounts pay interest multiple times per year, which increases the compounding effect. An account earning 6% annual interest compounded monthly would pay one-twelfth of the interest each month, effectively a monthly interest rate of 0.5%. Thus, after t years, the account would be worth P( I.005) t. The general formula is given below. Compound Interest Formula If an account begins with principal P and earns an annual interest rate r compounded n times per year, then the value of the account in t years is n nt EXAMPLE Compound Interest Bernie deposits $500 in an account earning 5.% annual interest. Find the value of the account in 0 years if the interest is compounded (a) annually; (b) quarterly; (c) monthly. (a) 500( , so the account is worth $ (0) (b) , so the account is worth $7.55. O.05 (c) , so the account is worth $8.8. Now Try Eercises 5 and 6. The Number e You may have noticed in Eample that the value of the account becomes greater as the number of interest payments per year increases, but the value does not appear to be increasing without bound. In fact, a bank can even offer to compound the interest continuously without paying that much more per year. This is because the number that serves as the base of the eponential function in the compound interest formula has a finite limit as n approaches infinity: n r n 00

4 6 Chapter Prerequisites for Calculus y = I/X)X You can get an estimate of the number e by looking at values of + increasing values of, as shown graphically and numerically in Figure.. This number e, which is approimately.788, is one of the most important numbers in mathematics. Like 7, it is an irrational number that shows up in many different contets, some of them quite surprising. You will encounter several of them in this course. One place that e appears is in the formula for continuously compounded interest qooo sooo [-0, 0 by [-5, Figure. A graph and table of values for f() = ( + l/) both suggest that as + 00, f() e.78. Continuously Compounded Interest Formula If an account begins with principal P and earns an annual interest rate r compounded continuously, then the value of the account in t years is = Pe rt EXAMPLE 5 Continuously Compounded Interest Bernice deposits $500 in an account earning 5.% annual interest. Find the value of the account in 0 years if the interest is compounded (a) daily; (b) continuously (0) (a) , so the account is worth $ , so the account is worth $90.0. Notice that the difference between "daily" and "continuously" for Bernice is 7 cents! Now Try Eercise 7. Quick Review. (For help, go to Section..) Eercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Eercises, evaluate the epression. Round your answers to decimal places.. 5/.+6 In Eercises 6, solve the equation. Round your answers to decimal places. = = In Eercises 7 and 8, find the value of investing P dollars for n years with the interest rate r compounded annually. 7. P = $500, r =.75%, n = 5 years 8. P = $000, r = 6.%, n = years In Eercises 9 and 0, simplify the eponential epression. ( Y) ab- ac- - CIO, (y) b Section. Eercises Eercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Eercises, graph the function. State its domain and range..y=-x+. In Eercises 5 8, rewrite the eponential epression to have the indi - cated base r, base 6. 6, base base base

5 Section. Eponential Functions 7 In Eercises 9, use a graph to find the zeros of the function. 9. f( ) 0. f() = e-. f() = X 05. f() - In Eercises 8, match the function with its graph. Try to do it without using your grapher.. Y = - 6.y = (a) (c) 5, y = (8.» =.5X - (b) (d) 5. Doubling Your Money Determine how much time is required for an investment to double in value if interest is earned at the rate of 6.5% compounded annually. 6. Doubling Your Money Determine how much time is required for an investment to double in value if interest is earned at the rate of 6.5% compounded monthly. 7. Doubling Your Money Determine how much time is required for an investment to double in value if interest is earned at the rate of 6.5% compounded continuously. 8. Tripling Your Money Determine how much time is required for an investment to triple in value if interest is earned at the rate of 5.75% compounded annually. 9. Tripling Your Money Determine how much time is required for an investment to triple in value if interest is earned at the rate of 5.75% compounded daily. 0. Tripling Your Money Determine how much time is required for an investment to triple in value if interest is earned at the rate of 5.75% compounded continuously.. Cholera Bacteria Suppose that a colony of bacteria starts with bacterium and doubles in number every half hour. How many bacteria will the colony contain at the end of h?. Eliminating a Disease Suppose that in any given year, the number of cases of a disease is reduced by 0%. If there are 0,000 cases today, how many years will it take (a) to reduce the number of cases to 000? (b) to eliminate the disease; that is, to reduce the number of cases to less than? (e) In Eercises 9, use an eponential model to solve the problem. 9. Population Growth The population of Knoville is 500,000 and is increasing at the rate of.75% each year. Approimately when will the population reach million? 0. Population Growth The population of Silver Run in the year 890 was 650. Assume the population increased at a rate of.75% per year. (a) Estimate the population in 95 and 90. (b) Approimately when did the population reach 50,000?. Half-life The approimate half-life of titanium- is 6 years. How long will it take a sample to lose (a) 50% of its. titanium-? (b) 75% of its titanium-? c Half-life The amount of silicon- in a sample will decay from 8 grams to 7 grams in approimately 0 years. What is the approimate half-life of silicon-?. Radioactive Decay The half-life of phosphorus- is about days. There are 6.6 grams present initially. (a) Epress the amount of phosphorus- remaining as a function of time t. (b) When will there be gram remaining?. Finding Time If John invests $00 in a savings account with a 6% interest rate compounded annually, how long will it take until John's account has a balance of $50? Group Activity In Eercises 6, copy and complete the table for the function. y=r-. y = + 5. Y' y y 9 Change ( Ay) Change ( Ay) Change ( Ay)

6 8 Chapter Prerequisites for Calculus 6. y = e y Ratio (Yi/Yi-l ) 7. Writing to Learn Eplain how the change Ay is related to the slopes of the lines in Eercises and. If the changes in are constant for a linear function, what would you conclude about the corresponding changes in y? 8. Bacteria Growth The number of bacteria in a petri dish culture after t hours is (a) What was the initial number of bacteria present? (b) How many bacteria are present after 6 hours? (c) Approimately when will the number of bacteria be 00? Estimate the doubling time of the bacteria. 9. The graph of the eponential function y = a b X is shown below. Find a and b. Standardized Test Questions You may use a graphing calculator to solve the following problems.. True or False The number - is negative. Justify your answer.. True or False If = Y, then a = 6. Justify your answer.. Multiple Choice John invests $00 at.5% compounded annually. About how long will it take for John's investment to double in value? (A) 6yr (B) 9yr (C) yr (D) 6yr (E) 0 yr. Multiple Choice Which of the following gives the domain of Y = e - (A) ( 00, 00) (B) [, 00) (C) [, 00) (D) ( 00, ] 5. Multiple Choice Which of the following gives the range of y = K'? (A) ( 00, 00) (B) (-00, ) (C) 00) (D) (-00, ] (E) all reals 6. Multiple Choice Which of the following gives the best approimation for the zero of f() e? = -.86 (B) = 0.86 (C) =.86 (E) There are no zeros. 0. The graph of the eponential function y = a below. Find a and b. (, o. 5) b X is shown Eploration 7. Let = and Y (a) Graph and Y in [ 5, 5] by [, 0]. How many times do you think the two graphs cross? (b) Compare the corresponding changes in and Y as changes from I to, to, and so on. How large must be for the changes in Y to overtake the changes in? (c) Solve for : (d) Solve for : < Etending the Ideas In Eercises 8 and 9, assume that the graph of the eponential function f() = k a passes through the two points. Find the values of a and k. {8. (,.5), (-, 0.5) (9. (,.5), (-, 6) Quick Quiz for AP* Preparation: Sections.. You may use a graphing calculator to solve the following problems.. Multiple Choice Which of the following gives an equation for the line through (, l ) and parallel to the line y=-+? 7 5. Multiple Choice If f() = + I and g() = which of the following gives (f o g )()? (C)9 (D) 0 (E) 5. Multiple Choice The half-life of a certain radioactive substance is 8 hr. There are 5 grams present initially. Which of the following gives the best approimation when there will be gram remaining? (B) 0 (C) 5 (D) 6 (E) 9. Free Response Let f() (a) Find the domain of f. (b) Find the range of f. (c) Find the zeros of f.

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