College Algebra Twelfth Edition Chapter 4 Inverse Exponential
College Algebra Twelfth Edition Chapter 4 Inverse, Exponential, and Logarithmic Functions Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 1
Section 4. 2 Exponential Functions • Exponents and Properties • Exponential Functions • Exponential Equations • Compound Interest • The Number e and Continuous Compounding • Exponential Models Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 2
Exponents and Properties (1 of 5) if a is a real Recall the definition of number, m is an integer, n is a positive integer, and is a real number, then For example, and Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 3
Exponents and Properties (2 of 5) In this section we extend the definition of to include all real (not just rational) values of the might be evaluated by exponent r. For example, with the rational approximating the exponent numbers 1. 7, 1. 732, and so on. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 4
Exponents and Properties (3 of 5) Because these values approach the value of more and more closely, it seems reasonable that should be approximated more and more closely by and so on. These the numbers expressions can be evaluated using rational exponents as follows. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 5
Exponents and Properties (4 of 5) To show that this assumption is reasonable, see the graphs of the function with three different domains. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 6
Exponents and Properties (5 of 5) Using this interpretation of real exponents, all rules and theorems for exponents are valid for all real number exponents, not just rational ones. In addition to the rules for exponents presented earlier, we use several new properties in this chapter. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 7
Additional Properties of Exponents For any real number statements are true. the following is a unique real number for all real (a) numbers x. (b) if and only if b = c. (c) If a > 1 and m < n, then (d) If 0 < a < 1 and m < n, then Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 8
Properties of Exponents (1 of 2) Properties (a) and (b) require a > 0 so that is always defined. For example, This means that is not a real number if will always be positive, since a must be positive. In property (a), a cannot equal 1 because for every real number value of x, so each value of x leads to the same real number, 1. For property (b) to hold, a must not equal 1 since, for example, even though Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 9
Properties of Exponents (2 of 2) Properties (c) and (d) say that when a > 1, increasing the exponent on “a” leads to a greater number, but when 0 < a < 1, increasing the exponent on “a” leads to a lesser number. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 10
Example 1: Evaluating an Exponential Expression (1 of 4) find each of the following. If (a) Solution Replace x with − 1. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 11
Example 1: Evaluating an Exponential Expression (2 of 4) If find each of the following. (b) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 12
Example 1: Evaluating an Exponential Expression (3 of 4) If find each of the following. (c) Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 13
Example 1: Evaluating an Exponential Expression (4 of 4) If find each of the following. (d) Solution Use a calculator. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 14
Exponential Function (1 of 4) then the exponential If function with base a is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 15
Note We do not allow 1 as the base for an exponential function. If a = 1, then function becomes the constant function defined by which is not an exponential function. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 16
Exponential Functions We repeat the final graph (with real numbers as domain) and summarize important details here. • The y-intercept is (0, 1). • Because for all x and the x-axis is a horizontal asymptote. • As the graph suggests, the domain of the function is and the range is • The function is increasing on its entire domain, and is one-to-one. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 17
Exponential Function f of x equals a to the Power of x (1 of 6) Domain: Range: For • for a > 1, is increasing and continuous on its entire domain, Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 18
Exponential Function f of x equals a to the Power of x (2 of 6) Domain: Range: For • The x-axis is a horizontal asymptote as Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 19
Exponential Function f of x equals a to the Power of x (3 of 6) Domain: Range: For • The graph passes through the points Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 20
Exponential Function f of x equals a to the Power of x (4 of 6) Domain: Range: For • for 0 < a < 1, is decreasing and continuous on its entire domain, Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 21
Exponential Function f of x equals a to the Power of x (5 of 6) Domain: Range: For • The x-axis is a horizontal asymptote as Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 22
Exponential Function f of x equals a to the Power of x (6 of 6) Domain: Range: For • The graph passes through the points Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 23
Exponential Function (2 of 4) Recall the graph of reflected the graph of across the y-axis. Thus, we have the following. If then This is supported by the graphs shown. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 24
Exponential Function (3 of 4) The graph of typical of graphs of where a > 1. For larger values of a, the graphs rise more steeply, but the general shape is similar. When 0 < a < 1, the graph decreases in a manner similar to the graph of Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 25
Exponential Function (4 of 4) The graphs of several typical exponential functions illustrate these facts. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 26
Characteristics of the Graph of f of x equals a to the Power of x 1. The points are on the graph. 2. If a > 1, then f is an increasing function. If 0 < a < 1, then f is a decreasing function. 3. The x-axis is a horizontal asymptote. 4. The domain is and the range is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 27
Example 2: Graphing an Exponential Function (1 of 2) Graph Give the domain and range. Solution The y-intercept is (0, 1), and the xaxis is a horizontal asymptote. Plot a few ordered pairs, and draw a smooth curve through them. The function also has domain and range and is one-to-one. The function is increasing on its entire domain. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 28
Example 2: Graphing an Exponential Function (2 of 2) Graph Give the domain and range. Solution Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 29
Example 3: Graphing Reflections and Translations (1 of 3) Graph each function. Show the graph of for comparison. Give the domain and range. (a) Solution is The graph of that of reflected across the x-axis. The and the domain is range is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 30
Example 3: Graphing Reflections and Translations (2 of 3) Graph each function. Show the graph of for comparison. Give the domain and range. (b) Solution is the The graph of translated graph of 3 units to the left. The domain is and the range is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 31
Example 3: Graphing Reflections and Translations (3 of 3) Graph each function. Show the graph of for comparison. Give the domain and range. (c) Solution The graph of is the graph of translated 2 units to the right and 1 unit down. The and the domain is range is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 32
Example 4: Solving an Exponential Equation (1 of 2) Solve Solution Write each side of the equation using a common base. Definition of negative exponent. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 33
Example 4: Solving an Exponential Equation (2 of 2) Solve Solution Write 81 as a power of 3. Set exponents equal. Multiply by − 1. The solution set of the original equation is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 34
Example 5: Solving an Exponential Equation Solve Solution Write each side of the equation using a common base. Write 8 as a power of 2. Set exponents equal. Subtract Divide by − 2. Check by substituting 11 for x in the original equation. The solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 35
Example 6: Solving an Equation with a Fractional Exponent (1 of 2) Solve Solution Notice that the variable is in the base rather than in the exponent. Radical notation for Take fourth roots on each side. Remember to use Cube each side. Check both solutions in the original equation. Both check, so the solution set is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 36
Example 6: Solving an Equation with a Fractional Exponent (2 of 2) Solve Solution Alternative Method There may be more than one way to solve an exponential equation, as shown here. Cube each side. Write 81 as Take fourth roots on each side. Simplify the radical. Apply the exponent. The same solution set, results. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 37
Compound Interest (1 of 3) Recall the formula for simple interest, where P is principal (amount deposited), r is annual rate of interest expressed as a decimal, and t is time in years that the principal earns interest. Suppose t = 1 yr. Then at the end of the year the amount has grown to If this balance earns interest at the same interest rate for another year, the balance at the end of that year will increase as follows. Factor. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 38
Compound Interest (2 of 3) After the third year, this will grow in a similar pattern. Factor. Continuing in this way produces a formula for interest compounded annually. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 39
Compound Interest (3 of 3) If P dollars are deposited in an account paying an annual rate of interest r compounded (paid) n times per year, then after t years the account will contain A dollars, according to the following formula. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 40
Example 7: Using the Compound Interest Formula (1 of 3) Suppose $1000 is deposited in an account paying 4% interest per year compounded quarterly (four times per year). (a) Find the amount in the account after 10 y r with no withdrawals. ea Solution Compound interest formula Let P = 1000, r = 0. 04, n = 4, and t = 10. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 41
Example 7: Using the Compound Interest Formula (2 of 3) Suppose $1000 is deposited in an account paying 4% interest per year compounded quarterly (four times per year). (a) Find the amount in the account after 10 y r with no withdrawals. ea Solution Simplify. Round to the nearest cent. Thus, $1488. 86 is in the account after 10 y r. ea Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 42
Example 7: Using the Compound Interest Formula (3 of 3) Suppose $1000 is deposited in an account paying 4% interest per year compounded quarterly (four times per year). (b) How much interest is earned over the 10 -y r period? ea Solution The interest earned for that period is Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 43
Example 8: Finding Present Value (1 of 5) Becky must pay a lump sum of $6000 in 5 y r. (a) What amount deposited today (present value) at 3. 1% compounded annually will grow to $6000 in 5 y r? ea ea Solution Compound interest formula Let A = 6000, r = 0. 031, n = 1, and t =5. Simplify. Use a calculator. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 44
Example 8: Finding Present Value (2 of 5) Becky must pay a lump sum of $6000 in 5 y r. (a) What amount deposited today (present value) at 3. 1% compounded annually will grow to $6000 in 5 y r? ea ea Solution If Becky leaves $5150. 60 for 5 y r in an account paying 3. 1% compounded annually, she will have $6000 when she needs it. We say that $5150. 60 is the present value of $6000 if interest of 3. 1% is compounded annually for 5 y r. ea ea Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 45
Example 8: Finding Present Value (3 of 5) Becky must pay a lump sum of $6000 in 5 y r. ea (b) If only $5000 is available to deposit now, what annual interest rate is necessary for the money to increase to $6000 in 5 y r? ea Solution Compound interest formula Let A = 6000, P = 5000, n = 1, and t = 5. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 46
Example 8: Finding Present Value (4 of 5) Becky must pay a lump sum of $6000 in 5 y r. ea (b) If only $5000 is available to deposit now, what annual interest rate is necessary for the money to increase to $6000 in 5 y r? ea Solution Divide by 5000. Take the fifth root on each side. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 47
Example 8: Finding Present Value (5 of 5) Becky must pay a lump sum of $6000 in 5 y r. ea (b) If only $5000 is available to deposit now, what annual interest rate is necessary for the money to increase to $6000 in 5 y r? ea Solution Subtract 1. Use a calculator. An interest rate of 3. 71% will produce enough interest to increase the $5000 to $6000 by the end of 5 y r. ea Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 48
Continuous Compounding (1 of 4) The more often interest is compounded within a given time period, the more interest will be earned. Surprisingly, however, there is a limit on the amount of interest, no matter how often it is compounded. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 49
Continuous Compounding (2 of 4) Suppose that $1 is invested at 100% interest per year, compounded n times per year. Then the interest rate (in decimal form) is 1. 00 and the interest rate period is According to the formula (with P = 1 ), the compound amount at the end of 1 y r will be ea Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 50
Continuous Compounding (3 of 4) A calculator gives the results shown for various values of n. The table suggests that as n increases, the value of gets closer and closer to some fixed number. This is indeed the case. This fixed number is called e. (Note that in mathematics, e is a real number and not a variable. ) Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 51
Value of e Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 52
Continuous Compounding (4 of 4) If P dollars are deposited at a rate of interest r compounded continuously for t years, the compound amount A in dollars on deposit is given by the following formula. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 53
Example 9: Solving a Continues Compounding Problem Suppose $5000 is deposited in an account paying 3% interest compounded continuously for 5 y r. Find the total amount on deposit at the end of 5 y r. ea ea Solution Continuous compounding formula Let P = 5000, r = 0. 03, and t = 5. Multiply exponents. Use a calculator. Check that daily compounding would have produced a compound amount about $0. 03 less. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 54
Example 10: Comparing Interest Earned as Compounding Is More Frequent (1 of 3) In Example 7, we found that $1000 invested at 4% compounded quarterly for 10 y r grew to $1488. 86. Compare this same investment compounded annually, semiannually, monthly, daily, and continuously. ea Solution Substitute 0. 04 for r, 10 for t, and the appropriate number of compounding periods for n into Compound interest formula and also into Continuous compounding formula The results for amounts of $1 and $1000 are given in the table. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 55
Example 10: Comparing Interest Earned as Compounding Is More Frequent (2 of 3) Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 56
Example 10: Comparing Interest Earned as Compounding Is More Frequent (3 of 3) Comparing the results, we notice the following. • Compounding semiannually rather than annually increases the value of the account after 10 y r by $5. 71. ea • Quarterly compounding grows to $2. 91 more than semiannual compounding after 10 y r. ea • Daily compounding yields only $0. 96 more than monthly compounding. • Continuous compounding yields only $0. 03 more than monthly compounding. Each increase in compounding frequency earns less and less additional interest. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 57
Exponential Models The number e is important as the base of an exponential function in many practical applications. In situations involving growth or decay of a quantity, the amount or number present at time t often can be closely modeled by a function of the form where is the amount or number present at time t = 0 and k is a constant. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 58
Example 11: Using Data to Model Exponential Growth (1 of 6) Data from recent past years indicate that future amounts of carbon dioxide in the atmosphere may grow according to the table. Amounts are given in parts per million. Year Carbon Dioxide (ppm) 1990 353 2000 375 2075 590 2175 1090 2275 2000 Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 59
Example 11: Using Data to Model Exponential Growth (2 of 6) (a) Make a scatter diagram of the data. Do the carbon dioxide levels appear to grow exponentially? Solution The data appear to resemble the graph of an increasing exponential function. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 60
Example 11: Using Data to Model Exponential Growth (3 of 6) (b) One model for the data is the function where x is the year and Use a graph of this model to estimate when future levels of carbon dioxide will double and triple over the preindustrial level of 280 ppm. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 61
Example 11: Using Data to Model Exponential Growth (4 of 6) (b) Solution A graph of y = 0. 001942 close to the data points. shows that it is very Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 62
Example 11: Using Data to Model Exponential Growth (5 of 6) (b) We graph on the same coordinate axes as the given function, and we use the calculator to find the intersection points. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 63
Example 11: Using Data to Model Exponential Growth (6 of 6) (b) The graph of the function intersects the horizontal lines at approximately 2064. 4 and 2130. 9. According to this model, carbon dioxide levels will have doubled by 2064 and tripled by 2131. Copyright © 2017, 2013, 2009 Pearson Education, Inc. All Rights Reserved Slide - 64
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