354 CHAPTER 5 Exponential and Logarithmic Functions ‘Are You Prepared?’ Answers 1. $15 2. 13 1 3 % (a) How long will it take to increase an initial investment of $1000 to $4500 at an annual rate of 5.75%? (b) What annual rate is required to increase the value of a $2000 IRA to $30,000 in 35 years? (c) Give a derivation of this formula. 68. Time to Reach an Investment Goal The formula = − t A P r ln ln can be used to find the number of years t required for an investment P to grow to a value A when compounded continuously at an annual rate r. Problems 69–72 require the following discussion. The consumer price index (CPI) indicates the relative change in price over time for a fixed basket of goods and services. It is a cost-of-living index that helps measure the effect of inflation on the cost of goods and services.The CPI uses the base period 1982–1984 for comparison (the CPI for this period is 100). The CPI for April, 2024 was 313.55. This means that $100 in the period 1982–1984 had the same purchasing power as $313.55 in April, 2024. In general, if the rate of inflation averages r% per annum over n years, then the CPI after n years is ( ) = + r CPI CPI 1 100 n 0 where CPI0 is the CPI at the beginning of the n-year period. Source: U.S. Bureau of Labor Statistics 69. Consumer Price Index (a) The CPI was 257.97 for 2020 and 308.42 for 2024.Assuming that annual inflation remained constant for this time period, determine the average annual inflation rate. (b) Using the inflation rate from part (a), in what year will the CPI reach 400? 70. Consumer Price Index If the current CPI is 234.2 and the average annual inflation rate is 2.8%, what will be the CPI in 5 years? 71. Consumer Price Index If the average annual inflation rate is 3.1%, how long will it take for the CPI to double? (A doubling of the CPI means purchasing power is cut in half.) 72. Consumer Price Index The base period for the CPI changed in 1998. Under the previous weight and item structure, the CPI for 1995 was 456.5. If the average annual inflation rate was 5.57%, what year was used as the base period for the CPI? Explaining Concepts 73. Explain in your own words what the term compound interest means. What does continuous compounding mean? 74. Explain in your own words the meaning of present value. 75. Critical Thinking You have just contracted to buy a house and will seek financing in the amount of $100,000. You go to several banks. Bank 1 will lend you $100,000 at the rate of 4.125% amortized over 30 years with a loan origination fee of 0.45%. Bank 2 will lend you $100,000 at the rate of 3.375% amortized over 15 years with a loan origination fee of 0.95%. Bank 3 will lend you $100,000 at the rate of 4.25% amortized over 30 years with no loan origination fee. Bank 4 will lend you $100,000 at the rate of 3.625% amortized over 15 years with no loan origination fee. Which loan would you take? Why? Be sure to have sound reasons for your choice. Use the information in the table to assist you. If the amount of the monthly payment does not matter to you, which loan would you take? Again, have sound reasons for your choice. Compare your final decision with others in the class. Discuss. Monthly Payment Loan Origination Fee $485 $709 $492 $721 Bank 1 Bank 2 Bank 3 Bank 4 $450 $950 $0 $0 Retain Your Knowledge Problems 76–85 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 81. If ( )= + − f x x x 5 4 8 2 and ( )= − g x x3 1, find ( ) f g x ( ) . 82. Find the domain and range of ( ) = − − + f x x x 2 8 1. 2 83. For ( ) = − − − f x x x x 2 5 4 7 , 2 find all vertical asymptotes, horizontal asymptotes, and oblique asymptotes, if any. 84. If ( ) = − − f x x x4 3, 2 find an equation of the secant line containing the points ( ) ( ) f 3, 3 and ( ) ( ) f 5, 5. 85. Find the difference quotient for ( ) = − f x x3 5. 76. Find the remainder R when ( ) = + + − f x x x x 6 3 2 11 3 2 is divided by ( ) = − g x x 1. Is g a factor of f? 77. The function ( ) = − f x x x 2 is one-to-one. Find −f .1 78. Find the real zeros of ( ) = − − − − − f x x x x x x 15 21 16 20 5 4 3 2 Then write f in factored form. 79. Solve: ( ) ( ) + = − x x log 3 2 log 3 2 2 80. Factor completely: + − − x x x x 2 6 50 150 4 3 2
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