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M1410301 - 3.01 CHAPTER 3 EXPONENTIAL AND LOG FUNCTIONS...

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3.01 CHAPTER 3: EXPONENTIAL AND LOG FUNCTIONS SECTION 3.1: EXPONENTIAL FUNCTIONS AND THEIR GRAPHS PART A: THE LEGEND OF THE CHESSBOARD The original story takes place in the Middle Ages and involves grains of wheat. Instead, we shall transport ourselves to the distant realm of Seattle, where a smart programmer is haggling with King Bill. The programmer agrees to work for King Bill for 63 days, starting tomorrow. After seeing a large chessboard engraved into King Bill’s floor, the programmer comes up with a scheme for his salary. For now, the programmer tells King Bill to place a check for $1 on “Square 0” on his chessboard. With each new workday, King Bill is to place twice as much money on the corresponding square as the day before. The chortling King Bill, who has forgotten all of his math, agrees. What will happen? The chessboard: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 The amount of money (in dollars) placed on square x is given by f x ( ) = 2 x .
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3.02 Here are some sample values: Square x f x ( ) (in $) 0 1 1 2 2 4 3 8 4 16 5 32 6 64 7 128 8 256 9 512 10 1024 20 Over 1 million (i.e., 10 6 ) 30 Over 1 billion (i.e., 10 9 ) 40 Over 1 trillion (i.e., 10 12 ) 63 Over 9 quintillion (i.e., 9 × 10 18 ) The amount of money on Square x + 10 ( ) will be over 1000 times the amount of money on Square x 0 x 53 ( ) , because the multiplier is 2 10 = 1024 . Challenge: How much money should be on the entirety of the chessboard after Day 63? Hint: Experiment with the first few days. We will see a relevant formula in Chapter 9 , when we get to finite geometric series. Remember that our national debt is “only” in the trillions. No wonder we associate this kind of exponential growth with “rapid growth” in our language!
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3.03 PART B: BASIC EXPONENTIAL GRAPHS We call b a “nice base” if b > 0 and b 1 . Basic exponential functions have the form f x ( ) = b x , where b is nice. Example Graph f x ( ) = 2 x , our “payment” function from Part A . Solution The table on the previous page gives some sample points x , y ( ) . However, the domain of 2 x is assumed to be R , not just the nonnegative integers. Technical Note: Let’s look at 2 3/4 , for example. We may interpret 2 3/4 as 2 3 4 , or 8 4 . It is the real number whose fourth power equals 8. The idea makes sense, although the number, which is irrational, may be time- consuming to approximate by “trial-and-error” on a calculator. You will encounter helpful methods and tools such as Newton’s Method in Calculus I: Math 150 and series in
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