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chapter8Notes

# chapter8Notes - Chapter 8 Exponential functions In this...

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Chapter 8 Exponential functions In this chapter, we leave behind the power functions and polynomials, and explore a new type of function, exponential growth. We find that this function is closely related to (uncontrolled) growth of living replicating organisms, and study one example from the popular science-fiction literature. The inverse function, the logarithm, is also studied, and the properties of the two are related. Understanding how to manipulate expressions involving such functions will be a skill to be practiced 1 . 8.1 The Andromeda Strain ”The mathematics of uncontrolled growth are frightening. A single cell of the bacterium E. coli would, under ideal circumstances, divide every twenty minutes. That is not particularly disturbing until you think about it, but the fact is that bacteria multiply geometrically: one becomes two, two become four, four become eight, and so on. In this way it can be shown that in a single day, one cell of E. coli could produce a super-colony equal in size and weight to the entire planet Earth.” Michael Crichton (1969) The Andromeda Strain, Dell, N.Y. p247 1 Lab 4 of this calculus course will exploit that skill in introducing the student to the formula for the temperature of a cooling object. We will later have more to say about such applications, once additional tools of differential equations are learned in a later chapter. v.2005.1 - September 4, 2009 1

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Math 102 Notes Chapter 8 8.2 Powers of 2 n 2 n 0 1 1 2 2 4 3 8 4 16 5 32 6 64 7 128 8 256 9 512 10 1024 -4.0 10.0 0.0 1000.0 Figure 8.1: Powers of 2 including both negative and positive integers: Here we show 2 n for - 4 < n < 10. Note that 2 10 1000 = 10 3 . Note that 2 10 1000 = 10 3 . This is a useful approximation in converting binary numbers (powers of 2) to decimal numbers (powers of 10). v.2005.1 - September 4, 2009 2
Math 102 Notes Chapter 8 8.3 Growth of e. coli Mass of 1 E. coli cell : 1 nanogram = 10 - 9 gm = 10 - 12 kg. Mass of Planet Earth : 6 · 10 24 kg Size of E. coli colony equal in mass to Planet Earth: m = 6 · 10 24 10 - 12 = 6 · 10 36 In a period of 24 hours, there are many 20-minute generations. To be exact, there are 24 × 3 = 72 generations, with each one producing a doubling. This means that there would be, after 1 day, a number of cells equal to 2 72 . We can estimate it using the approximate decimal form as follows: 2 72 = 2 2 · 2 70 = 4 · (2 10 ) 7 4 · (10 3 ) 7 = 4 · 10 21 . The actual value is found to be 4 . 7 · 10 21 , so the approximation is relatively good. Apparently, the estimate made by Crichton is not quite accurate. However it can be shown that it takes less than 2 days to produce a number far in excess of the desired size. (The exact number of generations is left as an exercise for the reader.. but we will return to this in due time.) 8.4 The function 2 x From previous familiarity with power functions such as y = x 2 (not to be confused with 2 x ), we know the value of 2 1 / 2 = 2 1 . 41421 . . .

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