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2/15/2010 8‐A Unit 8A Growth: Linear versus Exponential Growth: Linear versus Growth: Linear versus Exponential Copyright © 2008 Pearson Education, Inc. Copyright © 2008 Pearson Education, Inc. Slide 8‐1 8‐A Slide 8‐2 Key Facts about Exponential Growth Growth: Linear versus Exponential Linear Growth occurs when a quantity grows by some fixed absolute amount in each unit of time. Exponential growth leads to repeated doublings. With each doubling, the amount of increase is approximately equal to the sum of all preceding doublings. g Exponential Growth occurs when a quantity grows by the same fixed relative amount—that is, by the same percentage—in each unit of time. Exponential growth cannot continue indefinitely. After only a relatively small number of doublings, exponentially growing quantities reach impossible proportions. Copyright © 2008 Pearson Education, Inc. Copyright © 2008 Pearson Education, Inc. Slide 8‐3 Slide 8‐4 8‐A Growth: Linear versus Exponential Which of the following rates describes an exponential relationship? a) Increasing at a rate 500 people per year b) Decreasing at a rate of $5000 per year c) Increasing at a rate of 24% per day d) Increasing at a rate of 3000 bacteria per day Copyright © 2008 Pearson Education, Inc. Slide 8‐5 8‐A 8‐A Growth: Linear versus Exponential A single bacteria is in a dish at 1:00 pm. It divides into 2 at 1:01, and continues to double every minute. Find the population at 1:20 pm. a) 220 b) 219 c) 221 d) 40 Copyright © 2008 Pearson Education, Inc. Slide 8‐6 1 2/15/2010 8‐B Unit 8B Doubling Time After a time t, an exponentially growing quantity with a doubling time of Tdouble increases in size by a factor of . The new value of the growing quantity is related to its 2t Tdouble initial value (at t = 0) by Doubling Time and Half‐ Doubling Time and Half Life Copyright © 2008 Pearson Education, Inc. new value = initial value × 2t Tdouble Copyright © 2008 Pearson Education, Inc. Slide 8‐7 Slide 8‐8 8‐B Exponential Decay and Half‐Life The Rule of 70 For a quantity growing exponentially at a rate of P% per time period, the doubling time is approximately 70 Tdouble ≈ P This approximation works best for small growth rates and breaks down for growth rates over about 15% Copyright © 2008 Pearson Education, Inc. For a quantity decaying exponentially at a rate of P% per time period, the half‐life is approximately 70 Thalf ≈ P After a time t, an exponentially decaying quantity with a half‐life time of Thalf decreases in size by a factor of . The new value of the decaying (1 2)t Thalf quantity is related to its initial value (at t = 0) by ⎛ 1⎞ new value = initial value × ⎜ ⎟ ⎝2⎠ Copyright © 2008 Pearson Education, Inc. Slide 8‐9 The Approximate Half‐Life Formula 8‐B 8‐B Slide 8‐10 Exact Doubling Time and Half‐Life Formulas Tdouble = 8‐B log 2 log (1 + r) Exponential Growth (given that r > 0) Thalf = This approximation works best for small decay rates and breaks down for decay rates over about 15% Copyright © 2008 Pearson Education, Inc. t Tdouble −log 2 log (1 + r ) Exponential Decay (given that r < 0) Slide 8‐11 Copyright © 2008 Pearson Education, Inc. Slide 8‐12 2