Download Growth: Linear versus Growth: Linear versus

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
ƒ Linear Growth occurs when a quantity grows by some fixed absolute amount in each unit of time.
ƒ Exponential growth leads to repeated doublings.
With each doubling, the amount of increase is approximately equal to the sum of all preceding doublings.
g
ƒ Exponential Growth occurs when a quantity grows by the same fixed relative amount—that is, by the same percentage—in each unit of time.
ƒ 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