Transcript The Mathematics of Population Growth 10.3 The Exponential Growth

The Mathematics of Population
Growth
10.3 The
Exponential Growth
Model
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The Mathematics of Population Growth
Outline/learning Objectives
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To recognize exponential growth models.
To apply exponential models to solve
population growth problems.
To differentiate between recursive and explicit
models of population growth.
To apply the general compounding formula to
answer financial questions.
To state and apply the arithmetic and
geometric sum formulas in their appropriate
contexts.
The Mathematics of Population
Growth
Exponential growth is based on the
idea of a constant growth rate– in
each transition the population
changes by a fixed factor called the
common ratio.
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The Mathematics of Population
Growth– The Power of Compounding
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Figure (a) plots the growth of the money in the
account for the first eight years. Figure (b) plots the
growth of the money in the account for the first 30
years.
The Mathematics of Population
Growth– The Power of Compounding
The Geometric Sequence
A sequence defined by repeated multiplication– every
term in the sequence after the first is obtained by
multiplying the preceding term by a fixed amount r.
The Common Ratio
It is the ratio of two successive terms in the sequence
and is denoted by the constant factor r.
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The Mathematics of Population
Growth– The Power of Compounding
Exponential Growth (r > 0)
 PN = r · P N-1 (recursive formula)
 PN = P0 · r N (explicit formula)
When r > 1, the terms of the sequence get
bigger and we have real growth (positive
growth), but when 0 < r < 1, the terms of the
sequence get smaller and we have a situation
known as exponential decay.
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Example: #19, Page 366
A population grows according to the exponential
growth model. The initial population is P0 = 11
and the common ratio is r = 1.25 (ie 25% growth
each generation).
a) Find P1
b) Find P9
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c) Give an explicit formula for PN
Solution
a) We can use the recursive formula to get
P 1:
PN = r · P N-1
P1 = 1.25(P0) = 1.25(11) = 13.75
b) P9 = 1.25(P8) = 1.25(??)
We don’t know P8 so we use the explicit
formula:
PN = P0 · r N
P9 = P0 · r 9 = 11(1.25) 9 = 81.956
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Financial Applications of
Exponential Growth
The classic application of exponential growth is
investing money in an interest-bearing account.
 Before we use the exponential growth formula, let’s
review how mark-ups and discounts are computed.
 Examples 10.8 and 10.9 page 348
 To increase a number C by xx% we multiply C by
1 + xx/100 (or 1.xx)
 To decrease a number C by xx% we multiply C by
1 – xx/100
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The Mathematics of Population
Growth– The Power of Compounding
Annual Compounding Formula
$PN = P0 · (1 + i) N
If we let P0 denote the principal; i the annual
interest rate (expressed as a decimal); and N
the number of years the money is left in the
account, the explicit formula for exponential
growth becomes the above formula.
Example Page 367 #29
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Solution
#29 Suppose you deposit $3250 in a savings
account that pays 9% annual interest, with
interest credited to the account at the end of
every year. How much will you have after four
years? After four and a half years?
$PN = P0 · (1 + i) N
$PN =(3250)(1 + 0.09) 4 =$4586.64 after 4 years
After 4 ½ years you will still have the same
amount!
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The Mathematics of Population
Growth– The Power of Compounding
General Compounding Formula
Nk
i
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$ PN  P0 1  
 k
The above formula is a general compounding
formula for computing the growth of P0 left in
an account that pay an annual interest rate i
compounded k times a year.
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The Mathematics of Population
Growth– The Power of Compounding
Periodic Interest Rate
The effective interest rate for the
compounding period.
Annual Yield
The percentage increase of an investment
over a one-year period.
Ex. #37 Page 367
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Page 367 #37
You have some money to invest. The GB Bank
offers accounts that pay 6% annual interest
compounded yearly. The FN Bank offers
5.75% annual interest compounded monthly.
The BofW offers 5.5% annual interest
compounded daily. What is the annual yield for
each bank?
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The Mathematics of Population
Growth– The Power of Compounding
Geometric Sum Formula
a  ar  ar 2  ...  ar N 1 
a  r N  1
r 1
To simplify the notation we are using a instead
of P0 for the seed.
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Example:
Middletown Police cited 70 people for speeding
in the year 2000. Due to growth they are
predicting the number of people cited will grow
at a rate of 4% per year. How many people will
be cited for speeding in 2025? How many
people will be cited for speeding during the
period 2000 – 2025?
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This is exponential growth. Use the explicit
formula to find the answer:
PN = P0 · rN
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P25 = 70 ·(1.04) = 186.6
About 187 people will be cited in 2025.
To get the total in the 26 year period use the
geometric sum formula:
N
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a(r -1) = 70(1.04 – 1) = 124.07288= 3102
r–1
1.04 – 1
0.04
About 3102 people will be cited during the
26 year period
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Activity
Do Activity “Grouping Sequences” and also turn
in the solutions to the following today:
Page 366 #20, 22, 26
Homework: Page 366 #19, 21, 25, 28, 32, 63
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