Predicting

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Predicting GROWTH
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GROWTH the BIG picture...

• We have looked at the growth of individual
  objects
• some objects grow in predictable ways
• ex. The gnomonic growth (scaled growth) of
  some plants and shells
• The growth of humans and animals is also somewhat
  predictable, but not gnomonic.

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GROWTH the BIG picture...

• Now we will look at the growth of populations
• some will grow in predictable ways that can be
  estimated using mathematical formulas.
• we will compare linear, exponential, and logistic
  growth.
• if a population sequence is plotted on a graph,
  the type of growth it is most closely related to
  might be more easily determined.

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Whats so big about growth decay?

• BUSINESS
• Making predictions about production costs and
  company profits
• ECONOMICS
• Investment/Banking applications

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Whats so big about growth decay?

• BIOLOGY/ECOLOGY
• Making predictions about human/animal population
  growth (or decline)
• Making predictions about the availability of
  resources over time
• Making predictions about the growth/decay of
  populations of bacteria, insects etc
• Radioactive dating to estimate the age of certain
  artifacts...

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GROWTH

• Some mathematical descriptions...

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TWO TYPES OF GROWTH

• Growth is a process that occurs over time.
• This growth can occur continuously, changing all
  the time. Each second it will be a little
  different than before.
• But population growth occurs in sudden changes,
  e.g. a birth or a death.
• This type of growth is called discrete growth.

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GROWTH SEQUENCES...

• Often, the growth of an object (or a population
  or the production of a certain item) will be
  documented over a period of time.
• The result is a sequence of values that can be
  analyzed.
• We will refer to this as a population sequence,
  or a growth sequence.

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Mathematically speaking...
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RECURSIVE DESCRIPTIONS of sequences

• a formula that allows you to calculate a value in
  the sequence based on previous values.
• Ex. The recursive definition of the Fibonacci
  Sequence is
• FN FN-1 F N-2 where N the position in the
  sequence and F01 F1 1
• You should be able to, fairly simply, state a
  recursive description in words before writing it
  mathematically.

The next number in the Fibonacci sequence is
found by adding the two previous values. You
can describe the entire sequence this way, as
long as you know the first two values in the
sequence -- since each value depends on the two
preceding values.
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EXPLICIT DESCRIPTIONS of sequences

• a formula that allows you to calculate a value in
  the sequence without needing previous values.
• The value you are trying to find can be directly
  calculated simply by knowing its position in the
  sequence.

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POPULATION GROWTH EXAMPLES
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KCs COOKIE COMPANY

• KCs Cookie Company is a new business. The
  company wishes to keep track of their growth by
  examining the total of orders they have filled.
• Initially, the company filled 80 orders
• After 1 month, they had filled a total of 205
  orders
• after 2 months 330 orders
• after 3 months 455 orders
• after 4 months 580 orders

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KCs COOKIE COMPANY

• Create a chart that describes this sequence
• Create a line graph that describes this sequence
• Let the horizontal axis be MONTH
• Let the vertical axis be total ORDERS filled
• Create a bar graph that describes the same
  information

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Company Growth for through the first 4 months

• Three ways to describe the same data

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KCs COOKIE COMPANY

• EXAMINE THE DATA
• You should notice difference between the total
  of orders during consecutive months is constant
• Number filled after month m - number filled
  after month (m-1)

• 205 - 80 125
• 330 - 205 125
• 455 - 330 125
• 580 - 455 125

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KCs COOKIE COMPANY

• When the differences between consecutive values
  in a population sequence (here our
  populations are total orders filled) are the
  same (or very nearly the same), a linear growth
  model can be used to describe the growth of the
  population.
• Notice that the line graph model for this
  sequence was a straight line that went uphill.

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Predictions???

• DESCRIBE ANY TREND YOU SEE IN THE DATA
• To determine the total number of orders filled
  after any month (m), add 125 to the total filled
  after the previous month (m-1), when 80 orders
  were filled initially.
• This is a word model for the growth sequence in
  this application.
• It is a RECURSIVE description, because it uses
  previous values in the definition.

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CREATE AN ALGEBRAIC MODEL

• USE THE SUGGESTED NOTATION to describe how you
  will refer to the variable quantities in the
  application.
• This is often referred to as defining the
  variables in order to create an algebraic model
  of the growth sequence.
• For this example, lets use the following...
• Let m of months since the initial of orders
  was taken
• Tm TOTAL orders filled after m months
  
• So the algebraic model for
• Total number filled after month m 125 more
  than total number filled after month (m-1)
• will look like...
• Tm Tm-1 125 where T0 80

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Putting it all together

• A recursive description uses previous values in
  the sequence to calculate a new value
• For this example, another way to state the word
  description of the growth sequence is
• You can find the number of orders filled during
  any month by adding 125 to the filled during
  the previous month, if you know that 80 orders
  were filled initially.
• Mathematically, the RECURSIVE DESCRIPTION
• Tm Tm-1 125 where T0 80
• where m the month number
• Tm the total orders filled after month m

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ANOTHER DESCRIPTION

• The recursive description is easy to state, and
  easy to use however, it becomes tedious if you
  must use it to calculate values farther and
  farther out in the sequence.
• Lets examine the sequence again, and see if it
  can be described in another way...

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Find the pattern

• The CONSTANT VALUES for this application are
• 80 initial orders
• and 125 additional orders each month

• Notice that to get from 80 to 205, you add 125
• to get from 80 to 330, you add 2(125)
• to get from 80 to 455, you add 3(125)
• Are you beginning to see a pattern???

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Create a word model

• So, the sequence can be described using those
  values that are constant to the application, as
  well as the of months since the initial of
  orders was filled
• To find the number of orders filled in any
  month, add 125 (the of months) to the 80
  initial orders.

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Create an algebraic model...

• To find the number of orders filled in any
  month, add 125 the of months to the 80
  initial orders.
• Since this description depends only on the
  POSITION in the sequence, it is an EXPLICIT
  DESCRIPTION of the growth for this application.
• For this example, the algebraic model would be
• Tm 80 125(m)
• with m and Tm defined as before

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Usefulness???

• Creating models for a growth sequence can be very
  useful.
• It can help a company to make predictions about
  what will happen in the future, should the
  current trend continue.
• For example, if the trend we have observed were
  to continue throughout the first year of
  business, the YEAR END TOTAL number of orders
  filled would be?

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Applying the models...

• This problem can be solved in a number of ways
• You might choose to continue the chart out to the
  12th month (this method makes use of the
  recursive description)
• or you might use the explicit formula to find the
  year end total.
• Make sure that you understand the NOTATION...

T12 means the total number of orders filled from
the start of the business through the 12th month.
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Scrumptious Cookie Company

• Scrumptious Cookie Company is also opening for
  business
• Based on word of mouth, the company has 40
  orders at startup.
• During the 1st month, they have filled an
  additional 60 orders
• During the 2nd month another 80 orders
• 3rd month 100 orders
• 4th month 120 orders

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Lets describe the pattern

• Here our values are orders per month.
• But notice that the growth is still linear, each
  month the of orders is 20 more than the
  previous month.
• So we will let Cm be the of orders in month m.
  
• Then C0 40, and our recursive description
• . . . Cm Cm-1 20
• To find our explicit description, multiply 20 by
  the number of months and add to our original 40.
• So Cm 20(m) 40.

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Suggested Problems

• Linear Growth, 10.2
• 3, 5, 6, 13, 15, 17