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The Trouble With Tribbles or ...



more about
exponential
modeling



          tribbles by flickr user tr67
Solve using natural logarithms ...
Consider the graph of       and sketch the graph of ...




Identify the asymptote of each graph above.
Properties of The Exponential and Natural Log Functions
                  Let's compare




Properties of The                 Properties of The
Exponential Function Natural Log Function
 Domain:                           Domain:
 Range:                            Range:
 Root(s):                          Root(s):
 y-intercept:                      y-intercept:
 Increasing of Decreasing:         Increasing of Decreasing:
 Concavity:                        Concavity:
 Asymptote(s):                     Asymptote(s):
Consider the graph of   and sketch the graph of ...
Exponential Modeling

The basic function:

How we model real life situations depends on what kind, or how much ,
information we are given:
 Case 1: Working with a minimal amount of information (A,Ao, ∆t).
 We will create a model in base 10 and base e ... base e is prefered.
                         is the original amount of quot;substancequot; at the
                      beginning of the time period.

                      A is the amount of quot;substancequot; as the end of the
                      time period.

                      Model is our model for the growth (or decay) of
                      the substancequot;, it is usually an exponential
                      expression in base 10 or base e although any
                      base can be used.

                      t is the amount of time that has passed for the
                      substancequot; to grow(or Decay) from      to A.
Example: The population of the earth was 5.3 billion in 1990. In 2000
it was 6.1 billion.
(a) Model the population growth using an exponential function.
                                        World Population Clock
(b) What is the population in 2009?
Example: The population of the earth was 5.3 billion in 1990. In 2000
it was 6.1 billion.

(a) Model the population growth using an exponential function.
                               http://www.poodwaddle.com/worldclock.swf
                                           World Population Clock
 (b) What is the population in 2009?
Case 2: Given lots of information (   , m, p)


                A is the amount of quot;substance quot; at the end of the time
                period.

                  is the original amount of quot;substancequot; at the
                beginning of the time period.

                m is the quot;multiplication factorquot;or growth rate.

                p is the period; the amount of time required to multiply
                by quot;mquot; once.

                t is the time that has passed.
Example 1: A colony of bacteria doubles every 6 days. If there were
3000 bacteria to begin with how many bacteria will there be in 15 days?
Example 2: The mass (in grams) of radioactive material in a sample is
given by:



where t is measured in years.
(a) Find the half-life of this radioactive substance.




 (b) Create a model using the half-life you found in (a). How much of a
 10 gram sample of the material will remain after 40 years?
The Trouble with Tribbles




                   Video Source
HOMEWORK
Is Spock telling the truth?
 Spock says:
 • total of 1 771 561 tribbles

 • stared with 1 tribble

 • each tribble produces 10
 tribbles/litter every 12 hours

 • they did this for 3 days
What if he's not lying?                HOMEWORK
What if a little more
                    than 3 days have passed?
How much time has actually passed?

Assume a total of 1 771 561 tribbles, how long would it
take 1 tribble to produce that many?

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The Trouble With Tribbles Exponential Growth Modeling

  • 1. The Trouble With Tribbles or ... more about exponential modeling tribbles by flickr user tr67
  • 2. Solve using natural logarithms ...
  • 3. Consider the graph of and sketch the graph of ... Identify the asymptote of each graph above.
  • 4. Properties of The Exponential and Natural Log Functions Let's compare Properties of The Properties of The Exponential Function Natural Log Function Domain: Domain: Range: Range: Root(s): Root(s): y-intercept: y-intercept: Increasing of Decreasing: Increasing of Decreasing: Concavity: Concavity: Asymptote(s): Asymptote(s):
  • 5. Consider the graph of and sketch the graph of ...
  • 6. Exponential Modeling The basic function: How we model real life situations depends on what kind, or how much , information we are given: Case 1: Working with a minimal amount of information (A,Ao, ∆t). We will create a model in base 10 and base e ... base e is prefered. is the original amount of quot;substancequot; at the beginning of the time period. A is the amount of quot;substancequot; as the end of the time period. Model is our model for the growth (or decay) of the substancequot;, it is usually an exponential expression in base 10 or base e although any base can be used. t is the amount of time that has passed for the substancequot; to grow(or Decay) from to A.
  • 7. Example: The population of the earth was 5.3 billion in 1990. In 2000 it was 6.1 billion. (a) Model the population growth using an exponential function. World Population Clock (b) What is the population in 2009?
  • 8. Example: The population of the earth was 5.3 billion in 1990. In 2000 it was 6.1 billion. (a) Model the population growth using an exponential function. http://www.poodwaddle.com/worldclock.swf World Population Clock (b) What is the population in 2009?
  • 9. Case 2: Given lots of information ( , m, p) A is the amount of quot;substance quot; at the end of the time period. is the original amount of quot;substancequot; at the beginning of the time period. m is the quot;multiplication factorquot;or growth rate. p is the period; the amount of time required to multiply by quot;mquot; once. t is the time that has passed.
  • 10. Example 1: A colony of bacteria doubles every 6 days. If there were 3000 bacteria to begin with how many bacteria will there be in 15 days?
  • 11. Example 2: The mass (in grams) of radioactive material in a sample is given by: where t is measured in years. (a) Find the half-life of this radioactive substance. (b) Create a model using the half-life you found in (a). How much of a 10 gram sample of the material will remain after 40 years?
  • 12. The Trouble with Tribbles Video Source
  • 13. HOMEWORK Is Spock telling the truth? Spock says: • total of 1 771 561 tribbles • stared with 1 tribble • each tribble produces 10 tribbles/litter every 12 hours • they did this for 3 days
  • 14. What if he's not lying? HOMEWORK What if a little more than 3 days have passed? How much time has actually passed? Assume a total of 1 771 561 tribbles, how long would it take 1 tribble to produce that many?