The document discusses exponential modeling and growth. It provides examples of using exponential functions to model population growth over time given initial and final population sizes. It also discusses modeling growth when given a multiplication factor and period. The document then presents the example of modeling the growth of tribbles from one initial tribble over a period of time to a final population of 1,771,561 tribbles. It questions whether the reported time of 3 days for this growth is accurate.

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?