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# Pre-Cal 40S April 29, 2009

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Exponential Modeling the Trouble With Tribbles.

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### Pre-Cal 40S April 29, 2009

1. 1. The Trouble With Tribbles or ... more about exponential modeling tribbles by ﬂickr user tr67
2. 2. Solve using natural logarithms ...
3. 3. Consider the graph of and sketch the graph of ... Identify the asymptote of each graph above.
4. 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. 5. Consider the graph of and sketch the graph of ...
6. 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. 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. 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. 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. 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. 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. 12. The Trouble with Tribbles Video Source
13. 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. 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?