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

Properties of The Exponential FunctionDomain: Range: Root(s): y-intercept: Increasing of Decreasing: Concavity: Asymptote(s):

Let's compare

Properties of The Natural Log FunctionDomain: Range: Root(s): y-intercept: Increasing of Decreasing: Concavity: 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 "substance" at the beginning of the time period.

A is the amount of "substance" as the end of the time period.

Model is our model for the growth (or decay) of the substance", 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 substance" 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.(b) What is the population in 2009? World Population Clock

World Population Clock(b) What is the population in 2009?

(a) Model the population growth using an exponential function.

Example: The population of the earth was 5.3 billion in 1990. In 2000 it was 6.1 billion.

http://www.poodwaddle.com/worldclock.swf

Case 2: Given lots of information ( , m, p)

A is the amount of "substance " at the end of the time period.

is the original amount of "substance" at the beginning of the time period.

m is the "multiplication factor"or growth rate.

p is the period; the amount of time required to multiply by "m" 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

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

Is Spock telling the truth? HOMEWORK

What if he's not lying?What if a little more than 3 days have passed?How much time has actually passed?

HOMEWORK

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