# Workshop on Geometry Software

Math 241
Spring 2015
Professor: Mike O'Sullivan

### Week 5: Exponential Functions

First Day: Exponential functions and carbon dating

Let f(x)= 2^x. We started the day by comparing the functions g(x)= f(x-p), h(x)= f(x)+q, k(x)= f(a*x), m(x)= c*f(x), and exploring how different values of p, q, c, a affect the graphs of the corresponding functions.

We then saw that the m(x) functions can also be represented in the form used in g(x), using rules of exponents. Furthermore a function like k(x) can also be expressed as an exponential with a different base.

Finally, we considered the mathematics behind carbon dating, and how exponential functions play a role. The assumption in carbon dating (in its rough form) is that when a living being dies, the proportion of Carbon-14 is the same as that in the atmosphere (which we assume is the same as in today's atmosphere). After that point, the the Carbon-14 decays with half-life about 5730 years, and the other forms of Carbon are stable. Consequently, the proportion of Carbon-14 in a dead being declines over time. This proportion can be used to infer the time since death.

Assignment:
1. Make a worksheet in which you present the following toy carbon-dating problem. On planet Frisbee, the proportion of Carbon-14 is 10% of the total carbon in the atmosphere and the rest is Carbon-12. An initial sample has 10,000 atoms of carbon. Assume carbon has a half-life of 6000 years (it is fairly close to that amount). Give the formula for the total amount of carbon in the sample as a function of time. Illustrate how one would use a measurement of the amount of the carbon in the sample (many years later) to figure out how much time has elapsed since the sample was created. Pretend you have discussed exponential functions with your class and explain how the graph relates to finding the age of the object.

Second Day: You will have time to work on all your projects.

Final Assignment:
1. Create a GeoGebra book with chapters for the topics we have covered. (Including the one above.) Make corrections to your previous worksheets and polish the spelling, grammar and presentation. Make good use of color and layout.