Geology 351 - Geomath

1. Geology 351 - Geomath Review of exponential growth problems and equation manipulation using isostacy tom.h.wilson tom.wilson@mail.wvu.edu Department of Geology and Geography West Virginia University Morgantown, WV

2. Agenda/objectives Return/Discuss problems 2.11, 2.13 Creating e Exponentials! Take the plunge and find out about Archimedes principle and isostacy. Why ice cubes float and what they share with mountain roots. In class problem use and substitute h=r+e to solve for e as a function of h, i and w. Hand that in along with answers to computer problems 2.11 and 2.12.

3. 2.11 D/t= 1856cm/4175yr= 0. Make sure you distinguish between rate and slope Tom Wilson, Department of Geology and Geography

4. 2.11 Be quantitative in your responses Tom Wilson, Department of Geology and Geography

5. A few minutes on e Open up Excel and work through this as we go For example, N=10. You can give N a variable name if you wish or just refer to the cell carrying the value N =(1+1/c4)^c4 or =(1+1/N)^N Tom Wilson, Department of Geology and Geography

6. Increase N from 10 to 100 to 1000 What do you think this expression will converge to when N approaches infinity? Note that you can get a reliable answer in Excel up to about 1e12 Tom Wilson, Department of Geology and Geography

7. Let’s consider this as a compound interest problem You can also use Excel for this Let’s say you have $1000 to invest and you purchase a stock that earns on average 6% per year. What is your value in one year? In 2 years? In 3 years? We just have $1000 (1+0.06)(1+0.06)(1+0.06) for the three years. In other words, each year the value of our investment increases 6%. However, in this example, we are just compounding annually Tom Wilson, Department of Geology and Geography

8. Calculate earnings at different fractions of the year What if we compound or give out earnings every half year? Our interest would be calculated at the half as $1000(1+0.03) = And at the end of the year as $1000(1+0.03)(1+0.03) = What would happen if we calculated the value every quarter, every month, or continuously. The formula you are evaluating is What is Tom Wilson, Department of Geology and Geography

9. Evaluate ex Evaluate this expression for large N and then compute ex This gives rise to the relationship that Tom Wilson, Department of Geology and Geography

10. This relationship looks very much like the population growth relationship Where P is the initial investment and r is the fractional interest per anum and t is the number of years Where Po is the population at reference time 0 and r is the fractional growth per anum Tom Wilson, Department of Geology and Geography

11. Make a calculation and examine it Use Po =1000 and =0.06 What is P in one year? and note that P increases by about 6.2% not 6% We’ll come back to this in a minute. Tom Wilson, Department of Geology and Geography

12. Questions about population growth? Why does the approximation of  as the fractional growth in one year give you an accurate result? Using the series approximation for the natural log Tom Wilson, Department of Geology and Geography

13. Why does the fractional increase in population  ?  actually equals ln(1+the fractional increase) or Have a look at the Excel file ln_Expansion.xlsx Tom Wilson, Department of Geology and Geography

14. It’s a busy presentation ln_Expansion.xlsx We’re letting x= the fractional increase in the above Tom Wilson, Department of Geology and Geography

15. It's nice to simplify, but be sure you understand why it works In this case, there is a very small error in making the one term approximation Tom Wilson, Department of Geology and Geography

16. Low order approximations are good for small x! Tom Wilson, Department of Geology and Geography

17. Dissecting the exponential decay relationship Let’s use our calculus for a minute to get another perspective on this relationship The derivative of N is This is actually the starting point for the radioactive decay problem Tom Wilson, Department of Geology and Geography

18. Integrate Working from that relationship as a starting point, divide both sides by N and multiply both sides by dt to get What do you get when you integrate? Integration yields - Tom Wilson, Department of Geology and Geography

19. The decay equation in logarithmic form Take a few minutes and transform this relationship into its exponential equivalent, to get Tom Wilson, Department of Geology and Geography

20. Half life The half life is just the time it takes for No to decay to No/2. We used the ln transform to solve for t1/2. Note that if the half-life were given, we could solve for the growth rate. Make sure you can apply these ln transforms and their inverse. Tom Wilson, Department of Geology and Geography

21. Isostacy We’ll talk more about isostacy on Thursday. To introduce the problem consider the ice cube floating in your favorite drink. Some basic terms … Tom Wilson, Department of Geology and Geography

22. Due dates Hand in the in-class problems before leaving 2.13 and 2.15 will be due Thursday Read chapter 3 and look over problems 3.10 and 3.11 (also see today’s handout) for discussion this Thursday