Exploring Derivatives and Integrals in Geology

1. Geomath Geology 351 - Calculus IV tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

2. On Tuesday we worked through some puzzles in which the end result was to guess a function that yielded a certain derivative. You just got started on them so let’s come back and see how you did.

3. Forwards & Backwards and forwards

7. What's the point! See limits.xls The derivative Derivatives are good for telling us what the slope is. It tells us the rate at which something is changing at any point we happen to know.

8. Sometimes we might want to know the area bounded by a function. How could you get excel to compute the area for you?

9. Total Area under this Curve ≈ Bring up Area.xls

10. Open up Area.xls from your common drive You can access this from the internet or from the common drive. Now bring up integ.xls

11. In Waltham’s integ.xls file set n = 2 Notice that in his file you can compare the approximation and exact numbers shown above. The question you are probably asking now is just what is that “exact” number?

12. For now just experiment with a few x’s to get a sense for how the size of the x in the computation of the area yields a number closer to, or further from, the value noted as the “exact” value.

13. One way to look at it is described outlined below. The area, as we have said is simply the sum of the product of the function, whatever it happens to be, (in this case x2) and the little x we use to define the length of the base of our rectangles. let then so

14. These kinds of identities are not the sort of thing we geologists relish spending our time on. However, we can pull one out of the stack and put it to practical use and help develop a more intuitive understanding of the integral http://en.wikipedia.org/wiki/Summation_identities

15. The reason we do this is to get an idea whether there is some elegant way to describe this result as some other function substitute

16. Like most calculus concepts we are looking at behaviors when the interval becomes very small and the number of intervals becomes very large, so in the result below we just assume that n is much larger than 1 or 2 and simplify. From above with substitution yields or

17. What we find is that this area we are after can be expressed by a simple analytical function. So any time you want the area from 0 to some value X – you’ve got it! You are probably noticing something else interesting about this result as well.

18. You can see that the derivative This is an easy one. We just use the power law to find that = So what we’ve demonstrated here is that the answers to this little game we played on Tuesday actually represent a sum of the function we are trying to find or the areas under the curves defined by those functions.

19. So all those reverse derivates you figured out are just areas under the curves representing those functions

20. Like the derivative, this discrete summation can be carried into continuous form (that’s basically what we did when we assumed n>>1 or 2). The discrete sum is represented by the capital sigma and the continuous sum is expressed using the integral sign.

21. Let’s look at this problem using an example that may be easier to relate to. Let’s say that you know how fast something travels as a function of time and you want to catch it. For example, something is floating down the river and you want to be there to catch it, but the velocity keeps changing with time. So let’s say you know that the velocity, although not constant, varies directly related to the time, for example. Let’s say you know that

22. Then you also know that . Thus . You want to find x. This is what you want to know. If you know how x varies with time, then you can arrange to be at a certain spot at a certain time and catch your prey. So the question is how do we find x when we know that

23. Well, think about it, you are asking yourself “what is it that I have to differentiate to get nt?” Just like the game we played earlier. I think maybe now, after playing our little game, you can guess pretty quickly that the antiderivative of nt would be The guess is easy to confirm by taking its derivative

24. So, then you can say to yourself – OK –Now I know exactly where it will be at some time t and I’ll just jump in my car and drive down the road apiece and have my box or lasso ready and waiting for this thing to come by at point x. There you go. Give me a time – say 4 – and we know that x = 8n

25. But then you say, wait a minute – I can differentiate and get a result, but, I can also differentiate and get the same result. What does this mean? What additional information do you need to help you know where to be in order to capture your prey?

26. This idea that we’ve just gone over illustrates the indefinite nature of the process. In other words, we can find lots of things to differentiate that will give us the same answer. …. whatever C is.

27. To use our new notation, then we’d say that This is referred to as an indefinite integral or as indefinite integration. In the problem of finding the moving target we have

28. Graphically, what does this represent? Basically it says the starting point can vary. To know where the thing is going to be at a certain time, you have to know where it started.

29. Areas under curves

30. Here are some more that you’ve already done Remember this one? When you have an indefinte integral, the limits of integration are not specified.

31. then if Thus When n = -1, what happens?

32. So this kind of integral is referred to as an indefinite integral. To overcome the problem we encountered with trying to find the moving target We need to have some additional information when we do the integration

33. We have to know where it jumped in the stream. Then we can specify the limits of our integration This is referred to as a definite integral. It’s evaluated as follows

34. More generally - Notice that when we introduce the indefinite nature of the result, as follows Those constants cancel out.

35. Questions? • Turn in problems 8.13 and 8.14 • Hand in the “derivatives in reverse” worksheets • Continue reading chapter 9 … and

36. The fun part! Geology Applications • Especially look over the applications section (9.6) and consider problems 9.7 and 9.8 • Also look over questions 9.9 and 9.10. They will be assigned … down the road