Geomath

1. Geomath Geology 351 - Calculus - derivatives and integrals tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University Tom Wilson, Department of Geology and Geography

2. Derivatives in Reverse Puzzles to Solve Tom Wilson, Department of Geology and Geography

3. y=x • What is the area? Area x Take the simple function What do we have to differentiate to get x? • What does the result represent? Area= Tom Wilson, Department of Geology and Geography

4. Don’t forget Waltham’s excel files You’ll find the file integ.xls Tom Wilson, Department of Geology and Geography

5. Do these look familiar? Forwards & Backwards and forwards Tom Wilson, Department of Geology and Geography

6. We did these the other day in the “forward” direction Tom Wilson, Department of Geology and Geography

7. Note that both functions below have the same derivative Tom Wilson, Department of Geology and Geography

8. Just review the problems we did previously and consider them in the reverse direction Tom Wilson, Department of Geology and Geography

9. In some cases you’ll get the starting function in a different format or or … Tom Wilson, Department of Geology and Geography

10. Both these functions have the same derivative Tom Wilson, Department of Geology and Geography

11. Discrete evaluation of the integral Sometimes we might want to know the area bounded by a function. How could you get excel to compute the area for you? See Waltham’s integ.xls Tom Wilson, Department of Geology and Geography

12. Add the areas of little rectangles Total Area under this Curve ≈ Tom Wilson, Department of Geology and Geography

13. 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? Tom Wilson, Department of Geology and Geography

14. For y = x2 what do we have to differentiate to get x2? If we evaluate that then we get the area under the above curve Tom Wilson, Department of Geology and Geography

15. One way to evaluate the integral is 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. or so Then let & Tom Wilson, Department of Geology and Geography

16. 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 Tom Wilson, Department of Geology and Geography

17. 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 Tom Wilson, Department of Geology and Geography

18. 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 Tom Wilson, Department of Geology and Geography

19. 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. Tom Wilson, Department of Geology and Geography

20. This analysis suggests a general integration rule: the power rule for integrals 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. Tom Wilson, Department of Geology and Geography

21. Derivative in reverse = integral (area in this case) So all those reverse derivates you figured out are just areas under the curves representing those functions Tom Wilson, Department of Geology and Geography

22. Using the integral notation 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. The power rule for integrals is Tom Wilson, Department of Geology and Geography

23. Position as a function of velocity 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 head down river to a certain point and 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: Tom Wilson, Department of Geology and Geography

24. 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 Tom Wilson, Department of Geology and Geography

25. Well, think about it, you are asking yourself “what is it that you 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 Tom Wilson, Department of Geology and Geography

26. so What do we have to differentiate to get dx? x so Tom Wilson, Department of Geology and Geography

27. OK – Now you know exactly where you have to get to at some time t. You jump in your car and drive down the road apiece, get your lasso ready and waiting for this thing to come by at point x. If you can get there by the time t= 4. That gives you an x = 8n Tom Wilson, Department of Geology and Geography

28. 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? Tom Wilson, Department of Geology and Geography

29. 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. Tom Wilson, Department of Geology and Geography

30. The little step dx has length ntdt To get from the start to the finish … We get lazy and this becomes And we know that this gives us Tom Wilson, Department of Geology and Geography

31. To use this 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 Tom Wilson, Department of Geology and Geography

32. 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. Tom Wilson, Department of Geology and Geography

33. Areas under curves Tom Wilson, Department of Geology and Geography

34. 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. Tom Wilson, Department of Geology and Geography

35. then if Thus When n = -1, what happens? Tom Wilson, Department of Geology and Geography

36. 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 Tom Wilson, Department of Geology and Geography

37. Definite integral Tom Wilson, Department of Geology and Geography

38. 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 Tom Wilson, Department of Geology and Geography

39. More generally - Notice that when we introduce specific begin and end points, for example: Those constants cancel out. This is a definite integral Tom Wilson, Department of Geology and Geography

40. Geology Applications • Hand in textbook problems 8.13 and 8.14. • Remember computer problems 8.13 and 8.14 are due this Thursday • Finish reading chapter 9 and look over the problems in the text discussed in the applications section 9.6. How would you approach problems 9.7 and 9.8 • Also look over questions 9.9 and 9.10. They will be assigned … down the road • Spend some more time working through those derivatives in reverse Tom Wilson, Department of Geology and Geography

41. What function is this the derivative of? Tom Wilson, Department of Geology and Geography