1. Tom Wilson, Department of Geology and Geography tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

2. Discussion of HW problems Tom Wilson, Department of Geology and Geography Can you do those derivatives? Showing steps and organization are synonymous Working the algebra: solving for . Other questions?

3. Today Tom Wilson, Department of Geology and Geography Review and introduction of a couple additional differentiation rules Basic review Problems 8.16-8.18 Some added slides on maxima and minima that will not be covered Calculus and the trendline: minimization used to estimated the trendline coefficients. Derivatives in reverse

4. Log rule In general we use the Chain rule For more complicated functions Basic rules for differentiating log functions

5. The derivative of natural log functions Given > For the more general case where the base is unequal to e Log 10 (e)=0.434 Other bases …

6. Another way to look at this is to recall the earlier definition for logs with arbitrary base Tom Wilson, Department of Geology and Geography so Recall that for arbitrary base, we have When b is base 10. So we treat derivatives of logs the same way we do lns, but have to multiply by the factor 0.434.

7. The factor 0.43 (log 10 e) suggests the slope on the log 10 curve will be less than that of the ln curve, which as the graph indicates – it should be

8. The derivative of exponential functions In general for If express a as e n so that then Likewise with derivatives exponential functions with bases different than e, the relationships is similar but requires expression in terms of the base e

9. Sinceand in general a can be thought of as a general base. It could be 10 or 2, etc. A similar looking result including the factor ln(a)

10. Refer back to the review sheet Tom Wilson, Department of Geology and Geography

13. And you’ll find answers to these problems in the back of the book for question 8.8. Double check your results.

14. Tom Wilson, Department of Geology and Geography 8.16 Thickness of the bottomset bed i. What is the rate of change of bed thickness (the gradient)?

15. Tom Wilson, Department of Geology and Geography 8.17 burial depth and sediment age relationship i. What is the age gradient (change of sediment age with change in depth)? ii. If depth is in error by an amount  z. What is an expression for the error in age  a. iii. For k = 3000 y/m and  z=0.1 m, what is the error in age  a.

16. Tom Wilson, Department of Geology and Geography 8.18 error in cliff height i. How high is the cliff if a measurement taken at a distance of 100 m gave an angle of 20 o ? What is the error in the cliff height if the angular measure is in error by 2 o ?  h x In the example at right, how do we determine h given x and  ?

17. Tom Wilson, Department of Geology and Geography 8.18 continued It’s important to remember that angles are generally expressed in radians unless otherwise noted. So – what is 2 o in radians?

18. We will not spend additional time on this in class, but for your additional reference - some more on maxima and minima Tom Wilson, Department of Geology and Geography Given a function like the cosine, what would be an easy to find the locations of the maximum and minimum values of the cosine? The file limits.xls may be of general help for visualization of functions and their derivatives

19. An example from reflection seismology Tom Wilson, Department of Geology and Geography Reflection travel times are hyperbolic

20. Seismic reflection events form hyperbolas in shot records and cmp gathers Tom Wilson, Department of Geology and Geography The hyperbola Its slope

21. At the minimum the 2 nd derivative is positive Tom Wilson, Department of Geology and Geography Original function Its derivative2nd derivative

22. Tom Wilson, Department of Geology and Geography What do we know about points on a curve where the derivative equals 0?

23. Consider the function noted on the previous slide Tom Wilson, Department of Geology and Geography Local minima and maxima

24. Tom Wilson, Department of Geology and Geography has two roots The derivative goes to zero (i. e. has 0 slope) at two points (the roots). It derivative

25. Higher order derivatives Tom Wilson, Department of Geology and Geography

26. Has roots at -1&1/3 and 0

27. Tom Wilson, Department of Geology and Geography At 0 is positive A-1 and 1/3 We have a local maximum and minimum -8 and is negative

28. Tom Wilson, Department of Geology and Geography The root of the 1 st derivative is -1/2 The 2 nd derivative is neither positive or negative at the point where the slope goes to 0. The 2 nd derivative is 0.

29. Those trendlines you obtain from Excel also rely on estimation of minima – the minimum error Tom Wilson, Department of Geology and Geography Let’s take a few moments and consider the evaluation of the minimization criterion used to obtain the “best fit” lines we worked with in lab a few weeks ago.

30. Tom Wilson, Department of Geology and Geography y = mx + b The best fit line is a line which minimizes the difference between the estimated and actual values of y.

31. Tom Wilson, Department of Geology and Geography is the estimate of y i. We want to minimize these differences for all y i.

32. Tom Wilson, Department of Geology and Geography.. and the best way to do this is to minimize the sum of the squares of these departures. Mathematically the sum of the square of the departures or differences is Let the sum of these squared differences = D. How can we minimize D? Minimize difference between actual and estimated values

33. Tom Wilson, Department of Geology and Geography Remember, when you want to find the minimum of something you compute its derivative (its tangents) and set the derivative equal to 0, i.e., find a tangent to the curve whose slope is zero. Where is the minimum of the function ? Take its derivative and solve for the roots.

34. Tom Wilson, Department of Geology and Geography Given there are two ways we could minimize this expression - one with respect to the slope m - and the other with respect to the intercept b. Although you have the summation notation this is fairly straight forward. You just use the sum rule, power rule, etc. Yield minimum error estimates of the slope (m) and intercept (b).

35. The result Tom Wilson, Department of Geology and Geography The intercept The slope Where s x 2 =variance of x

36. Tom Wilson, Department of Geology and Geography It also turns out that Where the covariance between x and y is or

37. The next topic for us – derivatives in reverse or integrals Tom Wilson, Department of Geology and Geography Puzzles to solve

38. The results of a previous exercise jeopardy – what is … Tom Wilson, Department of Geology and Geography What is ?

39. We’ll continue this next time. Start reviewing basic integration rules Tom Wilson, Department of Geology and Geography

40. To do items Tom Wilson, Department of Geology and Geography Review derivative-in-reverse operations and integral concepts Finish reading chapter 9 and …

41. Geologic applications Tom Wilson, Department of Geology and Geography 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