Tom Wilson, Department of Geology and Geography tom.h.wilson Dept. Geology and Geography West Virginia University

Hand back Option 1 or 2 & discuss Tom Wilson, Department of Geology and Geography If you’d like some extra credit you can do the remaining exercise for up to 10 points extra credit. This will be added to your score for the option you turned in. If you decide to do it please hand in the extra credit option before leaving for spring break. You can put the exercise in my mailbox (by Friday at noon) or bring to class on Thursday.

Bottom Set bed example You get t o and X from the trendline constants Tom Wilson, Department of Geology and Geography t o =19.37 X=1/1.49 or 0.67km t(2.5km)=19.37e -1.49(2.5) t(2.5km)=19.37e -2.5/0.67 ~0.47m thick

Guttenberg-Richter relationship you get b and c from the trendline constants Tom Wilson, Department of Geology and Geography b= m=6 c= logN=-bm+c logN (m=6)= N= /N=7.6 years

10 Points Extra credit Tom Wilson, Department of Geology and Geography i.Statement of problem (1 point) ii.Plot (2 points) iii.Plot of trendline along with trendline equation (2 points) iv.Explicitly note the values of the two constants (2 points) v.Show calculations for value noted (2 points) vi.State your result (1 point) For extra credit – Have it in my mailbox by noon this Friday.

Review questions from last class and discussion/questions about problem 8.13 Tom Wilson, Department of Geology and Geography Don’t just write down the answers in the back of the book. Show steps and describe implications of results Problem 8.13

Text problem 8.13 – Extensional basin Tom Wilson, Department of Geology and Geography i.Thickness immediately following rifting is thickness at time =0. Substitute and show calcs. ii.Thickness when t large – when … ? Substitute and show calcs. iii.Sedimentation rate immediately following rifting. You’ve done the derivative … substitute and show calcs. iv.Sedimentation rate for t large. v.Numerical example: Calculate sedimentation rate immediately after rifting. Make statements!

Review questions from last class and discussion/questions about problem 8.14 Tom Wilson, Department of Geology and Geography Problem 8.14

Problem 8.14 – salinity variations across a barred basin Tom Wilson, Department of Geology and Geography i.Solve for  given some starting conditions. (Note that we should be using ppt rather than ppm). ii.Given s o and  solve for s at x=X/2. iii.Prove that the rate of increase of salinity with distance from the inlet x (i.e. the salinity gradient) is given by ds/dx = s/(  X-x). iv.Evaluate the salinity at x=X/2 and estimate the salinity at points 1km to either side, given that X=10,000m.

At this point derivatives have been obtained for 8.13 and 8.14, complete the analysis and submit next time Tom Wilson, Department of Geology and Geography

Today Tom Wilson, Department of Geology and Geography Some derivative review Today, we will spend some time using the computer as another way for you develop an appreciation of the derivative. …

Objectives for the week Tom Wilson, Department of Geology and Geography Last week we focused on reviewing and developing basic differentiation rules. Make sure you have a good grasp of those rules. This week is focused on applications. with some additional practice, and some additional differentiation rules, Main applications center around use derivatives to solve problems in geology starting with problems 8.13 and 8.14, text and computer.

Use computer to compute the following derivative (see handout). Tom Wilson, Department of Geology and Geography The second equation above illustrates combined use of the rule for differentiating natural exponential functions and the chain rule.

Computer exercise: Using the computer to calculate the derivative of a natural exponential function Tom Wilson, Department of Geology and Geography

You can explore some additional computer computations using the excel file limits.xls Tom Wilson, Department of Geology and Geography

Refer to comments on the computer lab exercise. What is the porosity gradient at a depth of 1.5 km?

Tom Wilson, Department of Geology and Geography The slope of the secant is a little larger than that of the tangent at 1.5 km Between 1 and 2 kilometers the gradient is km -1 See limits.xls

Tom Wilson, Department of Geology and Geography As we converge toward 1.5km,  /  z decreases to km -1 between 1.49 and 1.51 km depths. The slope of the secant decreases slightly as we converge on 1.5km

The derivative of exponential functions Tom Wilson, Department of Geology and Geography What is ?

Go through the derivative computations outlined in today’s handout – pages Tom Wilson, Department of Geology and Geography Your name in the title

We could make the problem a little more complex by introducing a  o and a compaction coefficient c or Tom Wilson, Department of Geology and Geography

Evaluate explicitly for  0 =0.5 and c = 0.5km -1

Hand in before leaving Tom Wilson, Department of Geology and Geography

In review - we implement the chain rule when differentiating the exponential function h in this case would be ax and, from the chain rule, becomesor and finally since and

Tom Wilson, Department of Geology and Geography For functions like we follow the same procedure. Letand then From the chain rule we have hence

Tom Wilson, Department of Geology and Geography Thus for that porosity depth relationship we were working with - ______________ Make a copy of the multiply it times the function derivative of the exponent

Tom Wilson, Department of Geology and Geography The derivative of logarithmic functions Given > What does a plot of the logarithm look like? Visualize the derivative of this line – the ln(x)

Tom Wilson, Department of Geology and Geography For logarithmic functions like We combine two rules, the special rule for natural logs and the chain rule. Let Chain rule Log rule then and so

Tom Wilson, Department of Geology and Geography The derivative of an exponential function In general for But what do you do with

Tom Wilson, Department of Geology and Geography If express a as e n so that then Since

In summary Tom Wilson, Department of Geology and Geography Sinceand in general a can be thought of as a general base. It could be 10 or 2, etc.

Tom Wilson, Department of Geology and Geography What would you do for y=a bx ? express a as e n so that then

Tom Wilson, Department of Geology and Geography Derivatives of exponential and log functions

See today’s handout regarding additional comments on use of excel to visualize of what a function will look like Tom Wilson, Department of Geology and Geography

Questions about problems 8.13 and 8.14? Answers to text problems due next time. Will take with 2.5

Computer problems 8.13 and 8.14 due in my mailbox by Friday at noon Tom Wilson, Department of Geology and Geography

Excel Setup Reference Sheet Tom Wilson, Department of Geology and Geography

Timeline for current activities Tom Wilson, Department of Geology and Geography Problems 8.13 and 8.14: hand in solutions to text problems next class Extra credit problem Option I or II due this Friday at noon. Problems 8.13 and 8.14: computer solutions due this Friday. Bring any questions to class next time. Look over problems 8.16, 8.17 and Although these will not be assigned, they will be discussed after break and you should be able to do these kinds of problems. Begin reading Chapter 9. We should wrap up derivatives and begin discussion of integrals the week following spring break.

Priorities Tom Wilson, Department of Geology and Geography i.Concentrate on getting the text problems solved for next class ii.In the next class we will answer questions regarding set up of the computer problems 8.13 and 8.14 iii.Get started on that so that we can address questions iv.Additional questions about extra credit options 1 and 2 will also be addressed next time