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Geology 351 - GeoMath

Geology 351 - GeoMath. Derivatives (p2). tom.h.wilson tom.wilson@mail.wvu.edu. Dept. Geology and Geography West Virginia University. Objectives for the day. Reminders Product and quotient rules General comments about trigonometric functions Wrap up derivatives worksheet

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Geology 351 - GeoMath

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  1. Geology 351 - GeoMath Derivatives (p2) tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University Tom Wilson, Department of Geology and Geography

  2. Objectives for the day • Reminders • Product and quotient rules • General comments about trigonometric functions • Wrap up derivatives worksheet • Hand out some additional examples for practice Tom Wilson, Department of Geology and Geography

  3. At 2.5km, the slope is -0.05at 3.5 km the slope is -0.0259at 0.5, -0.191 With slope varying as Tom Wilson, Department of Geology and Geography

  4. 0, -0.707 (at 0.785), -1 (at 1.571), -0.707 (at 2.356) Tom Wilson, Department of Geology and Geography

  5. For y=x2 Tangent line Slope of the tangent line Slope or derivative = 2x Tom Wilson, Department of Geology and Geography

  6. Product and quotient rules How do you handle derivatives of functions like or ? The products and quotients of other functions Tom Wilson, Department of Geology and Geography

  7. Removing explicit reference to the independent variable x, we have Going back to first principles, we have Evaluating this yields Since dfdg is very small we let it equal zero; and since y=fg, the above becomes - Tom Wilson, Department of Geology and Geography

  8. The quotient rule is just a variant of the product rule, which is used to differentiate functions like Which (after division by x) is a general statement of the rule used to evaluate the derivative of a product of functions. Tom Wilson, Department of Geology and Geography

  9. The quotient rule states that The proof of this relationship can be tedious, but I think you can get it much easier using the power rule Rewrite the quotient as a product and apply the product rule to y as shown below Tom Wilson, Department of Geology and Geography

  10. We could let h=g-1 and then rewrite y as Its derivative using the product rule is just dh = -g-2dg and substitution yields Tom Wilson, Department of Geology and Geography

  11. Multiply the first term in the sum by g/g (i.e. 1) to get > Which reduces to the quotient rule Tom Wilson, Department of Geology and Geography

  12. Take a few moments to work through the remaining examples on your worksheet Tom Wilson, Department of Geology and Geography

  13. Hand in today or Thursday … Finish up the in-class problems Tom Wilson, Department of Geology and Geography

  14. Look over problems 8.13 and 8.14 • Bring questions to class this Thursday • Due date – Tuesday, March 25th Tom Wilson, Department of Geology and Geography

  15. Next time we’ll continue with exponentials and logs, but also have a look at question 8.8 in Waltham (see page 148). Find the derivatives of Tom Wilson, Department of Geology and Geography

  16. Due dates …. • Hand in the graphical analysis of slopes for the porosity- depth relationship, cosine and x2 functions. • Wrap up the basic in-class differentiation worksheet and hand in today or Thursday • Next time we’ll talk about derivatives of logs and exponential functions and use Excel to compute the derivative of exponential functions for in-class illustration and discussion • continue reading Chapter 8 – Differential Calculus Tom Wilson, Department of Geology and Geography

  17. Next time we’ll talk about differentiation of log and exponential functions We’ll use the computer this time to help us conceptualize the derivative or slope This is an application of the rule for differentiating exponents and the chain rule Tom Wilson, Department of Geology and Geography

  18. Exponential functions in the form See chapter 8 Recall our earlier discussions of the porosity depth relationship Tom Wilson, Department of Geology and Geography

  19. Derivative concepts Refer to graphical exercise and to comments on the computer lab exercise. Tom Wilson, Department of Geology and Geography

  20. Between 1 and 2 kilometers the gradient (slope) is -0.12 km-1 Tom Wilson, Department of Geology and Geography

  21. In the limit that our computations converge on a point we have the slope (derivative) at that point. As we converge toward 1km, /z decreases to -0.14 km-1 between 1 and 1.1 km depths. Tom Wilson, Department of Geology and Geography

  22. We’ll talk more about this and logs next time What is ? Tom Wilson, Department of Geology and Geography

  23. Due dates …. • Hand in the graphical analysis of slopes for the porosity- depth relationship, cosine and x2 functions. • Wrap up the basic in-class differentiation worksheet and hand in today or Thursday • Next time we’ll talk about derivatives of logs and exponential functions and use Excel to compute the derivative of exponential functions for in-class illustration and discussion • continue reading Chapter 8 – Differential Calculus Tom Wilson, Department of Geology and Geography

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