Geology Geomathematics. Introduction to differential calculus part 2. tom.h.wilson

Transcription

1 Geology Geomathematics Introduction to differential calculus part 2 tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

2 Last time Basic differentiation rules: Power rule Sum rule Chain rule Trig functions Product and quotient rules Exponential rules Logarithmic differentiation More in-class example problems and brief assessment activity

3 At 2.5km, the slope is at 3.5 km the slope is at 0.5, Porosity depth relationship =0.4e -z/ PHI Z

4 Slopes evaluated over z of 0.2km about a given depth 0.5 ~ 0.19 z 1.5 ~ 0.1 z 3 ~ 0.04 z Reviewed in last class You ll find the derivative d o e dz z Evaluation of the derivative at each of these points yields d z o e e dz e e The individual slope calculations provide approximate estimates of the slope at the midpoint. The derivative is exact and also provides an analytical expression that is easily plotted.

5 What is the intercept for the derivative? Porosity depth relationship =0.4e -z/ PHI d dz z o Z e The intercept would be o / or, in this case 0.267km -1

6 A brief look at derivatives of trig functions. Consider dsin()/d. Start with the following - sin( ) sin( ) identities sin( ) sin cos cos sin sin( ) sin cos cos sin cos( ) cos cos sin sin cos( ) cos cos sin sin Take notes as we go through this and the derivative of the cosine in class.

7 We end up with cos lim sin sin... 3! 5! 7! When is small (such as in ), sin~ We can also see this graphically using arclength relationships

8 Intro Continued trig functions and the chain rule y sin( 2ax) (the angle is another function 2ax) let h 2 ax so that sin(2 ax) sin( h) then Which reduces to dy dy dy dh. dh cos(2ax).2a or just dy 2acos(2ax)

9 In general if y f ( g( h( i(...( q( x))...)))) then dy dy df. df dg. dg dh dh dq.... di The chain rule is a tool that can be used explicitly by defining h and then its derivative. Alternatively, one often eliminates explicit definition of h and implements the chain rule using, for example, an outside-to-inside differentiation of terms. Either way one has to show all steps!

10 How do you handle derivatives of functions like ) ( ) ( ) ( x g x f x y? or ) ( ) ( ) ( x g x f x y The products and quotients of other functions

11 Removing explicit reference to the independent variable x, we have y fg Going back to first principles, we have y dy ( f df )( g dg) Evaluating this yields y dy fg gdf fdg dfdg Since dfdg is very small we let it equal zero; and since y=fg, the above becomes -

12 Product rule applied to straight line formula with its constants Product rule applied to y=ax+b y y '=xda/+a/ +db/ = a The slope is constant for all values of intercept b y=ax+b b is the intercept The slope a=y/x is a constant x b, the intercept is a constant that just gets added to the ax and shifts it up or down. The slope does not change.

13 dy df dg dy gdf fdg & g f Which is a general statement of the rule used to evaluate the derivative of a product of functions. The quotient rule is just a variant of the product rule, which is used to differentiate functions like f y g

14 The quotient rule states that d f g g df g 2 f dg 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 f 1 y fg g

15 We could let h=g -1 and then rewrite y as y fh Its derivative using the product rule is just dy df h f dh dh = -g -2 dg and substitution yields dy df g f dg g 2

16 Multiply the first term in the sum by g/g (i.e. 1) to get > dy g g df g f dg g 2 Which reduces to g df dy g 2 f dg the quotient rule

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18 Rule for exponential differentiation discussed earlier x de e x Basically indestructible in this form; cx For a function like Ae, this is not the case. Calculating the derivative becomes a little more complex. Rewrite the function Multiply times derivative of the exponent cx dae cx d( cx) cx Ae cae This is an application of the rule for differentiating exponents and the chain rule de h( x) 2 x de x e, etc. 2 e h( x) dh

19 Basic rule for differentiating exponential functions de x e x dae cx Ae cx ( ) Sketch and discuss d cx cae cx Rewrite the exponential function and multiply it by the derivative of the exponent a two-step process.

20 Second derivative? 2 cx d Ae 2 dae cx cae cx and 2 cx d Ae 2 dcae cx

21 Follow up on carrying the constants through Use the product rule to differentiate a simple function like y ax 2 dy dg df f g.

22 Derivative of exponential function with base other than e The preceding rules for differentiation of exponential functions are specific to base e. An exponential function such as N=k10 -bm must be converted into an exponential function with base e. This is easily done for any base. In the case of base 10 (or any base) simply express 10 as e where is the natural log of 10 (e.g. ln10). The above expression for N becomes bm dn d ke bm Thus b ke dm dm bm bm N k e or ke

23 Differentiating exponential functions of arbitrary base (cont.) Since e = 10 & also since =ln(10) we can rewrite the derivative dn dm dn dm dn dm d ke d ke bm dm bm dm d ke bm dm bm b ke bm bk10 bln(10) k10 bm This rule can be applied in general to any base and would be an approach you could use for problem 13.

24 If you are cross-checking using WolframAlpha note

25 Derivative of logarithmic functions The derivative of ln(x) is 1/x. The application to more complex functions follows the same general rule and incorporates use of other differentiation rules as needed. As a general example take ln (f(x)). given y ln( f ( x)) dy 1 df ( x) f ( x) This incorporates a chain rule application. Depending on the nature of f(x), the chained component may require use of other basic rules (e. g. product rule, sum rule, etc.).

26 If we finish these today hand in otherwise bring in for discussion next time

27 Next time we ll put the rule to the test using Excel d dz cz c0e In the lab exercise c = 1. derivative

28 Before leaving answer the following assessment questions Please do these on your own. These are not group problems. You will receive 7 points regardless, so work these independently. The purpose of this assessment activity is to determine where additional focus should be placed as we move forward. You will receive feedback.

29 Take about 10 minutes

30 Next time we ll spend some more time with derivatives of exponential and log functions See Waltham problem 8.8 Find the derivatives of 2 x ( i ) x. e ( ii) ( iii) z y 2 3.sin( ) x.cos(x) x 2.tan( x) 4 ( iv) B 3.ln( ) 17 2

31 Before leaving Turn in the in-class worksheet Hand in the assessment questions That s it for today

32 For next time - look over problems 8.13 and 8.14 Bring questions to class (8.13, 8.14, ) No due date set at present for these two problems. Keep reviewing materials in chapter 8 to refresh your memory of basic differentiation rules and applications. There are several good examples in the text.