tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University
Items on the to do list Finish reading Chapter 8 and look over problems 8.13 and 8.14. Problems 8.13 and 8.14 are tentatively due next week. Bring questions to class next time
Let s pick up with the quotient rule How do you handle derivatives of functions like y( x) f ( x) g( x) or y( x) f ( x) g( x) The products and quotients of other functions?
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 -
dy gdf fdg 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
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
We could let h=g -1 and then rewrite y as y Its derivative using the product rule is just dy df h fh f dh dh = -g -2 dg and substitution yields dy df g f dg g 2
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 dy g df g 2 f dg the quotient rule
A simple example for the quotient rule y x x 3 5 Remember it s always a good idea to define each variable the derivative formula f ( x3) g ( x5) df dg 1 1 dy g df f dg g State the rule, substitute and solve 2
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.
We end up with cos lim sin 0 3 5 7 sin... 3! 5! 7! When is small (such as in ), sin~ We can also see this graphically using arc length relationships
Functions of the type Ae cx 0.5 Porosity-Depth Relationship 0.4 0.3 0.2 0.1 0.0 Slope 0 1 2 3 4 5 Z (km) Recall our earlier discussions of the porosity depth relationship o e cz
o e cz 0.5 Porosity-Depth Relationship 0.4 0.3 0.2 0.1 Slope z? 0.0 0 1 2 3 4 5 Z (km) Between 1 and 2 kilometers the gradient is -0.12 km -1
Exponential functions 0.34 0.32 0.30 0.28 0.26 0.24 0.22 0.20 0.18 Gradient 1.0 to 1.1 km Porosity-Depth Relationship Gradient 1 to 2 km 0.16 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Z (km) As we converge toward 1km, /z decreases to -0.14 km -1 between 1 and 1.1 km depths.
0.34 0.32 0.30 0.28 0.26 0.24 0.22 0.20 0.18 Gradient 1.0 to 1.1 km Porosity-Depth Relationship Gradient 1 to 2 km 0.16 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Z (km) When we ask What is the porosity gradient at exactly 1km? We are asking What is d dz - the slope of the tangent line to the curve at that point?
Next time we ll use Excel to demonstrate that the rules noted below accurately characterize slope variations. 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 Take 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) e h( x) dh
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.
Second derivative? 2 cx d Ae 2 dae cx cae cx and 2 cx d Ae 2 dcae cx
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.
Next time we ll put the rule to the test using Excel d dz cz c0e In the lab exercise c = 1. derivative
Locating minima and maxima Consider the following function y x x x 3 2 2 3.5 20 2 Although it is not easy to visualize such a function, we can answer some basic questions about its shape. For example: where are the maxima and minima in the function y?
Also have a look at the limits.xls file for some illustrative examples 1.5 The cosine and its derivative amplitude 1 0.5 0-0.5-1 cos( q ) -sin( q ) point 1 Point 2 Secant deriv (slope) -1.5-90 0 90 180 270 360 theta Limits.xls Includes derivatives of trig and other functions
We know that at the maxima and minima the slope or derivative is 0 As an easy to visualize example, consider the cosine- It s derivative is dcos x sin x Sketch the cosine and look at how its slope varies over one cycle. When the derivative (the slope) is 0 we know we are at a maximum or a minimum.
The cosine has maxima and minima at maximum maximum 0 minimum 0, and 2 2
We see that the derivative, sinx, is 0 at the maxima and minima sinx We know there is a maximum or a minimum, but we don t know which
We use the second derivative to help us determine this d 2 cos 2 x Take the second derivative, or take the derivative of sinx to get d( sin x) cos x We can evaluate cosx at 0-slope locations
If the 2 nd derivative is negative then the function is maximum At 0 radians the 2 nd derivative is a a negative number and at radians we get a positive number maximum The cosine The 1 st derivative The 2 nd derivative minimum d cos x 2 2
The function y x x x 3 2 2 3.5 20 2 The function and its derivatives y x x x 3 2 2 3.5 20 2 ' 2 y x x 6 7 20 y " 12x7
The roots of the first derivative (a quadratic) locate 0-slope points on the function of interest
The roots correspond to the 0 s in the 1 st derivative x 2 b b 4ac 2a 7 49 4(6)( 20) 12 7 49 480 12 7 23 16 30 or 12 12 12 1.33 or 2.5
Note that the 2 nd derivative is linear. We are just interested in the sign of this function y x x x 3 2 2 3.5 20 2 ' 2 y x x y " 6 7 20 12x7 Its negative value at the negative root indicates a maximum at this value and the positive value for the positive root indicates a minimum in the input function
You should be finishing your reading of Chapter 8 I m handing out the text and computer problems associated with problems 8.13 and 8.14. They will tentatively be due later next week. Bring questions to class next time. Note that you will actually have to go to the text to find out what is required for the text problems. Basically, in this handout, I am asking you to add some description of what you find. What does it mean geologically. Look over the computer lab component to these two problems as well. We will get started on that next time
Before leaving today take a look at the following questions. You get credit for doing them right or wrong for being here today Hand in before leaving