Geology 351 - GeoMath

Geology 351 - GeoMath

Tom Wilson, Department of Geology and Geography

[email protected]

Dept. Geology and GeographyWest Virginia University

Derivatives (p2)

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

At 2.5km, the slope is -0.05at 3.5 km the slope is -0.0259

at 0.5, -0.191


Z

0 1 2 3 4 5

PH

I

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4Porosity depth relationship

=0.4e-z/1.5

/1.510.4

1.5zd

edz

With slope varying as

0, -0.707 (at 0.785), -1 (at 1.571), -0.707 (at 2.356)


(radians)

cos

()

-1.0

-0.5

0.0

0.5

1.0Cosine function

For y=x2


x0 1 2 3 4 5 6

y

0

5

10

15

20

25

30

35

40

y=x2

Tangent line

Slope of the tangent line

Slope or derivative = 2x

Product and quotient rules


How do you handle derivatives of functions like

)()()( xgxfxy

?

or

)(

)()(

xg

xfxy

The products and quotients of other functions


fgy

Removing explicit reference to the independent variable x, we have

))(( dggdffdyy Going back to first principles, we have

Evaluating this yields dfdgfdggdffgdyy

Since dfdg is very small we let it equal zero; and since y=fg, the above becomes

-


fdggdfdy

Which (after division by x) 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

g

fy


2gdx

dgfdxdfg

g

f

dx

d

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

1 fgg

fy


fhy

We could let h=g-1 and then rewrite y as

Its derivative using the product rule is just

dx

dhf

dx

dfh

dx

dy

dh = -g-2dg and substitution yields

2gdx

dgf

gdx

df

dx

dy


2gdx

dgf

gdx

df

g

g

dx

dy

Multiply the first term in the sum by g/g (i.e. 1) to get >

Which reduces to

2gdx

dgfdxdfg

dx

dy

the quotient rule

Hand in before leaving


Finish up the in-class problems

Look over problems 8.13 and 8.14


•Bring questions to class this Thursday

•Due date – Tuesday, March 25th


Next time we’ll continue with exponentials and logs, but also have a look at question 8.8 in Waltham (see page 148).

xexi . )( 2

)sin(.3 )( 2 yii

)tan(.xx.cos(x) )( 2 xziii 24 17)ln(.3 )( Biv

Find the derivatives of

Due dates ….


Spend some time going over 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 before leaving

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

Next time we’ll talk about differentiation of log and exponential functions


xxdee

dx

( )cxcx cxdAe d cx

Ae cAedx dx

This is an application of the rule for differentiating exponents and the chain rule

We’ll use the computer this time to help us conceptualize the derivative or slope


Z (km)0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4

0.5Porosity-Depth Relationship

Slope

cxAeExponential functions in the form

czoe Recall our earlier discussions of the

porosity depth relationship

See chapter 8


Z (km)0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4


Slope

czoe

?z

Refer to graphical exercise and to

comments on the computer lab exercise.

Derivative concepts


Z (km)0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4


Slope

czoe

?z

Between 1 and 2 kilometers the

gradient (slope) is -0.12 km-1

In the limit that our computations converge on a point we have the slope (derivative) at that point.


Z (km)

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

0.16

0.18

0.20

0.22

0.24

0.26

0.28

0.30

0.32


Gradient1 to 2 km

Gradient1.0 to 1.1 km

As we converge toward 1km, /z decreases to -0.14 km-1 between 1 and 1.1 km depths.


Z (km)

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

0.16

0.18

0.20

0.22

0.24

0.26

0.28

0.30

0.32


Gradient1 to 2 km

Gradient1.0 to 1.1 km

We’ll talk more about this and logs next time

What is ?d

dz

Due dates ….


Spend some time going over 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 before leaving

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

Geology 351 - GeoMath

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