tom.h.wilson [email protected] dept. geology and geography west virginia university

47
tom.h.wilson [email protected] .edu Dept. Geology and Geography West Virginia University

Upload: kerry-blankenship

Post on 13-Dec-2015

214 views

Category:

Documents


1 download

TRANSCRIPT

Page 1: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

[email protected].

eduDept. Geology and

GeographyWest Virginia University

Page 2: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

X

0 2 4 6 8 10 12 14

y = x3

0

500

1000

1500

2000

X

0 5 10 15 20

y = x2

0

50

100

150

Slope

X2 X1 del x del y slope7 5 2 24 128 4 4 48 12

10 2 8 96 12

X2 X1 del x del y slope7 5 2 218 1098 4 4 448 112

10 2 8 992 124

Estimating the rate of change of functions with variable slope

Page 3: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

The book works through the differentiation of y = x2, so let’s try y =x4.

4)( dxxdyy

multiplying that out -- you get ... 432234 )()(4)(64 dxdxxdxxdxxxdyy

Page 4: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

432234 )()(4)(64 dxdxxdxxdxxxdyy

Page 5: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

432234 )()(4)(64 dxdxxdxxdxxxdyy

Remember the idea of the dy and dx is that they represent differential changes that are infinitesimal - very small.

So if dx is 0.0001 (that’s 1x10-4) then (dx)2 = 0.00000001 (or 1x10-8) (dx)3 = 1x10-12 and (dx)4 = 1x10-16.

So even though dx is very small, (dx)2 is orders of magnitude smaller

Page 6: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

432234 )()(4)(64 dxdxxdxxdxxxdyy

so that we can just ignore all those terms with (dx)n where n is greater than 1.

dxxxdyy 34 4

Our equation gets simple fast

Also, since y =x4, we have dxxydyy 34

dxxdy 34

and then -

Page 7: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

34xdx

dy

Divide both sides of this equation by dx to get

dxxdy 34

This is just another illustration of what you already know as the power rule,

Page 8: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

1 nnaxdx

dyis

Just as a footnote, remember that the constant factors in an expression carry through the differentiation.This is obvious when we consider the derivative -

baxy 2

which - in general for

naxy

Page 9: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

bdxxadyy 2)(

bdxxdxxadyy )2( 22

axdxbaxdyy 2)( 2 axdxydyy 2

)2( xadx

dy

Examining the effects of differential increments in y and x we get the

following

Page 10: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

Don’t let negative exponents fool you. If n is -1, for example, we still have

1 nnaxdx

dy

2 axdx

dy

or just

Page 11: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University
Page 12: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

)()()( xgxfxy Given the function -

what is dx

dy?

dx

dg

dx

df

dx

dy

We just differentiate f and g individually and take their sum, so that

Page 13: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

Take the simple example )()( 42 baxcxy

- what is

dx

dy?

baxcxy 42We can rewrite

Then just think of the derivative operator as being a distributive operator that acts on each term in the sum.

Page 14: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

Where

then -

On the first term apply the power rule

What happens to

baxcxy 42

2 4( )dy d

x c ax bdx dx

2 4dy dx dc dax db

dx dx dx dx dx an

d2dx

dxdc

dx?

Page 15: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

Successive differentiations yield

Thus -

342 axxdx

dy

2 4

32 0 4 0

dy dx dc dax db

dx dx dx dx dxdy

x axdx

Page 16: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

)()( 42 baxcxdx

d

dx

dy

Differences are treated just like sums

so that

is just

342 axxdx

dy

Page 17: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University
Page 18: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

Differentiating functions of functions -

Given a function

22 )1( xy we consider

)()1( 2 xhx write 2hy compute

hhdh

d

dh

dy22

Then compute

xxdx

d

dx

dh212 an

d

take the product of the two, yielding dx

dh

dh

dy

dx

dy.

Page 19: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

xxdx

dh

dh

dy

dx

dy2).1(2. 2

)1(4 2 xx

22 )1( xy

We can also think of the application of the chain rule especially when powers

are involved as working form the outside to inside of a function

Page 20: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

22 )1( xyWhere

xxdx

dy2.)1(2 12

Derivative of the quantity squared viewed from the outside.

Again use power rule to differentiate the inside term(s)

Page 21: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

Using a trig function such as )2sin( axy

let axh 2

then dx

dh

dh

dy

dx

dy.

Which reduces to aaxdx

dy2).2cos( or just

)2cos(2 axadx

dy

(the angle is another function

2ax)

Page 22: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

In general if

))...))))((...(((( xqihgfy

then

dx

dq

di

dh

dh

dg

dg

df

df

dy

dx

dy........

Page 23: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

How do you handle derivatives of functions like

)()()( xgxfxy

?

or

)(

)()(

xg

xfxy

The products and quotients of other functions

Page 24: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

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 -

Page 25: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

fdggdfdy

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

g

fy

Page 26: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

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 fg

g

fy

Page 27: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

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

Page 28: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

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

Page 29: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University
Page 30: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

Z (km)0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4

0.5Porosity-Depth Relationship

Slope

cxAeFunctions of the type

czoe Recall our earlier discussions

of the porosity depth relationship

Page 31: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

Z (km)0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4

0.5Porosity-Depth Relationship

Slope

czoe

?z

Refer to comments on the computer lab exercise.

Derivative concepts

Page 32: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

Z (km)0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4

0.5Porosity-Depth Relationship

Slope

czoe

?z

Between 1 and 2 kilometers the gradient is -0.12 km-1

Page 33: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

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

0.34Porosity-Depth Relationship

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.

Page 34: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

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

0.34Porosity-Depth Relationship

Gradient1 to 2 km

Gradient1.0 to 1.1 km

What is the gradient at 1km?

What is ?d

dz

Page 35: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

xxdee

dx

( )cxcx cxdAe d cx

Ae cAedx dx

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

rule

Page 36: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

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

Page 37: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

[email protected].

eduDept. Geology and

GeographyWest Virginia University

Page 38: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

Differentiating exponential and log functions

The log and its derivative

-10

-5

0

5

10

0 10 20 30 40 50

X

Yln(x)

1/x

Z (km)0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4

0.5Porosity-Depth Relationship

Slope

Page 39: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

axey ( ) axdy d axe

dx dx axae

Returning to those exponential and natural log cases – recall we implement the chain rule when differentiating

h in this case would be ax and, from the chain rule,

dx

dh

dh

dy

dx

dy. become

s dx

dh

dh

de

dx

dy h

. or

dx

dhe

dx

dy h. and finally

axaedx

dy

since

axh and

adx

dh

Page 40: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

For functions like 2axey

we follow the same procedure.

Let 2axh and then

From the chain rule we have dx

dh

dh

dy

dx

dy.

axdx

dh2

hh eedh

d

dh

dy

22. axaxe

dx

dh

dh

dy

dx

dyhence

Page 41: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

Thus for that porosity depth relationship we were working with

- /

0)( zez

?

)( /0

dz

ed

dz

zd z

/0 ze

Page 42: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

•The derivative of logarithmic functions

Given >

)ln(xy

xdx

dy 1

We’ll talk more about these special cases after we talk about the chain

rule.

Page 43: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

For logarithmic functions like )ln( 2xy

We combine two rules, the special rule for natural logs and the chain rule.

Let 2xhdx

dh

dh

dy

dx

dy.Chain

rule

Log

rule xdx

xd 1)(ln

then

2

1

xdh

dy an

dx

dx

dh2

soxx

x

dx

xd 22)ln(2

2

Page 44: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

•The derivative of an exponential

function

xey

xedx

dy

Given >

In general for axey xedx

axd

dx

dy )( axae

xay If express a as en so that nxxn eey

then nxnx needx

d

dx

dy

)ln()ln( aen n Note

Page 45: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University

nxnx needx

d

dx

dy

Since nxx ea and

)ln(an

xaadx

dy . )ln(in general

a can be thought of as a general base. It could be 10 or 2, etc.

Page 46: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University
Page 47: Tom.h.wilson tom.wilson@mail.wvu.edu Dept. Geology and Geography West Virginia University