geology 351 - geomath
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Geology 351 - GeoMath. Derivatives (p2). tom.h.wilson [email protected]. 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 - PowerPoint PPT PresentationTRANSCRIPT
Geology 351 - GeoMath
Tom Wilson, Department of Geology and Geography
Dept. Geology and GeographyWest Virginia University
Derivatives (p2)
Objectives for the day
Tom Wilson, Department of Geology and Geography
• 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
Tom Wilson, Department of Geology and Geography
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)
Tom Wilson, Department of Geology and Geography
(radians)
cos
()
-1.0
-0.5
0.0
0.5
1.0Cosine function
For y=x2
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
How do you handle derivatives of functions like
)()()( xgxfxy
?
or
)(
)()(
xg
xfxy
The products and quotients of other functions
Tom Wilson, Department of Geology and Geography
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
-
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
Hand in before leaving
Tom Wilson, Department of Geology and Geography
Finish up the in-class problems
Look over problems 8.13 and 8.14
Tom Wilson, Department of Geology and Geography
•Bring questions to class this Thursday
•Due date – Tuesday, March 25th
Tom Wilson, Department of Geology and Geography
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 ….
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
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
Tom Wilson, Department of Geology and Geography
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 graphical exercise and to
comments on the computer lab exercise.
Derivative concepts
Tom Wilson, Department of Geology and Geography
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 (slope) is -0.12 km-1
In the limit that our computations converge on a point we have the slope (derivative) at that point.
Tom Wilson, Department of Geology and Geography
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.
Tom Wilson, Department of Geology and Geography
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
We’ll talk more about this and logs next time
What is ?d
dz
Due dates ….
Tom Wilson, Department of Geology and Geography
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