introduction to differential calculus part 2
TRANSCRIPT
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Geology 351 - Geomathematics
Tom Wilson, Department of Geology and Geography
Dept. Geology and GeographyWest Virginia University
Introduction to differential calculus – part 2
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Last time
Tom Wilson, Department of Geology and Geography
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
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At 2.5km, the slope is -0.05
at 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
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Reviewed in last class
Tom Wilson, Department of Geology and Geography
0.5~ 0.19
1.5~ 0.1
3~ 0.04
z
z
z
Slopes evaluated over z of
0.2km about a given depth
You’ll find the derivative z
ode
dz
Evaluation of the derivative at each of these points yields
0.51.5
1.51.5
1.51.5
0.40.19
1.5
0.4 0.098
1.5
0.4 0.036
1.5
zod
e edz
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.
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What is the intercept for the derivative?
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
ozd
edz
The intercept would be o/ or, in this case 0.267km-1
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A brief look at derivatives of trig functions.
Consider dsin()/d.
Start with the following -
Tom Wilson, Department of Geology and Geography
sin( ) sin( )
sin( ) sin cos cos sin
sin( ) sin cos cos sin
cos( ) cos cos sin sin
cos( ) cos cos sin sin
identities
Take notes as we go through this and the derivative of the cosine in class.
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We end up with
Tom Wilson, Department of Geology and Geography
0
sincos lim
3 5 7
sin ...3! 5! 7!
When is small (such as in ), sin~
We can also see this graphically using arc-length relationships
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Intro Continued – trig functions and the chain rule
Tom Wilson, Department of Geology and Geography
)2sin( axy
let 2 so that sin(2 ) sin( )h ax ax h
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)
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Tom Wilson, Department of Geology and Geography
In general if
))...))))((...(((( xqihgfy
then
dx
dq
di
dh
dh
dg
dg
df
df
dy
dx
dy........
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!
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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
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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 -
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Product rule applied to straight line formula
with its constants
Tom Wilson, Department of Geology and Geography
y=ax+b
y
x
b is the intercept
The slope a=y/x is a constant
y'=xda/dx+adx/dx +db/dx = a
b, the intercept is a constant that just gets added to the ax and shifts it up or down. The slope does not change.
Product rule applied to y=ax+b 0 01The slope is
constant for all
values of intercept b
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Tom Wilson, Department of Geology and Geography
&dy df dg
dy gdf fdg g fdx dx dx
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
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Tom Wilson, Department of Geology and Geography
2g
dxdg
fdx
dfg
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
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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
2g
dxdg
f
g
dxdf
dx
dy
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Tom Wilson, Department of Geology and Geography
2g
dxdg
f
g
dxdf
g
g
dx
dy
Multiply the first term in the sum by g/g (i.e. 1) to get >
Which reduces to
2g
dxdg
fdx
dfg
dx
dy
the quotient rule
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Tom Wilson, Department of Geology and Geography
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Rule for exponential differentiation discussed earlier
Tom Wilson, Department of Geology and Geography
xxde
edx
( )
cxcx cxdAe d cx
Ae cAedx dx
This is an application of the rule for differentiating exponents and the chain rule
Basically indestructible in this form;
( )( )
h xh xde dh
edx dx
Rewrite the function
Multiply times derivative of the exponent
For a function like , this is not the case. Calculating the derivative becomes a little more complex.
cxAe
2
2, etc.
xxd e
edx
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Tom Wilson, Department of Geology and Geography
xxde
edx
( )cxcx cxdAe d cx
Ae cAedx dx
Basic rule for differentiating exponential functions
Sketch and discuss
Rewrite the exponential function and multiply it by the derivative of the exponent – a two-step process.
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Second derivative?
Tom Wilson, Department of Geology and Geography
2
2
cxd Ae
dx
cxcxdAe
cAedx
2
2
cx cxd Ae dcAe
dx dx and
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Follow up on carrying the constants through
Tom Wilson, Department of Geology and Geography
Use the product rule to differentiate a simple function like
2y ax
dy dg dff g
dx dx dx
….
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Derivative of exponential function with base
other than e
Tom Wilson, Department of Geology and Geography
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 e2.3026 where 2.3026 is the natural log of 10 (e.g.
ln10).
The above expression for N becomes 2.3026 2.3026 or bm
bmN k e ke
Thus
2.3026
2.30262.3026
bm
bmd kedN
b kedm dm
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Differentiating exponential functions of
arbitrary base (cont.)
Tom Wilson, Department of Geology and Geography
2.3026
2.30262.3026
bm
bmd kedN
b kedm dm
Since e2.3026 = 10 &
also since 2.3026=ln(10) we can rewrite the derivative
2.3026
2.3026 10
bm
bmd kedN
b kdm dm
This rule can be applied in general to any base and would
be an approach you could use for problem 13.
2.3026
ln(10 10)
bm
bmd kedN
b kdm dm
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If you are cross-checking using WolframAlpha
Tom Wilson, Department of Geology and Geography
note
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Derivative of logarithmic functions
Tom Wilson, Department of Geology and Geography
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)).
ln( ( ))
1 ( )
( )
given y f x
dy df x
dx f x dx
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.).
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Tom Wilson, Department of Geology and Geography
If we finish these today
hand in otherwise bring in
for discussion next time
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Next time we’ll put the rule to the test using Excel
Tom Wilson, Department of Geology and Geography
0
czdc e
dz
In the lab exercise c = 1.
derivative
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Before leaving answer the following
assessment questions
Tom Wilson, Department of Geology and Geography
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.
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Take about 10 minutes
Tom Wilson, Department of Geology and Geography
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Next time we’ll spend some more time with
derivatives of exponential and log functions
Tom Wilson, Department of Geology and Geography
xexi . )( 2
)sin(.3 )( 2 yii
)tan(.xx.cos(x) )( 2 xziii
24 17)ln(.3 )( Biv
Find the derivatives of
See Waltham problem 8.8
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Before leaving
Tom Wilson, Department of Geology and Geography
• Turn in the in-class worksheet• Hand in the assessment questions• That’s it for today
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For next time - look over problems 8.13 and 8.14
Tom Wilson, Department of Geology and Geography
•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.