Download Homogenization Rate of Diffusive Tracers in Chaotic Advection

Calculus
Weijiu Liu
Department of Mathematics
University of Central Arkansas
Overview
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What is calculus?
Calculus is a subject about the study of limiting processes
and it consist of three basic concepts: limit, differentiation,
and integration.
3
Who founded calculus?
Isaac Newton (1643 – 1727)
was the greatest English
mathematician of his generation.
He laid the foundation
for differential and integral
calculus. His work on optics
and gravitation make him one of
the greatest scientists
the world has known.
http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Newton.html
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Gottfried Wilhelm von
Leibniz (1646 – 1716)
was a German
mathematician
who developed the present
day notation for the
differential and integral
calculus though he never
thought of the derivative
as a limit. His philosophy
is also important and he
invented an early
calculating machine.
http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Leibniz.html
For a history of the calculus, see:
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html
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Why is calculus needed?
1. Geometrical problems.
y=f(x)
Slope of a sceant line:
Tangent line
Q(x,f(x))
Q(x,f(x))
f ( x)  f ( x0 )
Q(x,f(x))
P  x0 , f  x0  
x  x0
mPQ 
f  x   f  x0 
x  x0
Slope of a tangent line:
m  lim
f  x   f  x0 
x  x0
x0
x
=f   x 
x  x0
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The area A of a range under the graph of a function
y
y
y = f(x)
A3
A1
a
A2
y = f(x)
B4
A4
B1
b
x
a
A  A1  A2  A3  A4
B2
B3
B5
B6
B7
B8
x
b
A  B1  B2  B3  B4  B5  B6  B7  B8
b
A   f  x dx
a
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2. Physical problems.
s t 
s  t0 
s
Average velocity
s (t )  s (t0 )
va 
t  t0
Instantaneous velocity:
v(t0 )  lim
t t0
s  t   s  t0 
t  t0
 s  t 
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m
a
Work:
m
a
F
x
b
W  F b  a 
F  x
b
x
b
Work:
W   F  x  dx
a
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3. More advanced problems.
Description of particle motion:
Ordinary differential equations:
dx
 v1  x, y, t  ,
dt
dy
 v2  x, y, t  ,
dt
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Description of evolution of chemical concentration – partial differential equation:
Convection
Diffusion
Equation:
  2u  2u 
u
u
u
 v1  x, y, t   v2  x, y, t 
k 2  2 
t
x
y
 x y 
Convection
Diffusoin
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Description of the string vibration – the wave equation
2
 2u

u
2

c
t 2
x 2
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How to study calculus?
•
•
•
•
Read your textbook
Understand concepts clearly
Do your homework on time
Ask your instructor and classmates around you whenever
you have a question
• Learn from your classmate
• Form a group to discuss
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Tentative schedule
•
•
•
•
•
Chapter 1 – Limits, 3 weeks
Chapter 2 – Differentiation, 3 weeks
Chapter 3 – Application of differentiation, 3 weeks
Chapter 4 – Integration, 3 weeks
Chapter 5 – Application of definite integrals, 2
weeks
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Chapter 1-- Limits
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Problem of Limit
Tangent line problem
Slope of a sceant line:
y=f(x)
Tangent line
mPQ 
f  x   f  x0 
x  x0
Q(x,f(x))
Q(x,f(x))
f ( x)  f ( x0 )
Q(x,f(x))
P  x0 , f  x0  
x  x0
Slope of a tangent line:
m  lim
f  x   f  x0 
x  x0
=f   x 
x0
x  x0
x
Problem of Limit: As x gets closer and close to x0, to what number is a
function g(x) like g  x  
f  x   f  x0 
x  x0
though g(x) is not well defined at x0 ?
getting closer and closer to even
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Find limits of a function graphically
Consider the function
f  x 
sin x
x
As x>0 gets closer and closer to 0
f(x) is getting closer and closer to 1.
As x<0 gets closer and closer to 0
f(x) is getting closer and closer to 1.
We say the limit of f(x) as x approaches 0
We say the limit of f(x) as x approaches 0
from the left is 1, written
from the right is 1, written
sin x
lim
1
x 0
x
One-sided limits
sin x
lim
1
x  0
x
We say the limit of f(x) as x approaches 0 is 1, written
sin x
1
x 0
x
lim
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Find limits of a function numerically
x
0
sin x / x
-0.10000000000000
-0.01000000000000
-0.00100000000000
-0.00010000000000
-0.00001000000000
-0.00000100000000
-0.00000010000000
-0.00000001000000
-0.00000000100000
-0.00000000010000
0.99833416646828
0.99998333341667
0.99999983333334
0.99999999833333
0.99999999998333
0.99999999999983
1.00000000000000
1.00000000000000
1.00000000000000
1.00000000000000
1
sin x
lim
1
x 0
x
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x
0
sin x / x
0.10000000000000
0.01000000000000
0.00100000000000
0.00010000000000
0.00001000000000
0.00000100000000
0.00000010000000
0.00000001000000
0.00000000100000
0.00000000010000
0.99833416646828
0.99998333341667
0.99999983333334
0.99999999833333
0.99999999998333
0.99999999999983
1.00000000000000
1.00000000000000
1.00000000000000
1.00000000000000
1
sin x
lim
1
x 0
x
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In general, if f(x) is getting closer and closer to L as x gets closer and
closer to a, we say that the limit of f(x) as x approaches a is L, written
lim f  x   L
x a
Theorem.
lim f  x   L if and only if lim f  x   lim f  x   L
xa
xa
x a
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x
Example. Evaluate lim
x 0 x
x
x
lim  lim
 1
x 0 x
x 0  x
So
x
x
lim  lim  1
x 0 x
x 0 x
x
lim
x 0 x
does not exist!
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Example. Evaluate lim
x 1
x 1
x 1
x 1
lim
 
x 1 x  1
lim
x 1
So lim
x 1
x 1

x 1
x 1
does not exist!
x 1
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Graph of problem 4 of Exercises 1.2
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Intermediate Value Theorem
a
2.0000
2.0000
2.0000
2.1200
2.1800
2.2100
2.2100
2.2100
2.2100
2.2100
b
3.0000
2.5000
2.2500
2.2500
2.2500
2.2500
2.2300
2.2200
2.2200
2.2200
f(a)
-2.0000
-2.0000
-2.0000
-0.9519
-0.3598
-0.0461
-0.0461
-0.0461
-0.0461
-0.0461
f(b)
13.0000
3.6250
0.3906
0.3906
0.3906
0.3906
0.1696
0.0610
0.0610
0.0610
Midp
2.5000
2.2500
2.1200
2.1800
2.2100
2.2300
2.2200
2.2100
2.2100
2.2100
f(midp)
3.6250
0.3906
-0.9519
-0.3598
-0.0461
0.1696
0.0610
-0.0461
-0.0461
-0.0461
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a
-1.0000
-1.0000
-0.7500
-0.6300
-0.5700
-0.5400
-0.5400
-0.5400
-0.5400
-0.5400
b
f(a)
f(b)
Midp
f(midp)
0
1.0000 -2.0000 -0.5000 -0.1250
-0.5000 1.0000 -0.1250 -0.7500 0.5781
-0.5000 0.5781 -0.1250 -0.6300 0.2700
-0.5000 0.2700 -0.1250 -0.5700 0.0948
-0.5000 0.0948 -0.1250 -0.5400 0.0025
-0.5000 0.0025 -0.1250 -0.5200 -0.0606
-0.5200 0.0025 -0.0606 -0.5300 -0.0289
-0.5300 0.0025 -0.0289 -0.5400 0.0025
-0.5300 0.0025 -0.0289 -0.5400 0.0025
-0.5300 0.0025 -0.0289 -0.5400 0.0025
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