Lecture 15Section 9.7 Tangents to Curves Given
Parametrically
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1.1
Tangents to Parametrized curves
Tangents to Parametrized curve
Tangents to Parametrized curves
Tangent line
Let C = (x(t),
Lecture 14Section 9.6 Curves Given Parametrically
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1.1
Parametrized curve
Parametrized curve
Parametrized curve
Parametrized curve A parametrized Curve is a path in the xy-plane traced out b
Lecture 16Section 9.8 Arc Length and Speed
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1.1
Arc Length
Arc Length
Arc Length Formulas
Arc Length Formulas
Let C = (x(t), y(t) : t I . [0.5ex] The length of C is
b
L(C) =
2
x (t)
+ y (t)
Lecture 13Section 9.5 Area in Polar Coordinates
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1.1
Area of a Polar Region
Basic Polar Area
Area of a Polar Region
1
2
The area of the polar region generated by
3
r = (),
is
1
()
2
A=
2
d
P
Lecture 22Section 11.2 The Integral Test; Comparison Tests
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1.1
The Integral Test
The Integral Test
The Integral Test
Let ak = f (k), where f is continuous, decreasing and positive on [1, ),
Lecture 23Section 11.3 The Root Test; The Ratio Test
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Comparison Tests
Basic Series that Converge or Diverge ak converges
k=1
i
k=j
ak converges, j 1.
In determining whether a series conve
Lecture 26Section 11.6 Taylor Polynomials and Taylor Series
in x a
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1.1
Taylor Polynomials in x a
Taylor Polynomials in x a
Taylor Polynomials in Powers of x a
Taylor Polynomials in Powers o
Lecture 2711.7 Power Series
11.8 Dierentiation and
Integration of Power Series
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1.1
Power Series
Geometric Series and Variations
Geometric Series
Geometric Series:
k=0
xk
1
, if |x| < 1,
1x
Lecture 17Section 10.1 Least Upper Bound Axiom
10.2 Sequences of Real Numbers
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Real Numbers
1.1
Review
Basic Properties of R: R being Ordered
Classication
N = cfw_0, 1, 2, . . . = cfw_natur
Lecture 20Section 10.7 Improper Integrals
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Improper Integrals
What are Improper Integrals?
1
b
1
1
dx =?,
x2
0
1
dx =?
x2
b
b
1
1
1 1
dx =
= , 0 < a < b,
x2
x a
a b
a
a
the interval of int
Lecture 18Section 10.3 Limit of Sequence
Section 10.4 Some
Important Limits
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Limit of Sequence
1.1
Properties of Limits
Properties of Limits
Properties of Limits: 1
Let lim an = L and lim bn
Lecture 21Section 11.1 Innite Series
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Innite Series
What is an Innite Series? 1 1
1
Let the sequence ak = 21 : 2 , 1 , 8 , 16 , . Form the partial sums
k
4
1
1
s1 = a1 =
ak =
2
k=1
2
1 1
3
s
Lecture 19Section 10.5 Indeterminate Form (0/0)
Section
10.6 Other Indeterminate Forms (/), (0 ),
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Indeterminate Forms
1.1
Indeterminate Form (0/0)
Example the
What is 1. Indeterminate Form
b) MATLAB code of =0.1 clear all clc eta=0.1; syms t x=exp(-eta*t)*(cos(1-eta^2)^(0.5)*t)-exp(eta*pi/(2*(1-eta^2)^(0.5) ezplot(x) grid on
MATLAB code of =2 clear all clc eta=2; syms t x=exp(-eta*t)*(c
Gkalp Grsel 503091141
PREY PREDATOR MODEL
Prey Predator Models are generally modelling a population growth. However, sometimes this model can take the forms of parasite host, tumor cells immune system
Lecture 24Section 11.4 Absolute and Conditional
Convergence; Alternating Series
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Convergence Tests
Basic Series that Converge or Diverge
Basic Series that Converge
Geometric series:
xk ,
if