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), y(t) : t I . For a time t0 I, assume x (t0 ) = 0.
slope
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 by the point (x(t), y(t) as the parameter t ranges over
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)
2
dt
a
d(Pi1 , Pi ) =
[x(ti ) x(ti1 )]2 + [y(ti ) y(ti1
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
Proof
Let P = cfw_0 , 1 , , n be a partition of [, ]. S
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, ), then
ak converges
i
f (x)dx converges
1
k=1
n
Since f
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 converges, it does not matter where the summation begins. Th
Lecture 26Section 11.6 Taylor Polynomials and Taylor Series
in x a
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Taylor Polynomials in x a
Taylor Polynomials in x a
Taylor Polynomials in Powers of x a
Taylor Polynomials in Powers of x a
The nth Taylor polynomial in x a for a function f
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
x = 1 + x + x + x +
k=0
diverges,
if |x| 1.
Power Seri
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_natural numbers
Z = cfw_. . . , 2, 1, 0, 1, 2, . . . , = c
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 integration [a, b], 0 < a < b, is bounded,
Known:
f (x) dx
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 = M . Then
n
n
lim can = cL
n
lim (an + bn ) = L + M
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 (0/0)?
x2
1
1
1
x2
= lim
= lim
=
=
24
x2 (x 2)(x + 2)
Gkalp Grsel 503091141
ENGINEERING MATHEMATICS HOMEWORK 3
Roots of bessel functions: 1. Roots of bessel function I0 Matlab code:
clear all clc besseli0 = inline('besseli(0,x)'); for n=1:25 z(n)=fzero(besseli0,n); end disp(z)
Results: Columns 1 through 16 N
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)*(cos(1-eta^2)^(0.5)*t)-exp(eta*pi/(2*(1-eta^2)^(0.5) ezpl
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, resource consumer and susceptible infection interacti
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 |x| < 1
p-series:
1
,
kp
if p > 1
Basic Series that Div