NUMBER SETS
Natural, whole, rational, irrational and real numbers
IB
Syllabus
\
Natural Numbers
0 1 2 3 4 155
.
1000
Rational Numbers
- Can be written as quotients of integers
- Can be written as decimals
that terminate or repeat
Examples:
Irrational Numb
INTRODUCTION TO
SLOPE FIELDS
Defn: Differential Equation.
An equation involving a derivative is called a differential
equation. The order of a differential equation is the order
of the highest derivative involved in the equation.
Example 1
Find all functi
Introduction to Slope Fields
Name_
Matcheachdifferentialequationshownbelowwithitsslopefieldfromthebottomofthepage.
1.
y x
_
2.
y y
_
3.
y 2 y
_
4.
y x 1
_
5.
xy 1
_
6.
2 y y
_
7.
xy y
_
8.
yy x
_
9.
y x y
_
A.
B.
C.
D.
E.
F.
G.
H.
I.
Thegraphsont
.
5.
6.
7.
8.
1.
2.
3.
4.
Differential Equations:
3. Match each slope field to a differential equation. Be ready to justify your choice. We will review
solutions in 20 minutes.
2. Draw any homogenous solutions.
1. For each slope field, draw the solution c
The
Unit
Circle
BJECTIVES
Given
a real number that is an integral multiple of halves, thirds, fourths, or sixths of , tsw
identify the point on the unit circle determined by it
Given a real number that is an integral multiple of fourths, sixths, eights, o
TI LAB 1: Useful Stuf
IN THIS LAB, YOU WILL:
EVALUATE FUNCTIONS FROM THE GRAPH;
EVALUATE FUNCTIONS FROM THE HOME SCREEN;
DISPLAY A GRAPH AND TABLE SIDE-BY-SIDE;
SET UP THE TABLE USING VALUES YOU DETERMINE;
STORE A VALUE CALCULATED FROM A GRAPH.
1. Open th
Given thatf(x)= (3x5)4 , find,
5 f(X),
[3 marks]
b) the gradient ofthe curve y = f(x) at the point where X: 2.
[1 mark]
2 Given thatf(x)= VSXl find the value off(1).
[4 marks]
3 . . 1 . 7:
A curve IS defined such that y = Esm<6x), for 0 S x S 3.
Find th
Approximation
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Room #513
November 30, 2012
Dr. Tran Thai-Duong (IU HCMC)
Approximation
November 30, 2012
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CAUCHY THEOREM
If f and g are continuous functions on [a, b] which
are dierentiable in (a, b), then there is a point
c (a, b
Continuity
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October 12, 2012
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Continuity
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Denition
Suppose f is dened in an open interval that
contains a, then f is continuous at a if and only if
lim f (x) = f (a).
xa
In g
Mathematical English Help Sheet
Written
form
2+2=4 2plu32equals4
Reading
>
,3
E
w
a i b A plus or minus B
A times B
A multi lied by B
A divided by B
a / b A over B
1/2 (a) half
2/3 Two thirds
% Three quarters
it/3 nover3 0r nby3
(x) F of X
X squared
X cub
Numbers and Functions
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Room #513
February 22, 2013
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Numbers and Functions
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Intervals
Open interval (a, b) = cfw_x R|a < x < b
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Numbers and Functions
Febru
Elementary Functions
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Elementary Functions
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Exponential functions
f (x) = b x , b > 0, b = 1
where b is called the base
x y
Dr. Tran Thai-Duong (IU HCMC)
x+
MIN/MAX
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Room #513
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MIN/MAX
November 19, 2012
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Local Extremum
f has a local maximum at a if there exists > 0
such that for |a x| < ,
f (x) f (a)
Similarly, f has a local minimum at a
More Derivatives
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More Derivatives
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Second derivative
Second derivative of a function f at a point a is
f (a) = f
(2)
d 2f
(a) = 2
dx
=
x=a
d (f )
dx
x=a
Se
Riemann Integrals
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Riemann Integrals
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Riemann Sum
f is dened on an open interval (a, b)
n1
f (ti )(xi+1 xi )
i=0
where (xi )0in are chosen so that
x0 = a
Other Techniques of Integration
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Other Techniques
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Integration by parts
u(x)v (x)dx = u(x)v (x)
u (x)v (x)dx
Example
u(x) = x n , u (x) = nx n1,
v (x) =
sin1 x or arcsin x
sin x: D (sin) = , , R (sin) = [1, 1]
2 2
sin1 x: D sin1 = [1, 1], R sin1 = ,
2 2
Cancellation equalities
sin sin1 x = x,
1 x 1,
sin1 (sin x) = x,
x
2
2
How to express trigonometric functions through each other
cos x 0,
x ,
,
2 2
cos
Calculus 2-BA: Lecture 3&4:
MAXIMA and MINIMA, TOTAL DIFFERENTIALS
Dr. Pham Huu Anh Ngoc
July, 2012
Maximum and minimum: Motivation Examples
Example 1: The prot from the sale of x units of radiators for
automobiles and y units of radiators for generator i
VIETNAM NATIONAL UNIVERSITY AT HO
CHI MINH CITY
INTERNATIONAL UNIVERSITY
Chapter 4: Integration
Calculus 1
Lecturer: Nguyen Minh Quan, PhD
[email protected]
Nguyen Minh Quan (HCMIU-VNU)
Chapter 4: Integration
December 2012
1 / 78
Contents
1
Areas under
CALCULUS I
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Calculus I
February 6, 2013
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Textbook
James Stewart
Calculus, 6th edition
Early Transcenentals
1 FUNCTIONS AND
MODELS
2 LIMITS AND DERIVATIVES
3 DIFFERENTIATION
Math 113Q
Section 4.7
9. A farmer with 750 ft. of fencing wants to enclose a rectangular area and then divide
it into four pens with fencing parallel to one side of the rectangle. What is the largest
possible total area of the four pens?
(a) Draw several
CALCULUS I
Lecturer: Dr. Nguyen Minh Quan
Department of Mathematics
HCMC International University
February 16, 2013
Dr. Nguyen Minh Quan (HCMIU)
Calculus I
February 16, 2013
1/8
Textbook
Dr. Nguyen Minh Quan (HCMIU)
Calculus I
February 16, 2013
2/8
Refere
Mathematical terminology
(Calculus I)
English
Vietnamese
English
Vietnamese
Limit
gii hn
continuous
lin tc
continuity
tnh lin tc
analysis
gii tch
algebra
i s
domain
min xc nh
range
min gi tr
real number
s thc
real line
ng thng thc
independent variable bin
CHUYN TCH PHN
B ng cng th c tch phn b t nh :
0dx = C
n
x dx =
dx = x + C
x n +1
+C
n +1
1
x dx = ln x + C
n 1
ax
C
ln a
cos xdx = sin x + C
x
x
e dx = e + C
x
a dx =
sin xdx = cos x + C
1
cos
1
sin
dx = tan x + C
dx = cot x + C
2
x
x
u( x)
1
1
x
VIETNAM NATIONAL UNIVERSITIES AT HO CHI
MINH CITY
INTERNATIONAL UNIVERSITY
Chapter 5: Applications of Integration
Calculus 1
Lecturer: Nguyen Minh Quan, PhD
Nguyen Minh Quan (HCMIU-VNU)
Chapter 5: Applications of Integration
December 2012
1 / 49
Contents
Macroeconomics
Practice exam
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
FIGURE 24-1
1) If the economy in Figure 24-1 is currently producing Y0 level of output, then to attain full-employment leve
Student Solution Manual Calculus By James Stewart 6th Edition In (PDF Documents) provides by softx32.com and hosted
at docs58.softx32.com
Student Solution Manual Calculus By James Stewart 6th Edition In
Table of Contents
1. Solution Manual - Der Keiler Co
Derivatives
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Derivatives
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Denition
Derivative of a function f at a point a is
f (a + h) f (a)
f (x) f (a)
= lim
xa
h0
x a
h
f (a) = lim
Dr. Tran Thai-Duong
Means and Improper Integration
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Means
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Lorentz curves
For 0 x 1, L(x) is the percentage of total
income earned by a group of lowest-paid people.
The propo
Areas and Volumes
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December 23, 2012
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Areas-Volumes
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Net Change
The derivative f (x) is the rate of change of f
when the variable is near x. The net change of a
quantity Q f