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
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
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) =
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
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
MIN/MAX
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MIN/MAX
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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
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
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x+
Numbers and Functions
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Numbers and Functions
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Intervals
Open interval (a, b) = cfw_x R|a < x < b
Dr. Tran Thai-Duong (IU HCMC)
Numbers and Functions
Febru
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
Continuity
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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
Approximation
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Approximation
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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
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
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Contents
1
Areas under
CALCULUS I
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Calculus I
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Textbook
James Stewart
Calculus, 6th edition
Early Transcenentals
1 FUNCTIONS AND
MODELS
2 LIMITS AND DERIVATIVES
3 DIFFERENTIATION
Convergence
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Convergence
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x
We say f (x) converges to L as x and write
lim f (x) = L
x
if, for every > 0, there exists a number M such
that |f (x) L| < for a
Areas and Volumes
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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
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
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
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
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
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Contents
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
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
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
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
Numbers and Functions
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Numbers and Functions
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Intervals
Open interval (a, b) = cfw_x R|a < x < b
Dr. Tran Thai-Duong (IU HCMC)
Numbers and Functions
Septe