80
CHAPTER 2
Nonlinear Functions
vertically 3 units, as compared to the graph of y 5 22x. Since y 5 22x would have yintercept 1 0, 21 2 , this function has y-intercept 1 0, 2 2 , which is up 3 units. For negative
values of x, the graph approaches the line

Parametric Equations and Polar Coordinates
In this section , we extend the concepts of calculus to
curves described by parametric equations and polar
coordinates. For instance, in order to study the motion of
an object such as an airplane in two dimension

Basic Differentiation Formulas
In the table below, ? oe 0 B and @ oe 1B represent differentiable functions of B Derivative of a constant Derivative of constant multiple Derivative of sum or difference Product Rule Quotient Rule
.B . .B . .B
oe ! (-?) oe -

774
CHAPTER 9
.
Parametric Equations and Polar Coordinates
9-60
y2
x2
+
= 1,
35. In example 6.9, if the shape of the reector is
124 24
how far from the kidney stone should the transducer be
placed?
the spectator. If the objects distance from the spectator

764
CHAPTER 9
.
Parametric Equations and Polar Coordinates
problem has parametric equations x (t ) = r (t ) cos (t ) and
y (t ) = r (t ) sin (t ). Graph the solution in the cases
(a) a = 1, r0 = 1 , 0 = 0; (b) a = 1, r0 = 3 , 0 = 0; (c) your
2
2
choice of

786
CHAPTER 10
.
Vectors and the Geometry of Space
10-2
As you will see in the exercises in section 10.3, the banking of a road changes the role of gravity. In effect, part of the weight of the car is diverted into a force that helps the car make its turn

742
CHAPTER 9
.
Parametric Equations and Polar Coordinates
9-28
EXPLORATORY EXERCISES
1. For the brachistochrone problem, two criteria for the fastest curve are: (1) steep slope at the origin and (2) concave down (note in Figure 9.16 that the positive y-a

716
.
CHAPTER 9
9.1
Parametric Equations and Polar Coordinates
9-2
PLANE CURVES AND PARAMETRIC EQUATIONS
We often nd it convenient to describe the location of a point (x, y) in the plane in terms of
a parameter. For instance, in tracking the movement of a

9-41
SECTION 9.5
9.5
.
Calculus and Polar Coordinates
755
CALCULUS AND POLAR COORDINATES
Having introduced polar coordinates and looked at a variety of polar graphs, our next step
is to extend the techniques of calculus to the case of polar coordinates. I

www.mathportal.org
Integration Formulas
1. Common Integrals
Indefinite Integral
Method of substitution
Integrals of Exponential and Logarithmic Functions
ln x dx = x ln x - x + C
n x ln x dx =
f ( g ( x) g ( x)dx = f (u )du
Integration by parts
x n +1 x

IIUM, Faculty of Engineering,
Department Engineering in Science
Engineering Calculus II, MTH 1212,
Chapter I.1: Parametric Equations
I. PARAMETRIC EQUATIONS & POLAR COORDINATES
1.1
1.2
1.3
1.4
1.5
Curves defined by parametric equations;
Tangents;
Area. Ar

IIUM, Faculty of Engineering,
Department Engineering in Science
Engineering Calculus II, MTH 1212,
Chapter I.3: Area, Arc Length & Surface Area
1.3 AREAS, ARC LENGTH AND SURFACE AREA IN PE
a) AREAS
In this section, we investigate areas defined parametrica

Vectors and Geometry in Space
To locate a point in a plane, two numbers are
necessary. We know that any point in the plane can be
represented as an ordered pair ( a, b) of real numbers,
where a is the x-coordinate and b is the y-coordinate.
For this reaso

IIUM, Faculty of Engineering,
Department Engineering in Science
Engineering Calculus II, MTH 1212,
Chapter I.4: Polar Coordinates
1.4 POLAR COORDINATE
A coordinate system represents a point in the plane by an ordered pair of numbers
called coordinates. Us

IIUM, Faculty of Engineering,
Department Engineering in Science
Engineering Calculus II, MTH 1212,
Chapter II.3: L&P in space. Quadric Surface
2.3 LINE and PLANES in SPACE. QUADRIC SURFACE
The Lines and Planes in Space:
We now take the question of finding

Problem F12-2 (Page 15)
A ball is thrown vertically upward with a speed of 15 m/sec. Determine
the time of ight when it returns to its original position.
dZV 15:675.
:v( : aft
1 v 1
fdt :[aw o _
O 16
_ x- <13
t \/ cfw_b ~O at
[(ISl'dt ~/dz
2 O
t : is). 1

Jim Lambers
MAT 169
Fall Semester 2009-10
Lecture 29 Notes
These notes correspond to Section 9.1 in the text.
Parametric Curves
There are many useful curves that cannot be described by an equation of the form = (), because
is a function and therefore req

IIUM, Faculty of Engineering,
Department Engineering in Science
Engineering Calculus II, MTH 1212,
Chapter II.2: Vectors & Geometry in Space
2.2 DOT and CROS PRODUCT of VECTORS
The Dot Product:
The vector versions of length, distance, and angle can all be

IIUM, Faculty of Engineering,
Department Engineering in Science
Engineering Calculus II, MTH 1212,
Chapter II.1: Vectors & Geometry in Space
2 VECTORS and GEOMETRY in SPACE
2.1
2.2
2.3
2.43
Vectors in plane and space. Definition. Application;
Dot ant Cros

IIUM, Faculty of Engineering,
Department Engineering in Science
Engineering Calculus II, MTH 1212,
Chapter I.2: Caculus & Parametric Equation
1.2 CACULUS AND PARAMETRIC EQUATIONS. TANGENTS, AREA, ARC
LENGTH & SURFACE AREA.
TANGENTS
a)
Slope
b)
Tangent lin

726
CHAPTER 9
9.2
.
Parametric Equations and Polar Coordinates
9-12
CALCULUS AND PARAMETRIC EQUATIONS
The Scrambler is a popular carnival ride consisting of two sets of rotating arms (see
Figure 9.8a). Suppose that the inner arms have length 2 and rotate

Descriptions of the Product
Associate Prof. Pham Huu Anh Ngoc
Department of Mathematics
International university
May, 2015
Other description of the product of a matrix and a
vector
Example Let
1
A= 2
1
0
2
0
1
0 ;
1
1
x= 0
1
Normally,
c1
(1 1) + (0 0

Calculus 2-BA
Instructor:
Associate Professor Pham Huu Anh Ngoc
E-mail: phangoc@hcmiu.edu.vn
Room: A2-610
2015
Course Agreements
Grading
The grading of the course will be calculated as follows:
Midterm Test: 20%
Home Work/Class Work Assignments: 20%
Final

Chapter 1: Finance Mathematics
Lecture 2: Continuous Money Flows
by Associate Professor Pham Huu Anh Ngoc
Motivation
Question: Given a changing rate of annual income and a certain rate of
interest, how can we nd the present value of income?
Earlier, we co

Finance Mathematics
Lecture 3: Annuities
by Associate professor Pham Huu Anh Ngoc
Motivation
Example
Suppose $1500 is deposited at the end of each year for the next 6 years
in an account paying 8% per year, compounded annually. How much will
be in the acc

Calculus 2-BA
Chapter 3: Dierential Equations
Associate Professor Pham Huu Anh Ngoc
Department of Mathematics, IU
What is a dierential equation?
What is a dierential equation?
Roughly speaking, a dierential equation is an equation that involves
an unknown

Chapter 3: Finance Mathematics
Exercises
by Associate Professor Pham Huu Anh Ngoc
Simple Interest and Compound Interest
Problem 1: You want to invest $8000 for 3 years. Which account should
you choose?
a) Account A earns 4% simple interest per year.
b) Ac

Calculus 2-BA
Lecture 3&4: Maxima and Minima, Total Dierentials
April, 2015
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 is given by
P(x, y ) = x

Determinant of square matrices
Associate Professor Pham Huu Anh Ngoc
International University
April 22, 2015
Return
Motivation
When does the linear system
ax + by
= e
cx + dy
= f
have a unique solution?
Solution 1: Multiply the rst equation by d and the s

FINAL REVIEW A (MAT132)
For what value(s) of k does the circle x 2k 2 ( y 3k ) 2 10 passes through the point (1,0).
Find the equation for the circle with the center at the mid-point of A(x,3) and B 1,1 and
the
radius 3 given that the distance between AB i