Maxima and Minima
Lecture 28
March 5, 2007
Lecture 28
Maxima and Minima
Second Derivative Test
Fact
Suppose the second partial derivatives of f are continuous on a disk
with center (a, b), and suppose that fx (a, b) = 0 and fy (a, b) = 0.
Let
f
f
D = xx x

Section P1
Algebra Expressions
and
Real Numbers
Algebraic Expressions
Algebraic Expressions are combinations of
variables and numbers using the operations of
addition, subtraction, multiplication, or division
as well as powers or roots.
Example:
73 = 7 7

Section P2
Exponents and Scientific Notation
The Product Rule
Example
Simplify:
( 7 x ) ( 9 x )
5
3
5 x3 y ) ( 4 x5 y 2 )
(
The Quotient Rule
Example
Simplify:
( 2 )
3
( 2 )
7
4
8 x y3
2 x
Example
Simplify:
48 x 5 y 3 z 2
2
16 x y
Zero as an Exponent
Exam

Section P3
Radicals and Rational Exponents
Square Roots
Definition of the Principal Square Root
If a is a nonnegative real number, the nonnegative number b
such that b 2 =a, denoted by b= a is the principal square root of a.
81 = 9
40 + 9 = 7
93
=
64 8
E

Section P4
Polynomials
How We Describe Polynomials
The Degree of ax n
If a 0, the degree of ax n is n. The degree of a
nonzero constant is 0. The constant 0 has no
defined degree.
Adding and Subtracting
Polynomials
Combine Like Terms
Example
Perform the i

Section P5
Factoring Polynomials
Common Factors
Factoring a polynomial containing the sum of monomials
mean finding an equivalent expression that is a product.
In this section we will be factoring over the set of integers,
meaning that the coefficients in

Section P6
Rational Expressions
Rational Expressions
A rational expression is the quotient of two
polynomials. The set of real numbers for which
an algebraic expression is defined is the domain
of the expression. Because division by zero is
undefined, we

Section 1.1
Graphs and Graphing Utilities
Points and Ordered Pairs
Plot points on the rectangular coordinate system. Plot the
x coordinate first then the y coordinate. This is an ordered
pair of real numbers. (x,y)
Plot the following points:
Example
(-4,3

Section 1.2
Linear Equations
and
Rational Equations
Solving Linear Equations in
One Variable
Definition of a Linear Equation
A linear equation in one variable x is an equation
that can be written in the form
ax+b=0
where a and b are real numbers, and a 0.

Section 1.3
Models and Applications
Problem Solving
with Linear Equations
Strategy for Solving Word Problems
Step 1: Read the problem carefully. Attempt to state the
problem in your own words and state what the problem is
looking for. Let any variable rep

Section 1.4
Complex Numbers
The Imaginary Unit i
The Imaginary Unit i
The imaginary unit i is defined as
i = -1, where i 2 = 1.
Complex Numbers and Imaginary Numbers
The set of all numbers in the form
a+bi
with real numbers a and b, and i, the imaginary u

Section 1.5
Quadratic Equations
Definition of a Quadratic Equation
A quadratic equation in x is an equation that can be written
in the general form
ax 2 + bx + x = 0,
where a, b, and c are real numbers, with a 0. A quadratic
equation in x is also called a

BMFG 1823
Chapter 4-Resultant of Non-Concurrent Forces
1.
The foot segment is subjected to the pull of the two plantarflexor
muscles. Determine the moment of each force about the point of
contact A on the ground.
2.
The 70-N force acts on the end of the p

VIDEO
Fabricating A
Toolbox
Teachers Notes
(Includes tooibox plan)
Duration: 20 minutes
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Chapter 4 Resultant of Non-concurrent Forces
Tutorial 4b
1. Determine the resultant moment produced by forces FB and FC about
point O. Express the result as a Cartesian vector.
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Chapter 4 Resultant of Non-concurrent Forces
Tutorial 4b
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Maxima and Minima
November 27
Lecture 28
Second Derivative Test
Suppose the second partial derivatives of f are continuous on
a disk with center (a, b), and suppose that fx(a, b) = 0 and
fy (a, b) = 0. Let
D=
fxx fxy
fyx fyy
= fxxfyy (fxy )2.
1. If D > 0

Chapter 6 Friction
Tutorial 6
1.
The coefficient of static friction between
the 180N crate and the ramp is s = 0.3.
What is the magnitude of the smallest
horizontal force the rope must exert on
the crate to prevent it from sliding down
the ramp?
Ans: (a)

STATICS AND STRENGTH OF MATERIALS
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SESI 2011/2012
BBBBBTBR B
Question 1
The planet Neptune has an equatorial diameter
of BBBBBB Bm and its mass is BBBBBB x BB BB BgB Bf the planet is
modelled as a homogeneous sphereB what is the
acceleration
due

The Oxford 3000 Word list
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Az albbi listt az Oxford University gondozsban, angoltanrok s egyb szakemberek kzremkdsvel lltottk ssze. Ezek szerint a szakemberek szerint
ezeket a szavakat mindenkinek ismernie kell, aki hasznlhat angol nyelvt

Chapter 2
1. The three supporting cables exert the forces shown on the sign. Represent each
force as a Cartesian vector.
2. Cable AB exerts a force of 80 N on the end of the 3-m-long boom OA. Determine
the magnitude of the projection of this force along t

Chapter 2
1. The three supporting cables exert the forces shown on the sign. Represent each
force as a Cartesian vector.
2. Cable AB exerts a force of 80 N on the end of the 3-m-long boom OA. Determine
the magnitude of the projection of this force along t

1
1. The screw eye is subjected to two forces, F1 and F2. Determine the magnitude and
direction of the resultant force.
(For the solution, please go to Example 2.1, page 25)
2. Determine x and y components of F1 and F2 acting on the boom. Express each for

Sheet Metal Design
Preface
What's New?
Getting Started
Basic Tasks
Workbench
Description
Customizing
Glossary
Index
Dassault Systmes 1994-99. All rights reserved.
Preface
The V5 CATIA - Sheet Metal Design is a new generation product offering an
intuitive

Section 1.6
Other Types of Equations
Polynomial Equations
A polynomial equation is the result of setting two
polynomials equal to each other. The equation is in
general form if one side is 0 and the polynomial on
the other side is in descending powers of

Section 1.7
Linear Inequalities
and
Absolute Value Inequalities
Interval Notation
Example
Express the interval in set builder notation
and graph:
( 3, 2]
[ 0, 4]
( , 2 )
Intersections and
Unions of Intervals
Example
Find the set:
( 2,3]
U [ 0,4 )
Example

Section 3.6
Polynomials and
Rational Inequalities
Solving Polynomial Inequalities
Example
Solve and graph the solution set on a real number
line: x 2 x 12
10
-5
0
5
10
Example
Solve and graph the solution set on a real number line:
x + x 17 x 15
3
2
10
-5

Section 3.7
Modeling Using Variation
Direct Variation
Example
The volume of a sphere varies directly as the
cube of the radius. If the volume of a sphere
is 523.6 cubic inches when the radius is 5
inches, what is the radius when the volume is
33.5 cubic i

Section 4.1
Exponential Functions
Example
The exponential equation f ( x ) = 13.49 ( .967 ) 1 predicts the number of O-rings
x
that are expected to fail at the temperature x o F on the space shuttles. The
O-rings were used to seal the connections between