Polyas Problem Solving Techniques
In 1945 George Polya published the book How To Solve It which quickly became
his most prized publication. It sold over one million copies and has been translated
into 17 languages. In this book he identies four basic prin
MATH 2412
Test No. 3 A (Total: 70 points)
Name: _
April 16,
SHOW ALL WORK FOR FULL CREDIT.
A. (5 points) Represent x y
B.
1 0
in the plane (i.e., draw the vector x y ).
1 3
1
and a = 2. Represent x and a x in the same
3
(4 points) Suppose x
plane.
C
MATH 2412
Test No. 2 A (Total Points: 50 )
Name: _
March 19,
SHOW ALL WORK FOR FULL CREDIT.
A.
(5 points) Solve the differential equation (by separation of variables):
dy
3 x , with
dx
x =1 when y =2
B. (4 points) Assume that the Leslie matrix for the pop
MATH 2412
Test 1 A (Total: 55 points)
Name: _
February 12,
SHOW YOUR WORK FOR FULL CREDIT.
A. (15 points) Evaluate the following integrals.
1]
4 x
e
x
dx
3
2
x
2]
3]
x sin x dx
x
dx
B. (15 points) Evaluate the following integrals.
Page 1 of 4
MATH 2412
Calculus Exam III Review
Quiz 7:
A Find the eigenvalues (only) of the rotation matrix
[
1 2
B The eigenvalues of B=
2 1
=1.
]
[
]
R90 = 0 1 .
1 0
are =3=1. Find an eigenvector corresponding to
Quiz 8:
A Find a vector
y which is perpendicular to
[ ]
x = 2
Math 1306
Test 3 Review
Session: Tuesday April 26
Time: 11:30-1:00pm
SI Leader: Fernando
Location: S-405
1.
f
The graph of
is given below:
Determine if f ' ' ( x ) >0 , f ' ' ( x ) <0 , or f ' ' ( x )=0 at:
a.
x=2
b.
x=1
c.
x=0
d.
x=1
e.
x=2
2. True or Fa
Section 2.4 USE TABLES TO ESTIMATE THE RATE OF CHANGE 0 w +;
LIMIT DEFINITION OF 0 w +
NOTE
Recall that 0 w B is the instantaneous rate of change of 0 B with respect to B: it tells
you the change in 0 B when B increases by " unit. In the Apple net sales e
4.5 INTRODUCTION TO MARGINAL ANALYSIS
NOTE
COST FUNCTIONS
Every company has costs in their business. In particular, understanding the cost to
produce a certain number of items will help a company decide pricing and production
levels. Usually there are fix
4.4 SECOND DERIVATIVES AND INFLECTION POINTS
NOTE
SECOND DERIVATIVES
Given a function 0 B, we have learned to find the rate of change function 0 w B,
which is also called the first derivative of 0 B.
We will now learn that it is meaningful to consider the
Section 1.1 Review of Functions
In everyday life, a person may say that one quantity is a function of another
quantity. For example, we might read that a person's income is a function of the
person's education level. This means that a person's income depe
4.2 EXTREME VALUES OF A FUNCTION
In this section, we will focus on the turning points of a function graph, and on the highest and
lowest points on the graph.
1. At a turning point - 0 - which is a hill point, its C-value is called a local maximum or
relat
Section 1.4 LINEAR FUNCTIONS HAVE A CONSTANT RATE OF CHANGE
NOTE
A linear function is the simplest of all functions. A linear function can be described in
terms of a starting point and a value added at regular input intervals. The output of a
linear funct
Section 2.5 & 3.1 THE DERIVATIVE 0 w B; SOME DERIVATIVE
FORMULAS
THE RATE OF CHANGE 0 w +
In section 2.4, we studied the limit definition of the rate of change 0 w + and learned
to calculate its value by using the 4 or 5 step rule.
LIMIT DEFINITION OF THE
Section 2.2 & 2.3 THE RATE OF CHANGE 0 w +
INSTANTANEOUS VELOCITY EXAMPLE
Let's consider two similar, but very different situations.
Case 1.
Michelle travels a distance of #%! miles by car in % hours.
The average velocity of the car during this time is ca
Section 1.2 FUNCTION BEHAVIOR
INTERVAL NOTATION
There is a shorthand notation called interval notation that is useful to describe sets
of real numbers. It is described in the following table.
In particular, when using interval notation, note that the smal
Name Answer Key
MATH 1306 CLASS ACTIVITY # 4 (Section 2.1, 2.2, 2.3)
1. (5 Pts) Find the average rate of change of 0 B $B#
work.
% from B " to B "
We can rewrite the formula as 0 $ #
%. Then
0 " $"#
%(
0 " 2 $" 2#
$" #2 2# %
$ '2 $2# %
$2# '2 (
% $"
2"
Math 1324 Section 8.2 Union, Intersection, and Complement of Events
Homework: p. 408 #s 13 37, 61 65, 79 - 83
Basic Ideas:
Mutually exclusive A and B are mutually exclusive events if A and B cannot occur at the same time.
Addition Rule P(A or B) = P(A) +
Math 1324 Section 7.3 Basic Counting Principles
Homework: p. 366 #s 11 31, 35 41, 45 51, 57, 59, 65
Basic Ideas:
Multiplication principle for counting: if one event can occur in m ways and a second event can occur in n
ways, the number of ways the two eve
Math 1324 Chapter 2 Part I
2.1 HW: p. 54 #s 75 - 79
Basic Ideas:
Difference Quotient:
The difference quotient for a function is given by
f ( x h) f ( x )
h
.
Examples:
Find and simplify the difference quotient for each function.
(a)
(b)
f ( x) 8 3x
f ( x)
Math 1324 Sections 2.5 and 2.6
Homework: p. 104 #s 29 39; p. 115 #s 1 31, 43 49, 61 - 65
Basic Ideas:
Logarithmic functions have the form y = logax, where x = ay, x > 0, a > 0, and a 1.
Common logarithm: log x means log10x
Product Rule:
log a MN log a M l
Math 1324 Section 8.1 Sample Spaces, Events, and Probability
Homework: p. 396 #s 7 25, 33- 37, 41 55, 79 85, 93, 97
Basic Ideas:
Probability experiment trial through which results are obtained.
Outcome the result of a single trial.
Sample space set of all
Math 1324 Section 7.4 Permutations and Combinations
Homework: p. 378 #s 31 39, 43 51, 63, 65, 71 - 75
Basic Ideas:
Factorial: For a natural number n,
n! = n(n 1)(n 2) . . . 21
0! = 1
Permutations: If we make r selections from a group of n items where repe
Math 1324 Section 8.5 Probability Distributions
Homework: p. 438 #s 7 13, 17, 25, 31, 35, 39, 41, 49, 51
Basic Ideas:
Random variable represents a numerical variable associated with each outcome of a probability
experiment.
Discrete probability distributi
Math 1324 Section 7.2 Sets
Homework: p. 359 #s 15 47, 73, 75 - 87
Basic Ideas:
A set is a well-defined collection of objects. The objects in the set are called the elements of the set.
The notation
a A
means a is an element of A. The notation
a A
means a
Math 1432
Exam 3 Review
1. Integrate:
3x 2 + 3x + 3
a.
dx
x2 +1
x2
b.
dx
( x + 1)( x 1) 2
c.
d.
x 2 + 5x + 2
( x + 1)(x 2 + 1) dx
x
2x 2
9 x2
2
dx
dx
9 + x2
5
f.
dx
36 + ( x 1) 2
1
g.
dx
2
4+ x
5 x + 14
h.
dx
( x + 1)( x 2 4)
e.
i.
3
2
0
j.
k.
l.
m.
Math 1432
Sequences & Series Worksheet
(n + 2)25 2 n
1. Given an =
, find lim an .
n
2 n+3 (n + 5)25
2. Determine if the sequence cfw_an converges when an =
3n 2 (2n 1)!
. If it converges, find
(2n + 1)!
the limit.
12n 2 4n 2 + 1
3. Determine if the