3
Groups
83. Let a and b belongto some group. Sup se that lat = m,1b| = n,
and m and n are relatively prime. If = b for some integer k,
prove that mu divides k.
84. For every integer n greater than 2, prove that the group Ulr2 1)
is not cyclic.
85. Prove
Grou ps
15.
16.
17.
18.
19.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
Let G be anAbelian group and let H = cfw_g E G | Igl divides 12.
Prove that H is a subgroup of G. Is there anything special about 12
here? Would your proof be valid if 12 were replaced by
31.
32.
33.
37.
38.
39.
41.
42.
47.
d | CyclicGroups 89
Let G be a nite group. Show that there exists a xed positive inmger
n such that a = e for all a in G. (Note that n is independent of (3.)
Determine the subgroup lattice for 212-
Determine the subgrou
31%
69.
70.
71.
72.
73.
74.
75.
76.
81.
82.
d | CyclicGroups 9"
. Let a and b belong to a group. If lal = 24 and Ibl = 10, what are
the possibilities for Ka) n (JIM?
. Prove that U(2) (n a 3) is not cyclic.
. Suppose that G is a group of order 16 and that
Groups
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
61.
. Given the fact that U(49) is cyclic and has 42 elements, deduce the
For each positive integer n, prove that 0*, the group of nonzero
complex numbers under multiplication, has exactly 4:01) elements
ST 6523/4523 Introduction to Probability
Homework: Chapter 2
Solution
Problems
Q1: Solution: Let r, g, b denote the red marble, the green marble, and the blue marble,
respectively.
(a) With replacement. S = cfw_(r, r ), (r, g), (r, b), ( g, r ), ( g, g),
Chapter 3
Conditional Probability and
Independence
3.1
Conditional Probabilities
Definition 3.1. Suppose P (B) > 0, the conditional probability of event A given that event
B has occurred is defined as
P (A | B) =
P (A B)
.
P (B)
Example 3.1. Roll two fair
Chapter 1
Combinatorics
1.1
Basic Principle of Counting
Example 1.1. A team of one boy and one girl is to be made from a group of 5 girls and 2 boys.
How many different possible teams are there?
Solution.
Proposition 1.1 (The Basic Principle of Counting.)
Chapter 2
Axioms of Probability
2.1
Sample Space and Events
Definition 2.1. Consider a random experiment and let S be the set of all possible outcomes.
S is called the sample space.
Example 2.1. Toss a fair 6-sided die. What is the sample space?
Solution.
What Is Coming Up Due?
Appendix D
Trigonometric Functions
Conversion from Degrees to Radians:
Multiply by
180
Conversion from Radians to Degrees:
180
Multiply by
Ex 1: Convert 300 to radians.
5
Ex 2: Convert
from radians to degrees.
6
Ex 3: Find the lengt
What Is Coming Up Due?
Section 1.1
Functions
Can you define the following terms?
1. Function
2. Domain
3. Range
4. Independent Variable
5. Dependent Variable
6. Graph
Ex 1: Function f is graphed.
a. Find f (1).
b. Estimate the value of f (1).
c. For what
Appendix D Trigonometric Functions
To convert from degrees to radians: multiply by
180 .
To convert from radians to degrees: multiply by
180
.
Example: Complete the following conversions.
1. Convert 300 to radians.
2. Convert
5
6
from radians to degrees.
What Is Coming Up Due?
Section 3.7
Optimization
Ex 1: Find the dimensions of a rectangle with perimeter 100 m
whose area is as large as possible.
Ex 2: If 1200 cm2 of material is available to make a box with a
square base and an open top, find the largest
Section 2.4 Derivatives of Trig Functions
Extremely Important Limits
These two limits are EXTREMELY important to the following proofs for the
trigonometric derivatives.
Proofs for each one can be found on pages 141 and 142 in your textbook.
Example:
1. Fi
Section 1.5 The Limit of a Function
Example: Use the graph and/or table to guess the value of each function or limit.
1.
2.
3.
f ( x )=
lim
x 0
x1
x 21
sin x
x
lim sin
x 0
x
Left and Right Handed Limits
Example: Use the graph to determine each of the foll
What Is Coming Up Due?
Section 3.2
The Mean Value Theorem
Ex 1: Verify that the function satisfies the three hypotheses of Rolles
Theorem on the given interval. Then find all numbers c that
satisfy the conclusion of Rolles Theorem.
f (x) = x3 x2 6x + 2, [
What Is Coming Up Due?
Section 2.6
Implicit Differentiation
We dont need to solve an equation of y in term of x to find the
derivative of y.
Ex 1: We will solve using implicit differentiation first and then
compare the result we obtain by solving the equa
What Is Coming Up Due?
Section 3.9
Antidifferentiation
Lets look at these two functions:
f (x) = x2 7
g(x) = x2 + 5
n +1
x
Rule: xn has an antiderivative of
n +1
Ex 1: Find the most general antiderivative.
a. f (x) = x 3
b. f (x) = 7x2/5 + 8x-4/5
Ex 2: Fi
Section 1.1 Functions
From prior classes you should be able to define the following terms:
1.
2.
3.
4.
5.
6.
Function
Domain
Range
Independent Variable
Dependent Variable
Graph
Example: Identify the following items on the given graph of the function f .
1
Section 1.6 Properties of Limits
Limit Laws
Direct Substitution Property
If f ( x) is a polynomial or rational function and a is in the domain of f
lim f ( x ) =f ( a)
x a
so long as f ( x) is continuous at a .
*After substitution, if you get
Example: Fin
Section 2.3 Differentiation Formulas
Example: Use the definition of the derivative to find the following.
1.
d 2
(x )
dx
2.
d n
(x )
dx
First Few Derivative Rules
Power Rule:
d n
( x )=n x n1
dx
Constant Rule:
d
( x ) =1
dx
d
( c )=0
dx
Example: Find the
What Is Coming Up Due?
Section 1.2
More Examples of Functions
Types of Graphs
Linear: y = mx + b
Polynomials: anxn + an-1xn-1 + + a1x + a0
Power Functions: xa
Rational Functions: two divided polynomials
Algebraic Functions: algebra on polynomials
Trigonom
Section 2.5 Chain Rule
The Chain Rule
Example: Find the derivative using the chain rule.
1.
2 100
f ( x )=( 4 xx )
2.
3.
f ( x )=
2
( 1+ sec x )2
f ( t )=( 3 t1 )4 ( 2 t+1 )3
4.
5.
f ( t )=
( 2t +1 )3
5
( 4 t5 )
s 2 +1
g ( s )= 2
s +4
6.
f ( x )=5 cos 2 (
What Is Coming Up Due?
Section 2.8
Related Rates
In a related rates problem we try to compute the
rate of change of one quantity in terms of the
rate of change of another quantity, which may
be more easily measured.
Find the equation that relates the two
What Is Coming Up Due?
Section 2.2
The Derivative as a Function
The Derivative of a function f (x) is given by:
In the previous section we found the value of the derivative of f at a particular point a.
We learned that this value was equal to the slope of
Section 2.1 Derivatives and Rates of Change
Lines and Graphs and Tangents, oh my!
How would you find the slope of the line PQ?
If
x began to move to the left and approach a , the
secant would move.
Example:
1. Find the slope of the tangent line to the par
What Is Coming Up Due?
Section 2.1
Derivatives and Rates of Change
How would you find the slope of the line PQ ?
If x began to move to the left and approach a, the secant would move.
The line t is tangent to the curve at the point P.
Ex1: Find the slope o
Section 2.2 The Derivative of a Function
The Derivative of a Function f ( x) is given by:
In the previous section, we found the value of the derivative of f at a particular point a .
We learned that this value was equal to the slope of the tangent line at
Calculus I Test 1 Review Problems
From Section 1.1
Be able to evaluate the difference quotient of a given function and simplify your answer:
a) () = 4 + 3 2
b) () = 5 2 3 + 8
From Appendix D
Be able to solve a trig equation for all angles x in the interva