Arithmetic Series Notes
series: an indicated sum of the terms of a sequence
EX 1: Consider the sequence an = 3n 2
a1 = 1
a2 = 4
a3 = 7
a4 = 10
a5 = 13
If you add the terms together, you have a series.
5
finite series:
sn = a1 + a2 + a3 + a4 + a5
OR
a
n =1
Limits of Sequences
EX 1: Write out the first 10 terms of the sequence an =
1
n
(HARMONIC SEQUENCE)
1, 1/2, 1/3, 1/4, .
Examine its end behavior.
Suppose n = 100, 1000, 100 000, etc. What does the an value (a y-value on a graph)
seem to approach?
In other
Pre-Calcuius | QtrZ Review Packet
Non-Caicuiator Exercises Ch 5 3 6 9 & Factormg
A. Answer each of the foiiowing
1. 1
" . The scale change S(x,y) > [x,5y] is applied to the function f(x) = 2):3 +x , find the equation of the
2' '3 r '
«ill 2 ag; at m
Multiplication Counting Principles Notes
MULTIPLICATION COUNTING PRINCIPLES:
The number of ways to choose one element from A and one element from B is N(A) * N(B).
Ex. You have a menu with 3 choices for beverages, 6 choices for entrees, and 4 choices for
7-4 Permutations
SELECTIONS WITHOUT REPLACEMENT THEOREM can be rewritten as PERMUTATIONS
THEOREM: There are n! permutations of n different elements.
*WE USE PERMUTATIONS WHEN ORDER OR POSITION MATTERS*
ORDER MATTERS
Ex. Top 5 students in the class of 2008
Probability Distributions
EX 1: Look at the probabilities when you toss two die and look at the sum of the
numbers:
x=
sum of
dice
P(x) =
2
3
4
5
6
7
8
9
10
11
12
1/36
2/36
3/36
4/36
5/36
6/36
5/36
4/36
3/36
2/36
1/36
x is a random variable its values are
Addition Counting Principles Notes
In order to develop a sample space for a probability problem, you have to accurately
count the items there.
Two sets:
disjoint / mutually exclusive = nothing in common
A I B = intersection of the two sets elements in bot
Basic Principles of Probability Notes
experiment: a situation that has several possible results
outcomes: results of an experiment
sample space the set of all possible outcomes
event: subset of the sample space.
Probability of an event: a number from 0 to
Combinations
Combination = collection of objects in which ORDER DOESNT MATTER
n choose r
Cr
n
n
1
= ( n Pr ) =
r r!
=
n!
(n r )!r!
Ex. At a restaurant, you can order pizza with any of 9 different toppings. How many
different pizzas are possible with exa
Formulas for Sequences
sequence: a function whose domain is the set of consecutive integers greater than or
equal to k. (usually k = 1).
In other words, an ordered list of numbers that follow a certain pattern
*formulas can be explicit or recursive
*Seque
Inverse Sine, Cosine, and Tangent Functions Notes
Graph y = sin (x). Notice it fails the HLT, so the inverse of the sine is not a function.
WE NEED TO LOOK AT ONLY A PART OF THE GRAPH OF THE
FUNCTION (RESTRICT THE DOMAIN ) SO THAT IT WILL PASS THE
HORIZON
PRECALC | CHAPTER 7 STUDY GUlDE
Mswuzsl
1. A school consists of 330 freshmen, 270 sophomores. 350 juniors and 295 seniors.
a. How many students are in the sample space? I 5
b. If one student is selected at random, what is the probability that the st
Conic Sections: Circles Notes
Circle the set of all points at a given distance from a central point
CENTER = (h, k)
RADIUS = r
Equation for circles:
r2 = (x h)2 + (y k)2
h, k, r are all NUMBERS
x, y are VARIABLES
*If it helps you to remember, this equatio
ELLIPSES Notes
( x h)2 + ( y k )2 = 1
a2
b2
(h, k) = center
a = how far to count out horizontally
2a = length of horizontal axis
b = how far to count out vertically
2b = length of vertical axis
If b > a, then the ellipse lies vertically.
If a > b, then th
Geometric Series
3 1 + 3 2 + 33 + 3 4 + 35
EX 1: Consider the sequence
This is a finite geometric series
(3, 9, 27, 81, 243)
s n = g1 + g 2 + g3 + g 4 + g5
g1 = 3
sn =
r=3
g1 (1 r n ) g1 (r n 1)
=
1 r
r 1
n=5
5
So find
3(3)
n 1
sn =
n =1
3(35 1) 3(242)
=
Conic Sections: HYPERBOLAS Notes
( x h) 2 ( y k ) 2
=1
a2
b2
OR
( y k ) 2 ( x h) 2 = 1
b2
a2
*lies horizontally
*lies vertically
*opens to the left and right
*opens up and down
WHICHEVER VARIABLE COMES FIRST AFFECTS DIRECTION OF HYPERBOLA!
(h, k) = center
PARABOLAS:
There are 2 ways an equation for a parabola can be written:
I. y = a ( x h ) + k
2
or
( y k ) = a ( x h)
Axis of symmetry:
( h, k )
Vertex:
2
x=h
If a is POSITIVE, then happy face parabola
IF a is NEGATIVE, then sad face parabola
II. y = ax 2 +
Independent Events Notes
INDEPENDENT EVENTS the result of one event does not affect the future results
Events A and B are independent if and only if P ( A B ) = P ( A) P ( B )
Ex. Roll 2 dice
A = first die is a 2
B = second die is a 5
P ( A B ) = 1/36
P(A