Mathematics (Pg: All )

By:ShaDe’ Phoenix

Are you ready to have your brain filled with knowledge?

Take notes; this course gets really complex and you will be required to answer questions after every section. I will choose at random.

Please ask billions of questions if you do not understand something.

Let’s start with the basics (Pg: 5-22)

Multiplication is an easier form for repeated addition

Instead of 3+3+3+3 you could say 3*4

The Multiplication Principle

When listing the possibilities for k items, the total number of entries in this list is given by n sub 1*n sub 2* n sub 3….*n sub kEx: in this definition, n sub 3 would be the number of possibilities for the third item

(Refer to example 1.1e)

Permutations

An arrangement of objects from a group where NO object can be used more than once and the ORDER of selection MATTERS.

Keep in mind with and without replacementwithout replacement: factorialswith replacement: exponents

Permutation formula: n!/(n-k)! k being the number of objects in the list that you want; n-k is you canceling out all the other objects

Factorials

The symbol ! read aloud (pertaining to mathematics) means factorials

Factorials are the product of all whole numbers starting from the number indicated down to one

Ex: 5! is 5*4*3*2*1 which equals 120

Combinations

An arrangement of objects from a group where NO object can be used more than once and ORDER of selection DOES NOT matter

Combinations formula: (n!/(n-k)!)/k!

Notation and Calculator input

Notation: (n over k) which is read n choose k

Calculator input type your n

click math arrow to your right to the fourth column, select either nPr (for permutation) or nCr (for combination) *(permutations only work for without replacement problems)*type your k press enter

Algebra (Pg: 23-77)

Again, we'll start off with something basic

Sequences Sequence- a list of objects presented in a

particular order-The objects in a sequence are called the terms of the sequence (usually denoted by the variables a or x)

-The terms of sequences aren't always numbers - they can be shapes or even words

-The position of a term in a sequence is called the index (usually denoted as i or k) of the term (written as the subscript of the term)

-Usually, the relationship between the index and the term of the sequence becomes apparent

Based on the pattern 2,4,6,8,10 what would you guess the next 3 numbers in this

sequence would be?

What about the next 1 in this sequence? Monday Tuesday Wednesday

Nice sequences: can be extended in a logical manner

Mathematically writing sequences { } tell you that what you are about to solve

is a sequence not an equationEx: {x sub i=2i}

when i is 1 the term (or x) is 1*2 or 2 and so on Now you try. Give me 4 terms of the

sequence {x sub i=i^3}-The name of this sequence is the sequence of cubed numbers

Now give me the sequence that will generate these numbers 3,6,9,12

What about 3,7,11,15,19?

Finite sequences

Finite sequences do not go on forever … so how do you limit a sequence?

Arithmetic and Geometric Sequences

Recursive formula

(in terms of a sequence) is a formula that declares the starting value or values for the sequence and how the other terms (subsequent terms) are made from the previous terms.-The formula alone only shows how to get from one term to the next and thus can not describe the sequence alone

The recursive formula for the sequence 3,6,12,24,48 would be x sub 1=3; x sub i=x sub x-I * 2 (direct your attention to the board for clarity)

Direct formulas

Generate a sequence directly

-Like x sub i=4i-1 . If you plug in any number for the index you can find the term

-whereas with a recursive formula you need the previous term in order to find the next term

Want to try?

What are the terms of the sequence given by the recursive formula:

A sub 1=1A sub 2=1;A sub i= a sub i-1+ a sub i-2 ? First ask yourself, “What is this problem

saying?”

Arithmetic Sequence A sequence where a fixed amount is

added to move from one term to the next

-there is a constant difference between consecutive terms

-The constant difference can also be negative (it will still remain arithmetic) The constant difference of an arithmetic sequence is usually represented by the

variable d The first term in the sequence is usually represented by the variable a

Recursive arithmetic formula: x sub i=a; x sub i=x sub i-1 + d

Direct formula: x sub k= a+(k-1)*d

Now apply it What are the next 3 terms in the arithmetic

sequence 41,38,35,32? What is the 201st term in the arithmetic

sequence 41,38,35,32?-Think: what key words tell you which formula to use?

What is the first term of the sequence with x sub 54= 136 and x sub 77=205?

Geometric Sequences A sequence with a constant ratio (usually

represented by the variable r) between consecutive terms -This tells you that you will now being using multiplication or division -The first term for a geometric sequence is usually represented by the variable a

What is the ratio for the geometrical sequence 36,12,4,4/3?-Think: how are you getting from one term to the next?

Recursive formula: x sub 1=a; x sub i=r*x sub i-1

Direct formula: x sub k= a*r^k-1

Try it out for size What are the next 3 terms for the

geometric sequence 2,4,8,16,32? What is the 20th term for the geometric

sequence 2,4,8,16,32? What are the first 6 terms of a geometric

sequence when the 2nd term is 2 and the 6th term is 8?

Series

The sum of the terms in a sequence

Solving Arithmetic Series

What is 1+2+3…+97+98+99+100?

Think: “Work smarter, not harder”

Notice that if you form pairs they all have a sum of 101 (1+100, 2+99, 3+98, etc.) and there are 50 pairs (100/2). So, the sum can be found by multiplying the common sum by the number of pairs (101*50).

However, if there is not a common sum among the pair, this process will not work

Solving Arithmetic Series with an Odd Number of terms Ways to do so:

Add an extra term to the sequence (giving it an even number of terms) and then subtract the term you added at the end Omit a term and then add it to the endTry to figure out the middle term that has no pair

Given this knowledge, how would you go about solving the arithmetic series 41+38+35+…+(-1)+(-4)+(-7)Hint: This series has a constant difference of -3 and 17 terms

You could solve that problem by adding a term (-10) to make 9 pairs and multiplying that but the common sum (41+-10=31, 31*9=279) then subtracting the term you added (279- -10 or 279+10)

But, there is a simpler way

Formula

DIRECT YOUR ATTENTION TO THE BOARD

Given what you have learned…

Solve: “An arithmetic series equals 624. This first term is 3, and the second term is 5. What is the last term in this series?”

Hint: Keep in mind the situation

Geometric Series

S (sum)= a*r^k-a / r-1a- the first term

r- the constant ratio between consecutive termsk- the number of terms

If you don’t have know the number of terms the equation would be S (sum)= x sub k+1 - x sub 1 / r-1 X sub k+1- the term after the last term in the series (this term is not included in the series)

x sub 1- the first term in the series r- the constant ratio between consecutive terms

This method would not work if the ratio was 1 (1-1=0, any number divided by 0 is undefined)

Infinite series

If an arithmetic series has an infinite number of terms, the series will equal positive or negative infinity depending on the constant difference (d). If d > 0 the series will equal positive infinity. If d < 0 the series will equal negative infinity.

Infinite Geometric Series

An infinite geometric series equals a finite number if x sub k (the “last term”) is approximately equal to 0, which would mean that the ratio would have to be causing the terms in the series to decrease (like r= ½)

Formula: S (sum)= a/1-r a- the first termr- the constant ratio between consecutive terms

Example: The value of the infinite geometric series 2, -4/3, +8/9, -16/27 is 6/5 because:the first term is 2the ratio is -2/3, meaning the values of each individual term, gets smaller and smaller as the sequence goes on, meaning x sub k is approximately equal to 0

Sigma Notation- a lot of board work

Mathematicians use this to write out complicated/ long sums in shorthand

Write the following in sigma notation 1+4+9+16+…+196+225+256

Formulas

Direct attention to board

Polynomials

Definition- an algebraic object consisting of terms

Each term is made up of a variable (usually x) which is raised to a non-negative

power and has a coefficient.

The highest power of x with a non-zero coefficient is called the degree of the polynomial

Knowing this, what is the degree of 3x^3+7x^2+5 ?

Polynomials can still be thought of as a series and written in sigma notation

Adding and Subtracting polynomials Combining like terms What sequence occurs if the following 2

sequences are added:

(2,4,6,8,10)+(5,10,15,20,25) Subtract the following polynomial:

(6+2x-2x^2+-6x^3)-(3x+12x^2+21x^3)

Multiplying Polynomials

I am sorry, I don’t like the method in the book. So, I’ll teach the book method and foiling.

What is (x^2+5x-6)*(3x^3-7x^2+4)The book: since there are 3 terms in each quantity you can tell that you will be looking at 9 terms (3*3) The highest power will be x^5 which will result from x^2*3x^3 and generate 3x^5the coefficient for x^4 will be generated by x^2*-7x^2=-7x^4 and 5x*3x^3=15x^4the terms that will generate x^3 are -6*3x^3=-18x^3 and 5x*-7x^2=-35x^3the terms that will generate x^2 are x^2*4=4x^2 and -6*-7x^2=42x^2The terms that will generate x are 5x*4=20xThe terms that will generate the constant term are -6*4=-24

When combined and simplified, you get the answer of 3x^5+8x^4-58x^3-38^2+20x-24

Now try it with either method

(x-2)(x^2*3x*3)(2x*-4)

Think: how many terms should you have?

The Binomial Expansion Theorem

Definition- taking a binomial to a power Binomials-polynomials with 2 terms

Let’s start simple: 2(5x-3)^2

What about (x+2)^3 ?

What about (x+1)^10?

That last one would take forever !

So, instead we will represent it in sigma notation

Try writing (4x+2)^20 in sigma notation

Formula

When expanding a binomial that is in the form (x+y)^n realize that each term in this expansion is in the form (n choose k) * x^k *y^n-k

Where:

x- is one term

y- is another term

n- the power the binomial is being raised to

k-the power the first term is being raised to

Apply it

What is the coefficient of z^5 in the expansion (z-2)^10?

Compound interest

Simple interest- fee fixed per time period Compound interest- interest charged on

the interest previously charged

Try it out!

Mrs. Mal borrows $1800 dollars from the bank to pay for her trip to Moscow. She is charged 5% annual interest but interest is compounded every quarter. Assuming Mrs. Mal does not pay back any of the money along the way, how much money will she owe after 2 years?Think: What is annual interest? How often is quarterly?

Formula/Apply it

Honey Boo Boo invests $1,250 in a certificate of deposit (CD) that earns 3.7% interest compounded monthly. When she goes back to the bank in 4.5 years and collects her money, how much will she have earned?

Annuities

Megan invests in an annuity that earns 3.6% interest compounded quarterly. If Megan deposits $160 a quarter (and does not withdraw any money from her account), how much money will she have after 15 years?

Solution

Only the first $160 deposited is in the account for all 60 quarters, which means it is compounded 59 times; from there, the amount of times each $160 is compounded decreases until the last $160 added is not compounded at all. This is an example of a sum of a geometric series and can be written using that formula.

Annuity Formula/Apply it

ShaDe’ invests $1200 a year in a retirement fund that earns 4% annual interest for 43 years. How much money does ShaDe’ have in her fund when she retires?

How much did she invest? How much did she earn?

Loans

Samantha Taylor borrows $150,000 at 7% annual interest compounded monthly to buy a house. If Samantha takes out a 20 year mortgage, what will her monthly house payment be?

How much will she end up paying the bank?

Now that you know the formula…

Tanairy is ready to buy a house. She feels she can afford a $1000 monthly mortgage. If current mortgage interest rates are 6% and she is interested in a 25 year loan, about how much money can Tanairy expect to be able to borrow?

Euler’s Constant

e (the natural number)- the number that (1+(1/n))^n approaches as n increases to infinity. The decimal approximation for e is 2.718281828

100 bacteria are placed in a Petri dish, and after 3 hours there are 185 bacteria. Assuming a continuous growth model, how many bacteria will there be in 8 hours?

Statistics (Pg: 78-148)

Ready or not here we go!

Descriptive statistics

Aimed towards describing a set of data without listing every data point

Mean

The average Find the sum of all the data values and then

divide by the number of data points in the set An outlier (a value that is significantly different

from the rest of the data set) can cause the mean to be potentially inaccurate.

Example: what is the mean of the data set 2,3,5,7,900,6,10,5,1 and why is it so large when most of the data is below 10?

Median

In a data set with an odd number of values, the median is the middle number. In a data set with an even number of values, the median is the mean of the 2 middle numbers.

Definition- the value with the same number of data points above as below the value.

In order to properly find the median, the data points must be organized from the smallest (in value) to the largest

Mode

The number that occurs most often in a data set

If each value in the data set occurs once, the set has no mode

Bimodal- when a data set has 2 numbers that occur the same number of times (and occurs the most in that set)

Try it out

Serenity would like to average a 92 in Decathlon. So far she has 3 grades in the class, which are 100,80,90. Mrs. Mal is only putting 3 more grades in before report cards. On average, what must Serenity get on those 3 assignments in order to get a 92 in the class?Think: Which measure of central tendency do you need to use?

The mean and median of a finite arithmetic sequence are always equal

Because.. The difference in each consecutive term is constant, making the series symmetrical on each side of the median

Range

The difference between the largest and smallest value in the data set

The range roughly shows how spread out the data is but could be thrown off by outliers (so, the range can not be the only method of showing how spread out data is)

What is the range of the data set {2,5,16,36,13}

Quartiles

A measure of how spread out data is The lower quartile or the first quartile is the median value of the date below the median in a setThe upper quartile or the third quartile is the median value of the data above the median in a set

25% of the data falls between the first data point and the lower quartile, 25% falls between the lower quartile and the median, 25% falls between the median and the upper quartile, and 25% falls between the upper quartile and the last data point

What numbers in the data set {15,15,16,17,17,18,18,19,20,20,20,21,22,22,23,23,24,24,26} make up the upper quartile? What is the median of the upper quartile?

Interquartile range (IQR)

The difference between the lower quartile and the upper quartile It is not susceptible to outliers in the same way that the range is

IQR test for outliers:x sub i>Qsub3+1.5*IQRx sub i<Qsub1-1.5*IQR

where:x sub i- a data point Qsub3- is the median of the upper quartileQsub1- is the median of the lower quartile

Apply it

Are there any outliers in the data set

{0,0,1,3,8,9,12,18,20,20,20,22,26,27,30,31,22,42,59}

Measure of variation

The sum of the difference of each data point and the mean of a set of data squared (refer to page 94 of your packet) divided by the number of points

What is the value of the variance of the data set {0,0,1,3,8,9,12,18,20,20,20,22,26,27,30,31,32,42,59}

Standard deviation

The square root of the variance A low standard deviation means the data is clustered or closer

together while a higher standard deviation means the data is more spread out.

You can use the range to approximate the standard deviation by dividing the range by 4 (this is not accurate so only use it when it is difficult to calculate the standard deviation or you don’t really need it you just want to know an estimate of what it is)

What is the standard deviation of the data set {0,0,1,3,8,9,12,18,20,20,20,22,26,27,30,31,32,42,59}? What is the approximate standard deviation of that set using the range?

Too many equations to remember

So, Ill take the liberty of teaching you how to do this stuff on our handy dandy calculators

Apply it

Z-score or standard score: the number of standard deviations a particular data value is above or below the mean (z scores equal 0 if the data value is the same as the mean, the sum of z scores for any data set is always 0)

Lakeeya scored an 80 on her math test and the average score was a 76, and an 88 on her bio test which had an average scored of a 85. Which test did she do better on?The math test had a standard deviation of 2 and the bio test had a standard deviation of 5

Assume the average height of people in Chicago is 67 inches, with a standard deviation of 3.5 inches. What is the z score for an individual with a height of 77 inches?

Assuming the information above is true, how tall would someone need to be to have a z score of -7?

Basic Probability

The probability of an event occurring is the number of ways that event can occur divided by the total number of possible outcomes.

Complement- the probability of the event not occurring (denoted as E’)

You are handed a bag of marbles and told to choose 1. The bag contains 12 purple, 10 green, 7 blue, and 6 white marbles:

What is the probability you will choose a blue marble?

What is the probability you will choose a purple marble?

What is the probability you will choose a marble that is not orange?

Events

Independent events- the occurrence or nonoccurrence of one event does not affect the other(s).

The product is found by multiplying the probability of each individual event by each other

Dependent events- the occurrence of one event does affect the occurrence of the other(s)The product of the probability of the first event occurring and the probability of the second event occurring given that the first event occurred

Independent or Dependent?

Before going to school Hallema picks an outfit. She has 15 shirts: 3 orange, 5 purple, and 7 black. She has 5 jeans: 2 white, 2 black, and 1 blue jean. She has 10 shoes: 2 black, 3 white, 2 purple, 1 pink, and 2 yellow. What is the probability Hallema will wear an orange shirt, white jeans, and black gym shoes?

What is the probability of drawing 3 jacks from a standard deck of cards?

Probability Distribution

Definition- a set of outcomes and the probability of those outcomes where the probability of each event is a number between 0 and 1 and the sum of the probabilities (from all the outcomes added up) equals 1.

Expected Value

The average payoff expected from the probability distribution over a long period of time

fair game- an expected value of 0 Multiply the probability of the outcome by the

number of plays

Then multiply that by the payout

Then divide that total number by the number of plays to determine the average payout per game

Try it A casino introduces a new game, Snake Eyes.

The gambler pays $2 to roll a pair of dice numbered 1-6. The payout is based on the sum of the dice rolled:

A sum of 2=$100A sum of 10=$2 (meaning the play was free)A sum of 11=$10A sum of 7=-$1Any other sum= nothing (a loss of $2) Should gamblers play? What is the average

expected value if the game is played 800 times?

Now find that same problem’s variance and standard deviation of probability

distribution

Keep in mind the formula

Binomial Distribution

Expected value= p*n Variance= n*p*q Standard deviation= square root of

variation

Apply it!

Lissette forgot to study for her band exam (assuming she didn’t know anything about the topic even though we all know she does). The band exam is 15 questions. If she guesses on every question (multiple choice,4 choices) what is the probability that she will pass the exam if a passing grade is 9 questions correct?

Normal distribution

a continuous probability distribution, meaning all possible outcomes are considered

Empirical rule: About 68% of data falls in the first standard deviation, 95% falls in the first 2 standard deviations, and about 99.7% falls in the third standard deviation in a symmetrical bell curve

Apply the idea of the bell curve

Assume the average height of people in Chicago is 67 inches with a standard deviation of 3 inches. What approximate percentage of the population is between 64 and 70 inches?

What is the probability of meeting someone 70 inches or taller?

How tall does someone need to be so they are taller than about 2.5% of the population?