Partial fractions video for funnnnnnn

Hi since next week is the math final, I did not put anything unrelated to the final for our fun post.

However, instead, I found a YouTube video for partial fractions that really helped me when I do partial fractions.

So check it out!

http://youtu.be/cyYJ4HP2RFw

Vectors in space

Vectors in Space

Solving a Unit Vector

-

The equation for a unit vector is:

-

A vector space  is a set that is closed under finite vector addition and scalar multiplication. The basic example is -dimensional Euclidean space , where every element is represented by a list of  real numbers, scalars are real numbers, addition is componentwise, and scalar multiplication is multiplication on each term separately.

For a general vector space, the scalars are members of a field , in which case  is called a vector space over .

Euclidean -space  is called a real vector space, and  is called a complex vector space.

In order for  to be a vector space, the following conditions must hold for all elements  and any scalars :

1. Commutativity:

(1)

2. Associativity of vector addition:

(2)

3. Additive identity: For all ,

(3)

4. Existence of additive inverse: For any , there exists a  such that

(4)

5. Associativity of scalar multiplication:

(5)

6. Distributivity of scalar sums:

(6)

7. Distributivity of vector sums:

10.5 Parametric Equations

Definition of a plane curve

If f and g are continuous functions of t on an interval l, the set of ordered pairs (f(t),g(t)) is a plane curve C. The equations:

 

        x=f(t)      and     y=g(t)

are parametric equations for C, and t is the parameter.

 

Eliminating the Parameter

1. Parametric equations

2. solve for t in one equation

3. substitute into second equation

4.rectangular equation (only 2 variable)

 

Example:

1.

x=t+3, y= -t^2

step 1:  subtract both side of x=t+3 by three:

                                x-3=t

then, take x-3 into another equation

                   get   y= - (x-3)^2

 

Aristotle and logic : creating something from nothing

Aristotelian logic, after a great and early triumph, consolidated its position of influence to rule over the philosophical world throughout the Middle Ages up until the 19th Century.  All that changed in a hurry when modern logicians embraced a new kind of mathematical logic and pushed out what they regarded as the antiquated and clunky method of syllogisms.  AlthoughAristotle’s very rich and expansive account of logic differs in key ways from modern approaches, it is more than a historical curiosity.  It provides an alternative way of approaching logic and continues to provide critical insights into contemporary issues and concerns.  The main thrust of this article is to explain Aristotle’s logical system as a whole while correcting some prominent misconceptions that persist in the popular understanding and even in some of the specialized literature.  Before getting down to business, it is important to point out that Aristotle is a synoptic thinker with an over-arching theory that ties together all aspects and fields of philosophy.  He does not view logic as a separate, self-sufficient subject-matter, to be considered in isolation from other aspects of disciplined inquiry.  Although we cannot consider all the details of his encyclopedic approach, we can sketch out the larger picture in a way that illuminates the general thrust of his system.  For the purposes of this entry, let us define logic as that field of inquiry which investigates how we reason correctly (and, by extension, how we reason incorrectly).  Aristotle does not believe that the purpose of logic is to prove that human beings can have knowledge.  (He dismisses excessive scepticism.)  The aim of logic is the elaboration of a coherent system that allows us to investigate, classify, and evaluate good and bad forms of reasoning.

Mathematical Induction

This is the section of my presentation!!

 1. Definition:

mathematical induction is a method for proving that statements involving natural numbers are true for all natural numbers.  

2. Theorem:

suppose that 他the following two conditions are satisfied with regard to a statement bout natural numbers:

CONDITIONＩ:The statement is true for  natural number 1

Condition II: if the statement is true for some natural number k, it is also true for k+1

Thus the statement is true for all numbers

3. sums of powers of integers

Examples:

1.

For any positive integer n, 1 + 2 + ... + n = n(n+1)/2.

           n=1  1 = 1(2)/2 which is clearly true.

          Here we must prove the following assertion: "If there is a k such that P(k) is true, then (for this same k) P(k+1) is true."

        assume there is a k such that 1 + 2 + ... + k = k (k+1)/2.  

         We must prove, for this same k, the formula 1 + 2 + ... + k + (k+1) = (k+1)(k+2)/2.

        1 + 2 + ... + k + (k+1) = k(k+1)/2 + (k+1) = (k(k+1) + 2 (k+1))/2 = (k+1)(k+2)/2.

2.

Write the first five terms of the sequence

a0=1 an=a(n-1)+2

a0=1

a1=3

a2=5

a3=7

Determinants of a matrix

Determinant: the associated algebraic number associated with a matrix

Matrix: an array of numbers set inside of a two brackets

Notations of determinants differ. There are 2 ways to do a determinant. 

How to do a 2x2 matrix determinant

This is quite a simple application of timesing the top left and bottom right numbers then subtracting the multiplication of the bottom left and top right numbers. 

Determinant of 3x3 matrices. 

These are hard yet simple. The easiest way is to take the first 2 columns, copy them, and move them to the end of the matrix. After that take the first 3 diagonals starting at the top left and going down and multiply those numbers in their individual columns. Then add the other 2 columns to the first. after that take the columns going diagonally up from the bottom right and across and multiply those together. Then subtract all three of those columns from the columns of the first three. This will give a person their matrix. 

Limits

Sometimes you can't work something out directly ... but you can see what it should be as you get closer and closer! We call that the limit of a function.

For example, what is the value of (x2-1)/(x-1) when x=1? (Notice in the graph there is actually a "hole" at that point). Let us substitute "1" for "x" and see the result: (12-1)/(1-1) = (1-1)/(1-1) = 0/0

But 0/0 is "indeterminate", meaning we can't determine its value, so instead of trying to work it out for x=1 let's try approaching it closer and closer:

x	(x2-1)/(x-1)
0.5	1.50000
0.9	1.90000
0.99	1.99000
0.999	1.99900
0.9999	1.99990
0.99999	1.99999
...	...

Now we can see that as x gets close to 1, then (x²-1)/(x-1) gets close to 2, so we say:

The limit of (x²-1)/(x-1) as x approaches 1 is 2

And it is written in symbols as:

We don't know the value at x=1 (it is indeterminate), but the limit has a value of 2

More Formal

But you can't just say a limit equals some value because it looked like it was going to. We need a more formal definition.

So let's start with the general idea

From English to Mathematics

Let's say it in English first:

"f(x) gets close to some limit as x gets close to some value"

If we call the Limit "L", and the value that x gets close to "a" we can say

"f(x) gets close to L as x gets close to a"

 

Calculating "Close"

Now, what is a mathematical way of saying "close" ... could we subtract one value from the other?

Example 1: 4.01 - 4 = 0.01 
Example 2: 3.8 - 4 = -0.2 

Hmmm ... negatively close? That doesn't work ... we really need to say "I don't care about positive or negative, I just want to know how far" The solution is to use the absolute value.

"How Close" = |a-b|

Example 1: |4.01-4| = 0.01 
Example 2: |3.8-4| = 0.2 

And if |a-b| is small we know we are close, so we write:

"|f(x)-L| is small when |x-a| is small"

And this animation shows you what happens with the function f(x) = (x2 - 1) / (x-1) as x approaches a=1, f(x) approaches L=2 So |f(x)-2| is small when |x-1| is small.

Delta and Epsilon

But "small" is still English and not "Mathematical-ish".

Let's choose two values to be smaller than:

	that |x-a| must be smaller than
	that |f(x)-L| must be smaller than

(Note: Those two greek letters, δ is "delta" and ε is "epsilon", are often

used for this, leading to the phrase "delta-epsilon")

And we have:

"|f(x)-L|<when |x-a|<"

That actually says it! So if you understand that you understand limits ...

... but to be absolutely precise we need to add these conditions:

1)	2)	3)
it is true for any >0	exists, and is >0	x not equal to a means0<|x-a|

And this is what we get:

"for any>0, there is a >0 so that |f(x)-L|<when 0<|x-a|<"

That is the formal definition. It actually looks pretty scary, doesn't it!

But in essence it still says something simple: when x gets close to a then f(x) gets close to L.

How to Use it in a Proof

To use this definition in a proof, we want to go

From:		To:
0<|x-a|<		|f(x)-L|<

This usually means finding a formula for  (in terms of ) that works.

How do we find such a formula?

Guess and Test!

That's right, you can:

1. Play around till you find a formula that might work
2. Test to see if that formula works.

Example: Let's try to show that

Using the letters we talked about above:

• The value that x approaches, "a", is 3
• The Limit "L" is 10

So we want to know:

How do we go from:

0<|x-3|<

to

|(2x+4)-10|<

Step 1: Play around till you find a formula that might work

Start with:	|(2x+4)-10|<
Simplify:	|2x-6|<
Move 2 outside:	2|x-3|<
Move 2 across:	|x-3|</2

So we can now guess that =/2 might work

Step 2: Test to see if that formula works.

So, can we get from 0<|x-3|< to |(2x+4)-10|< ... ?

Let's see ...

Start with:	0<|x-3|<
Replace:	0<|x-3|</2
Move 2 across:	0<2|x-3|<
Move 2 inside:	0<|2x-6|<
Replace "-6" with "+4-10"	0<|(2x+4)-10|<

Yes! We can go from 0<|x-3|< to |(2x+4)-10|<by choosing =/2

DONE!

We have seen then that if we choose a we can find a , so it is true that:

"for any, there is a  so that |f(x)-L|<when 0<|x-a|<"

And we have proved that