▀▄▀▄▀▄ 2nd ѕeмeѕтer ѕυммary ▄▀▄▀▄▀

Greetings, guest of Mathland!

This will be the last post of the year!~ 

Thank you so much for accompanying me throughout my journey.

Starting off in the beginning of the year, we began with lessons on pre-calculus. As the year progressed, we peeked into the world of Calculus through the lessons of parametric equations, limits, and etc.

Overall, the year was a lovely and I was able to learn about numerous new concepts. Although, sometimes a few were confusing, but later on through practice, I was able to master them! 

Something I enjoyed the most in math this year would be storytime - as it was very relaxing during stressful times (AP Week game strong).

There is nothing I didn't like about the class....other than webassign which was a bit stressful occasionally. (o~ o)

But overall, I loved the class!

Thank you again!

(¯`·.¸¸.-> °º тяιg яєνιєω ωєєк º° <-.¸¸.·´¯)

Hello, guests of Mathland!

This week, we had trig review week which went back to our trig identity proofing. 

To start off, we began with simplifying the trig expression in order to make the problem easier to do.  After that, we verified (or proved) each identity - meaning we proved that the left side equaled the right side.

To do such a thing, we used a trigonometric identity sheet that had all the identities on it:

Learning these identities again were a bit difficult, as these were taught near the beginning of last semester. However after a bit more practice, it was easier to do!

Below is an example of simplifying a trig function:

Below is an example of proving a trig identity:

That is all for today, good bye! See you next time!~

••.•´¯`•.•• ŖËPËÄȚÏŅĠ ĐËĊÏMÄĻŚ ••.•´¯`•.••

Salutations, guests of Mathland!

Today we will be learning a fairly simple topic - repeating decimals!

Examples of such decimals are:

.23232323....

.7777...

.417417417....

It is stated that every repeating decimal is the sum of an infinite geometric series.

To solve for the repeating decimal, you would first write the decimal as a geometric sum. Then you would find the sum.

The equation used in this case is a1/(1-r).

To find r, it would be the common difference or ratio between each number in the repeating decimal.

Below is an example:

That is all! See you next time!

ıllıllı ραяαмεтяιc εqυαтισηs ıllıllı

Greetings, guests of Mathland!

Today, we will be reviewing parametric equations!

These equations give you the starting point and direction a particular line will flow. Starting off, the first step to solving the parametric equation is by sketching a graph. To sketch the graph, you would need to make a table that used the variables of:

 t,x, and y (which are included in the given equations).

For example:

After that, you would usually solve the equation to eliminate the parameter. There are two different ways to approach, by using either elimination or substitution, or the trig identities.

Four important trig identities that should be memorized are:

Sometimes, instead of substituting regular whole numbers for t, the equation will ask for degrees, in which you will need to refer to the unit circle.

Below is an example of solving a parametric equation:

That is all. Ta ta!~

—(••÷[ ⓟⓐⓡⓣⓘⓐⓛ ⓕⓡⓐⓒⓣⓘⓞⓝⓢ ]÷••—

Welcome, guests of Mathland!

Today we will be learning about partial fractions. 

In starting off, there are a few set up steps that must be written out.

Here are the steps:

1) Divide if improper

2)Factor denominator

3) If linear (no exponent), it is A,B, C...

4) If quadratic then Ax+B, Bx+C, etc...

To solve for the equation:

1) Multiply by the LCD (Least Common Denominator

2) Group terms by powers of x

3) Solve the system of equations

Further more, there are 4 cases that encompass partial fractions. The first is linear, the second is quadratic, the third is squared on the outside, and the last is long division.

Below are examples of the different cases:

Case 1:

Case 2:

Case 3:

That is all. Farewell!~

▁ ▂ ▄ ▅ ▆ ▇ █ тower oғ нanoι reғlecтιon █ ▇ ▆ ▅ ▄ ▂ ▁

Starting off, within the concept of mathematical induction, there are two major steps. The first is "Show true for n=1" and the second is "Assume the statement is true for n=k, and prove true for k+1." 

Using such steps in an example...

If the equation or problem given is 1 + 2 + 3 +4... + (2n-1) = n^2 , one would first begin by proving that n=1. To do this, you would plug in a "1" for anywhere you see a "n" within the equation. Therefore it would look 2(1) - 1, which is equal to 1 - and then (1)^2, which is also equal to one. By doing that, we have proved that both the left hand side and right hand side are equal to 1, showing true that n=1.

After that, we would continue on to step 2, which is "assume the statement is true for n=k, and prove true for k+1." This means that first, we will plug in a "k" for anywhere we see a "n" in the problem. Upon doing so, the resulting equation would be:

 1 + 2 + 3 +4... + (2k-1) = k^2  (This is known as the recursive formula)

After you find the equation above, you then plus in a "k+1" for anywhere a k is present. Therefore, the next resulting equation would be:

 1 + 2 + 3 +4... + [2(k+1)-1] = (k+1)^2

You then simplify the equation and plug in the recursive formula (mentioned above). This indicates that the equation will become:

k^2 + 2k +2-1 = (k+1)^2

and then simplifying it down

k^2 +2k + 1 = (k+1)^2

which is

(k+1)^2 = (k+1)^2

Thus we have shown that k+1 is true. It follows from Mathematical Induction that the statement is true for every positive integer.

-----

Moving along, in class, we did a project called "The Tower of Hanoi." This particular project was to find the correlation between the number of rings and the amount of times it was moved. Starting with 2, then 3, then 4, and then maxing at 7 rings, with three poles; my group discovered that the equation for the correlation was:

T1 = 2^n - 1

and the recursive of it was:

Tn = 2Tn +1

An observation i concluded while experimenting with the discs is that if you move the lowest number to the second pole, and then move the second lowest number to the third pole, you can then put the one on the second pole on the third pole - therefore, giving room for the third to be moved to the second pole. By using such a strategy, it is easier to move all the discs from the first pole to the third pole.

—(••÷[ ŚËǬŮËŅĊËŚ ÄŅĐ ŚËŖÏËŚ ]÷••—

[I will be simplifying the blogs from today on so that I may save time to study for upcoming AP's ]

Greetings, dear guests of Mathland!~

Today I will be expanding on sequences and series. For sequences and series, there are two types:
-arithemetic
and
-geometric

For the arithemetic sequence, the formula used is an=a1 +(n-1)d. "D" is the ratio between each number - otherwise known as the common difference.
For the geometric sequence ,the formula used is an=r^(n-1)
In addition to using these equations, one is frequently asked to find the sum of the series, meaning what is the sum of the pattern added up to a specific number. To find the number, summation notation is used.
The arithmetic formula for this is Sn = n[ (a1+an)/2 ]
The geometric formula for this is Sn = [a1(1-r^n)]/1-r
Here is an example:

That is all for today! Good bye!~

★·.·´¯`·.·★ gяαρнιηg sүsтεмs σғ ιηεqυαℓιтιεs ★·.·´¯`·.·★

Salutations, fellow visitors of Mathland!

How have you all been? 

Good I hope.~

I must tell you, but I have just learned about the most fascinating piece of knowledge.  It has compelled me so much that I must share it with you all.

This fascinating thing is known as "graphing systems of inequalities".

Let us start, yes?~

In the beginning of this lesson, we must review the three equations of the three different graphs:

Line: y=mx + b

Parabola: y = (x-h)^2 + k

Circle: x^2 + y^2 = r^2

Here are the steps to graphing the equations:

1) Determine which graph is used and graph each equation.

2) Pick a test point that is not on the line.

3) Shade the plane containing the test point if the test point satisfies the equation. Shade the other plane if it does not.

Also make sure to pay attention to the signs. 

If the sign is greater or less than and equal to then the graphed line is solid. 

If the sign is only greater than or less than, then the graphed line is dashed.

Below is an example:

Well, that is all for now! See you next time.~

Goodbye.~

•´¯`•. ĊŖÄMËŖ'Ś ŖŮĻË .•´¯`•

 Greetings, fellow visitors of Mathland. 

I humbly welcome you to our newly created attraction - Cramer's Land.

In this park, we will be discussing and enjoying the simplicity and efficiency of solving for variables.

Starting off, for Cramer's Rule, there are three different variations of it.

Dx/D, Dy/D, and Dz/D

The D in the denominator represents the determinant while the Dx, Dy, and Dz, represent the matrices "row x/y/z" replaced with the solution (number after the equal sign) placed in the numerator. Once you find all the answers, you merely divide the numerator by the demons stir and you will get the answer for each variable. To check to see if you have done it right, plus the numbers back into one of the original equations and if it solves to the answer given, then you have solved the problem correctly.

Overall, using Cramer's Rule allows for one to easily find the answer to the numerical values of the digits.

I do hope you enjoyed your time here and learned a few things.~

Good day and farewell, dear guests!

׺°”˜`”°º× ѕуѕтємѕ σf єqυαтισиѕ ×º°”˜`”°º×

Hello fellow individuals of Mathland!

It has been some time since I last came upon you!

Why don't we have a short chat to reacquaint ourselves?~

I actually looked forward to introducing you all to a new topic, called System of Equations. 

Shall we begin?~

System of equations are two or three given equations with variables. In order to solve these, one must find what numbers the variables are. There are two ways in solving such a problem.

Here are 2 ways:

As you can see, both are able to be used. 

Also in solving system of equations, one has to take into consideration of the problem being either inconsistent or consistent.

Inconsistent means that there are no solutions, and if graphed, the lines would be parallel.

Consistent means that there is 1 solution (independent) or an infinite amount of solutions (dependent).

Below is an example, of a "system of equation problem" with three equations.

Well, that will be all for now! I do hope you enjoyed our discussion. 

Farewell.~

My Polar Equation Graph

ıllıllı ⓟⓞⓛⓐⓡ ⓒⓞⓞⓡⓓⓘⓝⓐⓣⓔⓢ ıllıllı

Hello, dear ladies and gentlemen of Mathland.

Today we will be having a free day among'st ourselves to explore wherever you would like to go. For those who would like to journey with me,  I will be making a delivery to the Singing Flower Garden. Along the way I will be talking about polar coordinates.

Choose wisely young ones.~

Well, lovely to have those of you who joined me for company. I give many thanks.

Now moving along.~ 

Polar equations are quite different from the usual way in graphing. 

Unlike the Cartesian Plane, the Polar Graph's shape is slightly varied. 

While the Cartesian Plane graphs points with the coordinates of (x,y) =, the polar plane graphs points using (r, θ).

Notes:

However, if one is confused, there are ways to transition between polar to rectangular and rectangular to polar coordinates. 

For polar to rectangular, one uses two equations:

x=rcos(θ)

y=rsin(θ)

For rectangular to polar:

r²=x²+y²

tan(θ)=y/x

Notes:

An important note for the theta, is that if it is greater then zero, then the graph will go counterclockwise, and if the theta is less than a zero, then the graph will go clockwise.

~~~~~~~~~~~~~~

Thank you for accompanying me on my visit to the Singing Flower garden and listening to my lecture.~

Farewell, ladies and gentlemen.

See you soon.~

—(••÷[ roтaтιng conιc ѕecтιonѕ ]÷••—

Salutations, guests of Mathland!

We will be upon our destination of the Mad Hatter's Tea Party in a few seconds!

.

.

.

I most humbly welcome you!

Today at the party we will be discussing the rotation of conic sections, otherwise known as "Axis Rotation".

Here are some notes regarding this process:

As demonstrated by the image above, in order to start, we must determine and match the coefficients with that of the first equation,  Second, we plug it into the equation and find what cot2(θ) is equal to. Next, we are able to replace the x with x' (x-prime) and y with y' (y-prime) in order to determine the coordinates on the rotated Cartesian Plane. Finally we plus it into the original equation and use algebra to simplify it.

Here is an example:

That is all! 

I do hope you enjoyed your time and our lovely conversation at the tea party! 

The March Hare seems to have enjoyed our company, for he tells us to visit again soon.

Farewell for now!

See you next time, lovely guests.~

••.•´¯`•.•• ραяαвσℓαѕ ••.•´¯`•.••

Greetings, visitors of Mathland.~

How have you all been? I have been quite lovely myself. Today, I will be taking you all on a tour through the woods to go visit the March Hare at the Mad Hatter's tea party. Along the way, I will be reviewing the concepts of Parabolas and its equations with you all.

Starting off, the base equation for parabolas is:

(x-h)_­­­_­² = 4c (y-k)

Therefore:

Vertex: (h,k)

Focus: (h, k+c)

Directrix: y=k-c

A.O.S: x=h

In order to find the "c" value in this equation, one merely sets the number in the place of "4c" equal to '4c' and solve for "c". If the "c" is greater then zero, then the graph is facing upward. However, if the "c" is less then zero, then the graph is facing downwards.

Notes:

On the other hand, if the graph is opening to the left or right, the equation switches. It changes to:

(y-kh)_­­­_­² = 4c (x-h)

Meaning:

Vertex: (h,k)

Focus: (h+c, k)

Directrix: y=h-c

A.O.S: y=k

Notes:

To find the "c" of this equation, it is still the same exact process as finding the "c" of the other equation. Only in this situation, if c is greater then zero, then the graph opens right. I fc is less then zero, then the graph opens left.

That will be all for today!~ 

We will soon arrive at the Tea Party.~

See you next time.~

•´¯`•. 2nd ѕeмeѕтer goalѕ .•´¯`•

Welcome back, dear guests of Mathland~

Forgive me, for I took a few weeks journey to the Clover Kingdom that situates itself north of the Heart Kingdom of the Red Queen.~

I hope you all have had a lovely Christmas break?

There are a few things I wanted to touch on as I have received fantastic advice from the Emerald Queen in the Kingdom of Clover during my visit. 

She recommends that I improve upon my tendency to procrastinate, as well as my manner to which I proceed with my work. Also, she recommends that I work on studying more as it will benefit me greatly. 

However, along with all such advice, she did in fact say that I have done well in the categories of being able to keep up with the workload and the classes, along with the memorization of formulas.

Ah, might I also regale you dear guests with a tale from my break?~

During such a lovely relaxing vacation, I was able to catch up on my lack of sleep - just a bit.~ 

I usually slept around 3-4 in the morning and awoke around noon. In addition to such, I usually fell asleep while talking with friends through what some refer to as "Skype".~ 

Quite an interesting experience I might say.~

Well that is all, I do hope you all enjoyed this first post of the new 2015 year.~