August 30, 20XX
In the meantime, the world on their co-existing planet is filled with certain areas.
Metro City is home to Officer Chun-Li and Street Fighting clubs, in which they clash with Shadaloo criminal organization in which they associate with "Shy Guys", Andross, and moguls alongside with greedy mercenaries.
Ancestral Steppe is a location for many huge amounts of different species as noticed by Sticks. She once commented about a kid named Alex that lived in Mt. Eternal.
Sonic once knew about a kid named Billy from Morning Land.
If any cross the Yurian Forest, there lies the Old Firewood Kingdom. Nightopia is the home of a certain group of magical knights.
Crescent Isle is where sky pirates run rampant. Dragons are often spotted in the area of the Submerged City.
The Lost Shore is home to warriors who are engineers.
Many fear the lair of the Astaroth's Castle.
The peaceful island of Kamiki Village is often noticed.
If one isn't busy, people usually visit the Movieland theater or Lakewood Plaza.
A maracas playing band is lead by a monkey.
Colorful slimes give amounts of Tetris-type entertainment.
There is an amount of rivalry between space mice and cats.
A wanderer in space named Harrier.
Continuing traditional heroic actions by high school students on tackling spiritual evils and monsters.
World Fighting Tournaments are hosting their fighting anniversaries.
Sentient space ships are invented.
Monkey Island is quite close to Mobian inhabitants.
Skating youth gangs combat against violent gangs within the streets with help.
It is noted by the G.U.N that a giant monkey and "Miss" Daisy help out their best Mobian agents. Rosalina often invites Blaze for tea. The Chaotix with Silver often clashed with the Wario Bros, Kamek, and Koopa royalty. Meanwhile, Tikal commented that the Echidnas often correspond with Lumas, and Star Spirits. Sticks talks about how Yoshis and Birdos are often around forest ranged areas. If Metal Sonic is notorious enough, Mario would be unfortunately a basis for a self aware metal being that based his appearance after him thus known as Metal Mario. Tails' uncle often corresponds with Mervin, Rosalina, Geno, Dr. Toadley, Merlumina, Merluvlee, Merlon, Locke, Princess Blaze the cat, Delphius' oracle, Forelock, Chief Kordon, Merlynx, Lazaar with his guardian, and Shaman Warpull with the "Enchanter Monarchs" to repel opposition forces such as Kamek, Kammy, Kooky Von Koopa, X-Nauts, Dr. Finitevus Ix Zachary, Kurse, Mesmer, a mysterious traveler, Omen, Fajar, Medusa, Mephistor, mystic genies, Walter Ixis Naugus, Mogul, and Wendy Naugus Witchcart. Hauntings of spirits or ghosts are often well known in case of Boos and Boom Boos.
Math Notes: (filled with academic musings)
Tangent Line
It is a line that just touches a graph of a single point.
Finding the Slope of a Line
How many points do you need normally to find the slope of a line?
Answer: 2
Limit
If you keep picking points that get closer and closer, it's really hard to get to see that original point that you got, that was right there.
But if you keep getting closer and closer and closer, won't you eventually get so close that it's almost the same slope?
So what comes with this is the idea of limit.
The idea of a limit is about how close you can get to a point without actually hitting that same point.
Points Q and P
If you let Q get really close to P, then the secant will be identical to the tangent.
Graphing a Function
There is a point right at the vertex, at the origin.
Points: (1, 1); (-1, 1); (2, 4); (-2, 4)
Point P is at (1, 1)
The goal is to find the equation of the tangent line of the curve, f(x) = x² at (1, 1).
Note: You might see this kind of question on your homework. You are given a graph and you will be told to find the equation of the tangent line at a point.
You do not know what Q is but call it (x, y) for now.
Can you represent y in terms of x?
Answer: Yes
Remember, f(x) can easily be replaced with y.
This works for any Q you choose.
Finding the Slope
Slope formula: m = (y2 - y1) / (x2 - x1)
Assume that Q is at (2, 4), find the slope.
Let x = 2
m = (2² - 1) / (2 - 1)
m = 3 / 1
m = 3
Let x = 1.5
m = (1.5² - 1) / (1.5 - 1)
m = 2.5
Let x = 1.1
m = (1.1² - 1) / (1.1 - 1)
m = 0.21 / 0.1
m = 2.1
Let x = 1.01
m = (1.01² - 1) / (1.01 - 1)
m = (1.0201 - 1) / 0.01)
m = 0.0201 / 0.01
m = 2.01
Let x = 1.001
m = (1.001² - 1) / (1.001 - 1)
m = (1.002001 - 1) / 0.001
m = 0.002001 / 0.001
m = 2.001
Let x = 1
What will happen?
m = (1² - 1) / (1 - 1)
m = 1 / 0
Answer: Undefined