June 24, 20XX
Jolting down, touring, and seeing more memorabilia and advertisements on the southern side's walls with lots of key sentences and quotes:
"Players can play through stylus-controlled levels!"
"Engage in flashy street basketball matches with over-the-top dunks and tricks!"
"Play the traditional historical board game Mahjong with various rule sets"
"Mario using dancing of beat and step on the dance pad to save the Mushroom Kingdom from Wario"
"Compete in a whimsical baseball tournament!"
"Train your tennis skills in the Tennis Academy and participate in challenging tournaments"
"Dr. Mario demonstrating the usage of combined colored capsules to eliminate viruses"
"Race against opponents in arcade-style action, featuring special items and tracks!"
"Snowboard down thrilling slopes, perform tricks, and compete against other riders in this extreme sports game!"
"Princess Peach saving Mario"
"A party game with a virtual board game filled with minigames and competitive challenges!"
"Race with iconic characters in a variety of tracks, utilizing new items and features!"
"Play intense soccer matches with a twist, featuring power-ups and special shots!"
"Mario and Luigi, teaming up!"
"Enjoy a collection of classic games in a portable format"
"Experience the timeless puzzle game with a twist, featuring various game modes!"
"Mario, rescuing Princess Peach from Bowser from an ambush"
"Engage in fast-paced basketball matches with characters, using touch controls and special moves"
"Mario gaining a power of an army of Mini-Marios"
"Helping Yoshi to protect Yoshi Island and friends"
"Race against opponents, featuring new tracks and characters"
"Play in the game where you can explore a 2D/3D hybrid world, flipping between dimensions to save the universe!"
"Compete in intense soccer matches with power-ups and strategic gameplay!"
"Join the virtual board game party, playing a variety of minigames and challenges!"
"Participate in a virtual board game where you buy and develop properties, competing against Mario characters!"
"Mario, traversing this planet's moons and collecting Power Stars"
"Compete in a variety of Olympic sports events as iconic characters, showcasing their athletic abilities and skills!"
"Join Mario and friends in a handheld party game at Princess Peach castle!"
"Mario engages in an epic fighting game"
"Dr. Mario teaching on how to eliminate colored viruses by strategically matching pills"
"Race against friends or computer-controlled opponents in a thrilling-themed kart racing experience!"
"Gather a team and compete in whimsical baseball matches with special moves and abilities"
"Enjoy a second compilation of classic games!"
"Dr. Mario using compact and portable format tools"
"Mario and Luigi, exploring the inside of huge Bowser's robot body and battling enemies"
"Compete in a variety of winter Olympic sports events!"
"Team up with friends in a cooperative 2D platforming adventure!"
"Mario, traversing this planet's moons and collecting Power Stars once again"
"Mario, Luigi, and Yoshi joining with Sonic, Shadow, and Silver to face off a dark threat that surrounds Mobian country of Solaris"
"Sonic in an Arabian Nights-themed adventure"
"Sonic's transformation into a Werehog during a full-moon party"
"Sonic with a sword in hand, teaching others how to use a blade"
"Sonic rescuing alien creatures known as the Wisps from Dr. Eggman"
Math Notes: (filled with academic musings)
Integral (sec^2 (theta) - sec(theta) tan(theta)) d(theta)
= tan(theta) - sec(theta) + C
tan = y/x = sin/cos = (square root 2 / 2) / (- square root 2 / 2) = -1
tan (37/4) / 3 + C = 0
3 (-1/3) + C = 3
C = 3 + 1/3 = 10/3
Finding the derivative input: f(x) = tan(3 Pi) / 3 + 10/3
Find the unique function Integral that satisfies f '(x) = sec^2 (3x) dx and f(Pi / 4) = 3
Integral sec ^2 (3x) dx whereas f '(x) so anti derivative
u=3x
dx = du/3
Integral sec^2 (u)(du/3)
=1/3 Integral sec^2 u du = 1/3 tan u + C
=1/3 tan (3x) + C
=1/3 tan (3x) + C
f(x) = 1/3 tan (3x) + C
3= 1/3 tan (3)(Pi/4) + C
3= 1/3 tan (3 Pi / 4) + C
Opposite of Chain Rule
Integral sec^2 u du = tan u + C
So: 1/3 tan (3x) + 10/3
f(x) = 1/3 tan (3)(Pi/4) _ C
3= 1/3 (-1) + C
C= 10/3
Find the unique function Integral that satisfies f '(x) = (8x^3 + 3x^2) dx and y(2)=0
f '(x) = (8x^3 + 3x^2) dx
Integral (8x^3 + 3x^2) dx
Integral (8x^4 / 4 + 3x^3 / 3)
Integral 8 (x^4 / 4) dx + Integral 3 (x^3 / 3) dx
8x^4 / 4 + 3x^3 / 3 + C becomes 2x^4 + x^3 + C
y(2)=0
0=2x^4 + x^3 + C
0= 2(2)^4 + (2)^3 + C
0=40 + C
C= -40
2x^4 + x^3 - 40
Theoretical Definition of Integral
f(x)=2x^2 + 5; [0, 2]
Integrating 2, 0 (2x^2 + 5) dx
Power Rule: 2x^2 becomes 2x^3 / 3
5 becomes 5x
=(2x^3 / 3 + 5x) | 2,0 = 16/3 + 10 = 16/3 + 30/3 = 46/3
Theoretical Definition of Integral
f(x)=2x^2 + 5; [0, 2]
Delta x = 2-0/n = 2/n
xn^delta = a+2k/n = 2k/n
f(x^delta (2)) = 2 (2k/n)^2 + 5 = 8k^2 / n^2 + 5
n E k=1 (8k^2 / n^2 + 5)(2/n) = n E k=1 (16k^7 / n^3 + 10 / n)
=16/n^3 (n E k=1) k^2 + 10/n (n E k=1) 1
=16/n^2 (n(n+1)(2n+1)/6) + 10/n (4)
=8(n+1)(2n+1)/3n^2 + 10
=16n/3 + 8/n + 8/3n^2 + 10
lim n to infinity: 16/3 + 8/n + 8/n+1 + 10 = 16/3 + 10 = 46/3
Partial Fraction Exposition
X k = 2/n k
f(x ' k) = 2 (12k/n)^2 + 5
= 8k/b + 5
f(x' k)/ E f(xk) times delta(x)
= E k=1 (8k^2/n + 5)(2/n)
f(x) = x^3 + 1; [0,2]
Integral b,a f(x)dx = F(b) - F(a)
Integral 2,0 (x^3 + 1) dx
Intergrated: (x^4 / 4 + x) | 2,0 = (2^4 / 4 +2) - (0^4 / 4 + 0)
=6
Determine the value of integral a,2 f(x)dx given that integral 2,5 f(x)dx=3 and integral 9,5 f(x)dx = 8
Integral 9,2 f(x)^x = - Integral a, b f(x)dx
Integral 2,5 = -Integral 5,2
So: [Integral 9,5 f(x)dx=8] + [-Integral 5,2 f(x)dx=3]
=5
Remember the Theoretical Definition of Integral
f(x) = 2x^2 + 5; [0,2]
Integral 2,0 (2x^2 + 5) dx
= (2x^3 / 3 + 5x) | 2,0 = 16/3 + 10 = 16/3 + 30/3 = 46/3
Riemann Sums: Find the area under the curve over the given interval in a set your solution using the limit as n goes to infinity of the upper sum.
A = Integral b,a f(x)dx
Integral 2,0 [x^2 +1] dx
=x^3 / 3 + x | 2,0
=[2^3 / 3 + 2] - [0^3 / 3 + 0]
=8/3 + 2/1
=4.67
=14/3
y=f(x) and A=lw and f(x) delta x
A= n E i=1 delta x f(x i)
f(x) x^2 + 1
Delta x = b - a / n
=2 - 0 / 4
=0.5
Delta x [f(0) + f(1/2) + f(1) + f(1.5)]
=0.5[1+1.25+2+3.25]
=0.5 [7.5]
=3.75
Integral 6,11 (6 g(x) - 10 f(x)) dx
Given that: Integral 11,6 f(x)dx= -7 and Integral 11,6 g(x)dx = 24
Integral 6,8 = -Integral 11,6 (6 g(x) - 10 f(x)) dx
-Integral 11,6 (214) dx
Integral 11,6: f(x)dx = -7 to 7
Integral 11,6: g(x)dx = 24 to -24
Integral 6,11: (6 g(x) - 10 f(x)) dx
Integral 6,11: 6 g(x) dx - Integral 6,11: 10 f(x) dx
6 Integral 6,11 g(x) dx - 10 Integral 6,11 f(x) dx
=6(-24) - 10(7)
= -214
Find F '(x)
F(x) = Integral x,-1 (-1^3 + 2t^2 -1) dt
If g(x) = Integral x,a f(t)dt, g '(x)=f(x)
d/dx [Integral x,a f(t) dt] = f(x)
d/dx [Integral x,0 square root t^2 + 4 dt]
f(t)= square root t^2 + 4
d/dx [Integral x,0 f(t)dt] = d/dx [F(t) | x,0]
d/dx [F(x) - F(0)] = f(x) - 0 = f(x)
f(x) = square root x^2 + 4
d/dx [Integral 4,x square root t^3 + 5 dt]
f(t)= square root t^3 + 5
d/dx [Integral 4,x f(t) dt] = d/dx [F(t) | 4,x]
d/dx [F(4) - F(x)] = 0 - f(x)
= -square root x^3 + 5