June 25, 20XX
Jolting down, touring, and seeing more memorabilia and advertisements on the western side's walls with lots of key sentences and quotes:
"Helping Mario navigate through puzzles and obstacles to save his toy factory"
"Engage in a variety of sports activities, including basketball, volleyball, hockey, and dodgeball!"
"Embark on a 3D platforming adventure"
"Compete in various Olympic events in a British themed sports!"
"Compete in exhilarating races, utilizing power-ups, and clever tactics to claim victory on a variety of imaginative tracks!"
"Engage in a virtual board game where players invest in real estate, manage stocks, and compete to accumulate wealth and become the richest player!"
"Join Mario and his pals in another party-style game at Princess Peach's castle!"
"Serve and volley your way to victory in intense tennis matches, utilizing different shots and strategies"
"Mario, collecting coins and power-ups in an attempt to rescue Princess Peach from Bowser once again"
"Mario, where he is using stickers play a crucial role in battles and puzzle-solving"
"Mario and friends, navigating through levels and facing off against Bowser to save the Mushroom Kingdom"
"Luigi as he explores haunted mansions, capturing ghosts with his trusty Poltergust 5000"
"Solve challenging puzzles by manipulating tiles in this unique and addictive puzzle game!"
"Mario and Luigi exploring the concepts of dreams from the Ancients"
"Mario and his friends, competing against AI and other players in a variety of race tracks"
"Participate in this season's winter Olympic events, with the showcasing your athletic skills!"
"Mario and his friends, using their unique abilities to rescue Sprixies from Bowser's clutches"
"Competing in mini-games and exploring various themed boards!"
"A compilation of mini-games that offer new challenges and twists on nostalgic titles!"
"Dr. Luigi receiving his degrees"
"A charming platforming adventure, utilizing Yoshi's unique abilities to overcome obstacles"
"Featuring a fresh selection of retro games and unique twists!"
"Tee off in an immersive golfing experience, competing in various courses, and utilizing special shots and power-ups"
"Race against friends or opponents in gravity-defying kart races, featuring anti-gravity sections and a vast selection of iconic tracks and characters!"
"Engage in intense battles with a roster of characters, utilizing their unique movesets and items in a crossover fighting game!"
"Experience an enhanced compilation of classic games with new challenges and features, putting your skills to the test in a variety of retro-inspired tasks!"
"Mario helping Captain Toad through puzzling levels, solving environmental puzzles, and collecting treasures"
"Unleash epic battles with a diverse roster of characters in this crossover fighting game, featuring stunning visuals and dynamic stages!"
"Create and share your own custom levels in a puzzle game, earning stars, and tipping other players for their creations!"
"Sonic with his hero saving career, offering a generation on how to be aware against dangers of recent warfare"
"Sonic in an adventure to save the lost world from the evil Dr. Eggman"
"Sonic with his forces fighting against the tyranny of Dr. Eggman and his army of robots"
"Sonic embarks on an open-world adventure in a vast and immersive environment, exploring new realms and facing off against powerful enemies in a new frontiers"
"Sonic experiencing a high-speed hoverboard racing in a thrilling competition against other racers in a variety of tracks"
"Sonic takes the hoverboard racing to new heights with gravity-defying mechanics and intense races, as Sonic and his allies uncover the secrets of ancient artifacts"
"Sonic utilizes the Kinect motion sensor to control the hoverboards and competes in fast-paced races, performing tricks and battling opponents in an immersive experience"
"Compete in a kart racing extravaganza featuring a diverse roster of characters, utilizing power-ups and unique tracks!"
"Sonic in a thrilling racing adventure that has him meeting alien-transforming vehicles that are often are cars, boats, and planes"
"Experience the racing game on the Game Gear, where you compete against other series in fast-paced kart races"
"Sonic and his friends compete in challenging races with new tracks and power-ups"
"Race against other characters in a 3D racing game, exploring vibrant environments and collecting power-ups to unlock secrets"
"Sonic and his rivals in a battle to reach the finish line first while utilizing special moves and abilities"
"Sonic returns to the racing rivalry in an intense racing action"
"Sonic and his friends in a cooperative racing adventure, working together in teams to win races and unleash team-based special moves in a variety of vibrant tracks"
"Mario and Sonic in a sports game as they compete in a variety of Olympic events"
"Play a mobile phone game that captures the excitement of the Olympic Games with various events and challenges"
"Mario and Sonic in a winter sports-themed game, participating in events such as skiing, figure skating, and snowboarding"
"Experience the thrills of the Olympic Winter Games!"
"Mario and Sonic experience British-themed Olympics"
"Mario, Sonic, and their friends in a winter sports competition"
"Mario and Sonic experience Brazilian-themed Olympics"
"Mario, Sonic, and others experience Japanese-themed Olympics"
"Take part in the excitement of the Olympics in this mobile game!"
Math Notes: (filled with academic musings)
F(x) = Integral x,-1 (-t^3 + 2t^2 - 1) dt
F(x) = Integral x,-1 (-t^4 / 4 + 2t^3 / 3 - t) | x,-1
[-x^4 / 4 + 2x^3 / 3 - x] - [-(-1)^4 / 4 + 2(-1)^3 / 3 -(-1)]
[-x^4 / 4 + 2x^3 / 3 - x] - [1/12]
Derivative
[-4x^3 / 4 + 6x^2 / 3 - 1]
Simplify
-x^3 + 2x^2 - 1
Replacing t with x. Integral x,a constant on lower bound, replace the original variables with x BUT: Integral -1,x (-t^3 + 2t^2 - 1) dt
F(x) = Integral x,2 (t^3 - 10t^2 + 33t -34) dt
F(x) = Integral x,2 (t^4 / 4 - 10t^3 / 3 + 33t^2 / 2 - 34t) | x,2 dt
[x^4 / 4 - 10x^3 / 3 + 33x^2 / 2 - 34x] - [(2)^4 / 4 - 20(2)^3 / 3 + 33(2)^2 / 2 -2]
[x^4 / 4 - 10x^3 / 3 + 33x/2 - 34x] - [35/3]
Derivative
4x^3 / 4 - 30x^2 / 3 + 66x / 2 - 34
Simplify
x^3 - 10x^2 + 33x -34
lim n to infinity: 1/n = u
Delta x = b-a / n
Integral b,a f(x)dx = lim n to infinity: n E i=1 f(xi*) delta x
xi = a + i delta x
A = lim n to infinity n E i=1 f(xi) delta x
A = lim n to infinity n E i=1 f(xi - 1) delta x
A = lim n to infinity n E i=1 (xi*) delta x
n E i=1 i = n(n+1) / 2
n E i=1 i^2 = n(n+1)(2n+1) / 6
n E i=1 i^3 = [n(n+1) / 2]^2
Integral b,a cdx = c(b-a)
Integral b,a [f(x) + g(x) = Integral b,a f(x) dx + Integral b,a g(x) dx
Integral b,a cf(x) dx = C Integral b,a f(x) dx
Integral b,a [f(x)-g(x)] dx = Integral b,a f(x) dx - Integral b,a g(x) dx
A= lim n to infintiy: n E k=1 f(xk*) delta x = Integral f(x)dx
Delta x = b-a / n
Xk:
Right Endpoints: x* k = a + delta x k
Midpoint: x * k = a + delta x (k - 1/2)
Left Endpoints: x* k = a + delta x (k-1)
n E(1) k=1 = n
n E(k^2) k=1 = n(n+1)(2n+1) / 6
n E (k^3) k=1 = [n^2 (n+1)^2 / 4
n E c f(k) = C n E k=1 f(k)
n E k=1 (f(x) + g(x)) = n E k=1 f(x) + n E k=1 g(x)
Indefinite Integrals
Integral cf(x) dx = c Integral f(x) dx
Integral kdx = kx + C
Integral x^n dx = x^n+1 / n+1 + C (n should equal to -1)
Integral e^x dx = e^x + C
Integral sin x dx = -cos x + C
Integral sec^2 x dx = tan x + C
Integral sec x tan x dx = sec x + C
Integral 1 / x^2 + 1 dx = tan^-1 x + C
Integral sin h x dx = cos hx + C
Integral 1/x dx = ln |x| + C
Integral a^x dx = a^x / ln (a) + C
Integral cos x dx = sin x + C
Integral csc^2 x dx = -cot x + C
Integral csc x cot dx = -csc x + C
Integral 1 / square root 1-x^2 dx = sin^-1 x + C
Integral cos hx dx = sin hx + C
n E(1) i=1 =n
n E(c) i=1 =n(c)
n E(i) i=1 =n(n+1) / 2
n E(i^2) i=1 = n(n+1)(2n+1) / 6
n E(i^3) i=1 = [n(n+1) / 2]^2
Integral 5,0 3 dx = 3x | 5,0 = 3(5) = 15
Integral 10,8 x dx = x^2 / 2 | 10,0 = 10^2 / 2 - 0 = 100 / 2 = 50
f(x) = 2x^2 + 5
Delta x = 2-0 / n = 2/n
Index = xk / xn = 2(k) / n
f(1/k) = 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^2 / n^3 + 10/n]
f(x k) delta x
n E k=1 16k^2 / n^3 + n E k=1 10/n = 16/n^3
n E k=1 k^2 + 10/n n E k=1 (1)
Remembering that (n+1)(2n+1) = 2n^2 + 2n + n + 1
= 8(2n^2 + 3n + 1) + 10 / 3n^2
= L n to 10: 10 + 16/3 + 8/n + 1/3k^2
Since L n to 10: 10 + 16/3 + 8/n + 1/3k^2
= 30/3 + 16/3
= 46/3
A= lim n to infinity
Rn = lim n to infinity
n to infinity: infinity / # = 0
n to any #: #/# = #
Integral 1,-2 (5z^2 - 7z + 3) dz
Top - Bottom
F(1) - F(-2)
Fraction Power Rule
Integral (5z^3 / 3 - 7z^2 / 2 + 3z / 1) | 1,-2
(5(1)^3 / 3 - 7(1)^2 / 2 + 3(1)/1) - (5(-2)^3 / 3 - 7(-2)^2 / 2 + 3(-2) / 1)
7/6 - 100/3 = 69/2
Integral 4,1 (8 / square root t - 12 square root t^3) dt
Power Rule
Integral 4,1 (8 / square root t dt - 12) Integral 4,1 square root t^3 dt
Integral 8t^-1/2 dt - 12 Integral t^3/2 dt
8t^1/2 / 1/2 |4,1 - 1 = t^5/2 / 5/2 |4,1
16t^1/2 |4,1 - 24/5 t^5/2 |4,1
Do Not Forget the Power Rule
Integral 4,1 (8 / square root t - 12 square root t^3) dt
Integral 4,1 (8 / square root t dt - 12) Integral 4,1 square root t^3 dt
Integral 4,1 8t^-1/2 dt - 12 Integral 4,1 t^3/2 dt
8t^1/2 / 1/2 |4,1 - 12 (t^5/2 / 5/2) |4,1
16 [t^1/2 |4,1] - 12(2/5t^5/2 |4,1)
(2-1) - 24/5 (2^5 - 1) = -664/5
Do Not Forget the Power Rule
Integral x^1 dx = x^2 / 2 + C
x^1+1 / 2
Integral x^2 dx = x^3 / 3 + C
Integral x^1/2 dx = x^1/2 / 1/2 + 1
Do Not Forget the Power Rule
Integral 4,1 (8 / square root t - 12 square root t^3) dt
Integral 4,1 (8 / square root t - 12 square root t^3) dt
Power Rule
(8t^1/2 / 1/2 - 12t^5/2 / 5/2) |4,1
(8(4)^1/2 / 1/2 - 12(4)^5/2 / 5/2) - (8(1)^1/2 / 1/2 - 12(1)^5/2 / 5/2)
= -608/5 - 56/5
= -664/5