June 7, 20XX
Today, I had the incredible fortune of witnessing an epic battle that unfolded before my very eyes. It was a clash of heroes and villains, a confrontation that will surely be recounted in the annals of history. I shall now recount the events in detail, as they unfolded on this eventful day.
I had the extraordinary privilege of being present at a monumental battle, one that surpassed all expectations and unfolded as a true testament to heroism and courage. The events that transpired on this fateful day will forever be etched in the annals of history. I shall recount every detail of this remarkable encounter that unfolded before my very eyes.
As the sun began its descent, casting an orange hue across the horizon, the once tranquil field transformed into a war zone. The atmosphere crackled with tension as the faint rumble of approaching aircrafts reached our ears. It was then that I spotted them, flying through the sky with remarkable agility and speed. It was none other than Mario, Sonic, Luigi, and Yoshi, the iconic quartet of heroes, returning from an intense air battle.
A once serene field transformed into a chaotic theater of war. The air crackled with anticipation, and the distant rumble of approaching aircraft grew louder with each passing moment. Suddenly, a spectacle of sheer exhilaration unfolded in the sky above.
Their faces were etched with determination, and their eyes gleamed with a fierce resolve. Mario, with his iconic red cap and blue overalls, led the charge with his trusty fireballs at the ready. Sonic, the blue blur, streaked through the air, his trademark red shoes leaving a trail of energy in his wake. Luigi, ever the loyal sidekick, followed closely behind, wielding his tools with ghostly prowess. And last but not least, Yoshi soared through the sky, unleashing devastating egg projectiles upon their foes.
As they landed gracefully on the ground, the heroes confronted their adversaries, the formidable Koopa unit leaders. Basilisx and the one-eyed Koopa leader, stood at the forefront, his menacing presence sending shivers down my spine. Beside him were the blue-eyed Koopa unit leader, exuding an air of icy cunning, and the glasses-wearing Goomba unit leader, who seemed to be the brains behind the operation. Lastly, there was the red turban-wearing Koopa unit leader.
With a forceful rallying cry, Mario initiated the battle, charging towards Basilisx with unwavering determination. Sonic, utilizing his lightning-fast speed, dashed around the battlefield, launching devastating spin attacks at his foes. Luigi, ever the strategist, skillfully maneuvered through the chaos, using his vacuum to suck up enemies and turn the tide of battle. And Yoshi, with his incredible agility and aerial prowess, unleashed a flurry of egg projectiles, striking fear into the hearts of their adversaries.
Mario, the fearless plumber clad in his iconic red cap and blue overalls, led the charge with unwavering determination etched across his face with a huge metal mallet hammer of his own. Fireballs danced at his fingertips, ready to be unleashed upon their adversaries. Sonic, the blue blur with blazing red sneakers, streaked through the sky, leaving a trail of energy in his wake as he prepared to strike with lightning speed.
Luigi, Mario's loyal and often unsung brother, followed closely behind. Armed with his trusty tools such as a huge hammer mallet, he exuded an air of resilience and determination. His green hat cast a shadow over his focused eyes, as if harboring a quiet intensity.
Yoshi, the dinosaur with a vibrant, eggshell-patterned back, soared through the air with unmatched grace. He expelled a flurry of egg projectiles, ready to rain chaos upon their foes.
Meanwhile, on the other side of the battlefield, a separate conflict was underway. Shadow, the enigmatic anti-hero, led the forces of G.U.N in a fierce face-off against a colossal blue spikey robot. The ground shook beneath their feet as powerful energy blasts and projectiles illuminated the battlefield. Shadow's red eyes burned with determination as he skillfully dodged the robot's attacks, retaliating with his Chaos powers.
Yet, amidst this incredible display of heroism, the battle was not fought solely by the renowned quartet. The dear friend of mine who was the benefactor that brought me and my British friend here, played an instrumental role in coordinating their efforts. With a calm demeanor and a mind sharpened by experience, he was reported to be standing on the sidelines, offering invaluable strategic guidance and support.
His presence elevated the heroes' performance, serving as a beacon of wisdom amidst the chaos. Through precise instructions and tactical insights, he ensured that every move was calculated, maximizing the heroes' chances of victory. His expertise and leadership were indispensable, forging a powerful alliance between her strategic mind and the might of Mario, Sonic, Luigi, Shadow, and Yoshi.
As this grand confrontation unfolded, a separate battle raged on another part of the battlefield. Shadow led the forces of G.U.N against a colossal blue spikey robot. The ground quaked beneath their feet as powerful energy blasts and projectiles illuminated the war-torn landscape.
Shadow's blood-red eyes burned with an unwavering determination as he skillfully evaded the robot's relentless attacks. His movements were swift and precise, a testament to his combat prowess and the chaos-infused powers of Chaos Control that coursed through his veins. Alongside him, G.U.N soldiers fought with unwavering loyalty, their weapons firing in synchronized harmony.
The battlefield itself became a maelstrom of chaos and heroism. Explosions ripped through the air, their concussive force threatening to shatter the very fabric of reality. Cries of battle, both triumphant and agonized, merged into a symphony of conflict. The clash of forces echoed through the field, a testament to the unwavering will of both heroes and villains alike.
Math Notes: (filled with academic musings):
f(x)=x^3 e^x / ln(x)
d/dx f(x)/g(x) = g(x) d/dx(f(x)) - f(x) d/dx(g(x)) / g(x)^2
(ln(x)) d/dx(x^3 e^x) - d/dx(ln(x)) (x^3 e^x) / (ln(x))^2
Remember: d/dx ln(x)=1/x and that e^x doesn't change
=(ln(x))(3x^2 e^x + x^3 e^x) - (1/x)(x^3 e^x) / (1/x)^2
...
f(b) = e^3b - 4 / 1+e^7b
g(x) d/dx f(x) - d/dx g(x) f(x) / g(x)^2
(1+e^7b) d/dx(e^3b - 4) - d/dx(1+e^7b)(e^3b - 4) / (1+e^7b)^2
Chain Rule is applied: e^3b = e^3b(3) or (3)e^3b
(1+e^7b)(e^3b (3)) - (e^7b (7))(e^3b - 4) / (1+e^7b)^2
=(1+e^7b)(3e^3b) - (7e^7b)(e^3b - 4) / (1+e^7b)^2
...
f(x)=x^4 e^x / ln(x)
g(x) d/dx (f(x)) - f(x) d/dx(g(x)) / g(x)^2
(ln(x)) d/dx(x^4 e^x) - (x^4 e^x) d/dx(ln(x)) / (ln(x))^2
(ln(x))(4x^3 e^x) - (x^4 e^x)(1/x) / (ln(x))^2
...
f(b) = e^4b - 5 / 2+e^8b
(2+eb^8b) d/dx(e^4b - 5) - d/dx(2+e^8b)(e^4b - 5) / (2 + e^8b)^2
=(2+e^8b)(4e^4b) - (8e^8b)(2+e^8b)^2
...
Implicit Differentiation
d/dx (x^2 + y^2 = 36_
= 2x+2y(dy/dx)=0
2y(dy/dx) = -2x/2y
dy/dx= -2x/2y
...
x^3 y^7 = 5
f(x) d/dx(g(x)) + g(x) d/dx(f(x))
x^3 (7y^6)(dy/dx) + y^7 (3x^2)=0
dy/dx= -3y^7 x^2 / 7x^3 y^6
...
d/dx (x^3 y^5 = 5)
x^3 (7y^6)(dx/dy) + 3x^2 y^6 = 0
...
x^4 y^8 = 6
(x^4)(8y^7)(dy/dx)+(4x^3)(y^8)=0
x^4 8y^7(dy/dx) + 4x^3 y^8 = 0
x^4 8y^7 (dy/dx) = -4x^3 y^8
= -4x^3 y^8 / x^4 8y^7
...
x^3 y^7 + x^3 y^7
x^3 (7y^6)(dy/dx) + (3x^2) y^7=0
x^3(7y^6)dy/dx / x^3 (7y^6) = -3x^2 y^7 / x^3 (7y^6)
dy/dx= -3x^2 y^7 / x^3 (7y^6)
...
sin(x) + cos(y) = e^4y
d/dx (sin(x) + cos(y) = e^4y)
cos(x)-sin(y) = 4e^4y (dy/dx)
cos(x)=4e^4y(dy/dx)+sin(y)(dy/dx)
cos(x)=dy/dx[4e^4y + siny] / 4e^4y + sin(y)
dy/dx = cos(x)/[4e^4y + sin(y)]
...
Derivatives are always first
dy/dx = f'(x) = triangle y / triangle x = slope
Plug it in
y= -x^2 +8x^2 - 20x + 14 at (2,-2)
y is m (slope)
f'(x) = 17 = m
f'(x)=17x-36
-2=mx+b
-2=3x+b
b=-36
Plug x to set slope = (m)
Use y=mx+b