June 4, 20XX
Today was an extraordinary day filled with adrenaline and excitement as I witnessed an epic battle unfold between two iconic teams in the gaming world. The Mushroom Kingdom and Mobius military forces joined in battle against the second wave of enemy targets as Mario, Sonic, Luigi, Shadow, and Yoshi joined forces to confront their arch-rivals who has employed the notorious Koopa Bros and the formidable Axem Rangers.
The sun shone brightly in the clear sky as the two teams faced each other on a vast open field. Mario, the courageous and charismatic plumber, took the lead, rallying his comrades for the impending fight. Sonic, with his trademark speed and determination, stood by Mario's side, ready to unleash his lightning-fast attacks. Luigi, the younger brother of Mario, showed a mix of bravery and trepidation, while Shadow, the dark hedgehog, exuded an aura of confidence and power. Yoshi, the familiar dinosaur that Mario is often friends with, brought his agility and unique abilities to complete the heroic team.
On the opposing side, the Koopa Bros, consisting of Red, Black, Yellow, and Green, exhibited their mischievous grins, their shells glistening with wicked intent. The Axem Rangers, a group of robotic warriors, comprised of Red, Black, Yellow, Green, and Pink, appeared unfazed, their metal bodies reflecting the sunlight with an intimidating glow.
As the battle commenced, chaos ensued. Mario and Sonic leaped into action, their combined strength creating a whirlwind of punches and kicks. Luigi, though initially hesitant, found his courage and cast his green magic, stunning the Koopa Bros with a burst of power. Shadow, with his supernatural abilities, showcased his chaos control, manipulating time and space to gain an upper hand. Yoshi, true to his nature, unleashed a barrage of egg projectiles, catching the Axem Rangers off guard.
The Koopa Bros retaliated with a barrage of explosive shells, each aiming to disrupt the unity of the heroes. But Mario's quick thinking, coupled with Sonic's speed, allowed them to dodge the onslaught. Luigi's lightning strikes countered the Axem Rangers' relentless attacks, temporarily disabling their weapons. Shadow utilized his darkness-infused powers, casting shadows that immobilized the Koopa Bros, rendering them unable to move.
Acknowledging the strength and valor each brought to the battle, the Mushroom Kingdom and Mobius remained safe, for now, thanks to the indomitable spirit of their champions.
The battle raged on, a true spectacle of strength, strategy, and resilience. The heroes and villains clashed with fury, each unleashing their special abilities and signature moves. The ground trembled with each impact, and the air crackled with energy.
Eventually, through a combination of teamwork, skill, and sheer determination, the heroes prevailed. Mario's iconic stomp sent the Koopa Bros flying, and Sonic's supersonic spin dashed through the ranks of the Axem Rangers. Luigi's green thunderbolts and Shadow's chaos-infused blasts provided the final blows, leaving the villains defeated and forced to retreat.
With certain civilians such as the nearby Yoshi village's inhabitants and my female British friend, we witnessed the multiple clashes between enemy forces and this world's protectors.
As the clash of heroes and villains erupted, chaos reigned over the once peaceful landscape. The vibrant colors of the Yoshi people's village became a stark contrast to the raging storm of combat that surrounded them. Their joyful expressions turned to fear and confusion as the ground shook beneath their feet.
Mario, with his unwavering determination to protect the innocent, took charge. He shouted instructions to his comrades, urging them to be mindful of the Yoshi people's safety. Sonic, his blue blur of motion, darted around the battlefield, creating sonic booms that temporarily shielded the Yoshi people from harm's way. Luigi, his bravery fueled by the need to protect those in need, formed a defensive barrier using his magic to safeguard the Yoshi people from stray projectiles.
Shadow, normally stoic, utilized his chaos control to teleport the Yoshi people to a safer location. With each flicker of his crimson eyes, he ensured their swift and secure escape from the encroaching danger. Meanwhile, Yoshi, ever loyal and resourceful, used his nimble movements to shield the Yoshi people from harm, his sturdy body acting as a living shield against the onslaught.
But even with the heroes' valiant efforts, the battle continued to rage, its ferocity threatening to engulf the Yoshi people's village. The Koopa Bros and the Axem Rangers, driven by their relentless pursuit of power, showed no mercy, launching a barrage of attacks that tore through the surroundings.
In the face of such imminent danger, Mario devised a daring plan. He called upon Sonic to distract the villains, luring them away from the Yoshi people's village. With his lightning-fast speed and agility, Sonic led the villains on a wild chase, his acrobatic maneuvers drawing their attention away from the innocent.
Meanwhile, Luigi, Shadow, and Yoshi worked together to create a diversion within the village itself. Luigi cast illusions, confusing the enemy and buying precious moments. Shadow used his darkness-infused powers to create a protective shield, guarding the Yoshi people against any stray attacks. Yoshi, relying on his bond with the Yoshi people, rallied them to a hidden safe haven, ensuring their safety throughout the chaos.
As the battle drew to its climax, the heroes and villains clashed with unprecedented ferocity. Sonic's speed and Mario's relentless determination pushed the Koopa Bros and the Axem Rangers to their limits. With each blow exchanged, the balance of power seemed to shift, but the heroes never wavered in their resolve to protect the Yoshi people.
Finally, the combined strength of Mario, Sonic, Luigi, Shadow, and Yoshi proved overwhelming. The villains, defeated and disarmed, retreated from the battlefield, their ambitions quelled for the time being.
Exhausted and relieved, the heroes turned their attention to the Yoshi people, who emerged from their safe haven, timid but grateful for the heroes' unwavering protection. Mario, Sonic, Luigi, Shadow, and Yoshi embraced the Yoshi people, their victory overshadowed by the reminder of the fragility of innocent lives caught in the turmoil of battle.
As the sun set on the battlefield, casting a warm glow over the wounded land, I couldn't help but reflect on the cost of conflict. The battle had been won, but the scars left behind.
To reduce my nervousness; I read my math notes again.
Math Notes: (filled with academic musings)
d/dx (c)=0
d/dx [cf(x)]=cf'(x)
d/dx [f(x) + g(x)]=f'(x)+g'(x)
d/dx [f(x)+g(x)]=f'(x)+g'(x)
d/dx [f(x)-g(x)]=f'(x)-g(x)
d/dx [f(x)-g(x)]=f'(x)-g'(x)
Product Rule: [f(x)g(x)]=f(x)g'(x)+g(x)f'(x)
Quotient Rule: d/dx[f(x)/g(x)]=g(x)f'(x)-f(x)g(x) / [g(x)]^2
Chain Rule: d/dx f(g(x))=f'(g(x))g'(x)
Power Rule: d/dx (x^n)=nx^n-1
...
d/dx (sinh^-1 x)= -1 square root 1+x^2
d/dx (cosh^-1 x)= -1 square root x^2 - 1
d/dx (tanh^-1 x)= -1/1-x^2
d/dx (csch^-1 x)= -1/1x1 square root x^2 + 1
d/dx (sech^-1 x)= -1/x square root 1-x
d/dx (coth^-1x)= 1/1-x^2
...
d/dx (e^1)= e^1
d/dx (a^1)= a^1 ln(a)
d/dx ln |x| = 1/x
d/dx (log(n)x)=1/xlna
...
d/dx (sin x)= cos x
d/dx (cos x)= -sin x
d/dx (tan x)= sec' x
d/dx (csc x)= -csc x cot x
d/dx (sec x)= sec x tan x
d/dx (cot x)= -csc^2 x
...
d/dx (sin^-1 x)= 1/square root 1-x^2
d/dx (cos^-1 x)= -1/square root 1-x^2
d/dx (tan^-1 x)= 1/1+x^2
d/dx (csc^-1 x)= -1/x square root x^2 - 1
d/dx (sec^-1 x)= 1/ x square root x^2 - 1
d/dx (cot^-1 x)= -1/1+x^2
...
d/dx (sin hx) = cos hx
d/dx (cos hx) = sin hx
d/dx (tan hx) = sec h^2 x
d/dx (csc hx)= -csc hx cot hx
d/dx (sec hx)= -sec hx tan hx
d/dx (cot hx) = -csch'x
...
1. dy/dx
y=-12x^7/4
dy/dx (-12x^7/4)
(7/4 X -12x^7/4-1)
-21x^3/4
2. f'(x) = (x^2) cos (x)
d/dx (x^2) cos (x)
f'(x) = 2x(-sin (x))
f(x) (d/dx g(x)) + g(x) (d/dx f(x))
f(x)=x^2
g(x)=cos(x)
f' = x^2 d/dx(cos(x))+(cos(x)) d/dx(x^2)
x^2 (-sin x) + cos(x) 2x
or x^2 (-sin x) + 2x cos(x)
3. y=(3t^4)/2t-5
g(t) dy/dt(f(t)) - f(t) dy/dt(2t-5) / (2t-5)^2
(2t-5)(12t^3) - (3t^4)(2) / (2t-5)^2
24t^4 - 60t^3 - 6t^4 / (2t-5)^2
=18t^4 - 60t^3 / (2t-5)^2
...
Derivatives
f(x)=x^-4 - 9x^-3 + 8x^-2 +12
f'(x)= -4x^-5 + 27x^-4 - 16x^-3
Derivatives
f(x)= square root x + 8 (3 square root) x + 10 (5 square root) x^3
f'(x)=1/2x^1/2-1 + 8/3^1/3-1 + 10(3/5)x^3/5-1
f'(x)=1/2x^-1/2 + 8/3(x)^-2/3 + 6x^-2/5
Derivatives
f(x)= square root x + 9 (3 square root) x + 11 (5 square root) x^4
f'(x)= 1/2(x)^1/2-1 + 3x^1/3-1 + 11(4/5)x^4/5-1
f'(x)= 1/2(x)^-1/2 + 3x^-2/3 + 44/5(x)^-1/5
Derivatives
f(y) = (y-4)(2y+y^2)^5
Chain Rule, Product Rule
f'(y)=(y-4) d/dx(2y+y^2)^5 + d/dx(y-4)(2y+y^2)^5
=(y-4)[5(2y+y^2)^4 (2+2y)] + (1)(2y+y^2)^5
Derivatives
(y-5)(3y+y^3)^6
f'(y)=(y-5) d/dx(3y+y^3)^6 + d/dx(y-5)(3y+y^3)^6
(y-5)[g(3y+y^3)^5 (3+3y^2)] + (1)(3y+y^3)^6
Derivatives
(y-6)(4y+y^4)^5
f'(x)=(y-5)[5(4y+y^4)(4+4y^3)] + (1)(4y+y^4)^5
f'(x)=(y-5)[5(4y+y^4)(4+4y^3)] + (1)(4y+y^4)^5
Derivatives
f(n)=4n^3 X tan(6n^2 -7n)
(4n^3) d/dx[tan(6n^2-7n)] + d/dx(4n^3)(6n^2-7n)
(4n^3)[sec^2(6n^2-7n)(12n-7)]+(12n^2)(tan(6n^2-7n)
Derivatives
f(n)=5n^4 [tan(7n^3-8n)]
(5n^4) d/dx[tan(7n^3-8n)] + d/dx(5n^4)[tan(7n^3-8n)]
(5n^4)[sec^2(7n^3-8n)(21n-8)]+(20n^3)[tan(7n^3-8n)]
Derivatives
f(t)= t^2/3 / t+t^3
(t+t^3) d/dx(t^2/3)-(t+t^3) d/dx(t+t^3) / (t+t^3)^2
(t+t^3) 2/3(t^-1/3)-(t+t^3)(1) / (t+t^3)^2
Derivatives
f(theta) = cos(2theta) / cos(17theta)
g(x)f'(x)-f(x)g'(x) / g(x)^2
cos(17theta) d/dx(cos(2theta))-(cos(2theta)) d/dx(cos(17theta)) / cos^2(17theta)
(cos(17theta)(-2sin(2theta))-(cos(2theta)(-17sin(17theta)) / cos(17theta)^2