June 13, 20XX
In the ongoing battle against the forces of evil as one hundred brave soldiers of the military embarked on a daring mission to take down two formidable adversaries. Their objective was to eliminate Mecha Mario, a towering robotic version of the Mushroom Kingdom's heroic guardian, while it engaged in combat with Robo Mario. Additionally, they also aimed to neutralize Metallix, a formidable metallic enemy, as it clashed with a powerful team comprising Mario the Red Plumber, Sonic the Hedgehog, Luigi the Green Plumber, Shadow the Hedgehog, and Yoshi the Dinosaur.
The operation began at dawn as the soldiers, armed to the teeth with state-of-the-art weaponry and advanced battle gear, assembled near the battleground. The tension in the air was palpable, but the resolve in their eyes spoke volumes about their determination and unwavering commitment to protect the Mushroom Kingdom and Mobius.
The clash between Mecha Mario and Robo Mario was awe-inspiring, with powerful blows echoing across the landscape. Our soldiers swiftly split into two groups, one assigned to engage Mecha Mario and the other focused on Metallix. I was part of the group targeting Mecha Mario, a towering metallic behemoth.
As we closed in, Mecha Mario's red eyes glowed ominously, and its metallic armor shimmered in the morning light. The soldiers around me tightened their grip on their weapons, ready to unleash a barrage of firepower. The plan was to exploit its weak spots, the joints and sensors, to disable and eventually dismantle the imposing machine.
With a battle cry that reverberated through the air, they charged forward with their gunfire raining upon Mecha Mario. The air filled with the sound of bullets, explosions, and the clanging of metal. The soldiers, exhibiting exceptional discipline and coordination, aimed for the targeted areas, hoping to slow down and eventually incapacitate the colossal enemy.
However, Mecha Mario proved to be a formidable opponent. Its advanced defensive systems deflected a significant portion of our attacks, and it retaliated with powerful energy blasts and devastating melee strikes. The sheer force behind each blow sent soldiers flying, prompting others to maneuver swiftly to avoid the impending danger.
Meanwhile, on the other side of the battlefield, the combined might of Yoshi, Luigi, Shadow, Sonic, and Mario engaged in an intense battle against Metallix. The ground shook as their attacks collided, unleashing shockwaves that reverberated across the landscape. Despite the odds, our heroes fought valiantly, exploiting their individual strengths to counter the metallic menace.
As many combative groups continued the assault on Mecha Mario, the soldiers adjusted their tactics. Some utilized heavy artillery to weaken its armor, while others focused on precision shots aimed at its joints. The sheer number of soldiers allowed us to coordinate our attacks effectively, overwhelming the robotic giant with a relentless onslaught.
With each passing minute, Mecha Mario's movements grew sluggish, its armor dented and scorched. We knew victory was within their grasp, and the soldiers' determination surged as they pressed the attack. Exploiting a brief moment of vulnerability, they concentrated our firepower on Mecha Mario's central power core.
The battle against the mechanical menace, Mecha Mario, quickly transformed into a horrifying massacre. Their initial plan to neutralize the towering robotic adversary alongside Robo Mario took a devastating turn, leading to the loss of countless lives. Mecha Mario, driven by an unknown force, turned against our soldiers, ruthlessly slaughtering them with unmatched brutality.
The day began with a sense of determination as our allied soldiers prepared to engage Mecha Mario in combat. Little did we know the dark fate that awaited us. As they closed in on the colossal machine, its red eyes flickered with an unholy gleam, and the once-mechanical noises emanating from it transformed into an eerie, malevolent robotic sound. In an instant, it began its merciless assault.
The soldiers, caught off guard, found themselves facing an enemy unlike anything they had ever encountered. Mecha Mario's attacks were swift and lethal, tearing through t their ranks with terrifying efficiency. The air was thick with the stench of burning metal and the anguished cries of our fallen comrades. It was a scene of unimaginable horror, one that will forever haunt those who survived.
Amidst the chaos, a group of brave individuals rallied to provide aid and support to the heavily wounded soldiers. Lee, the Duplighost known for his shape-shifting abilities, used his powers to create makeshift stretchers and transport the injured to safety. The Toad Town Dojo's Master, a seasoned warrior, provided medical assistance and offered words of advice to those in need. Goompa the Goomba, Chan the Buzzy Beetle, and Rawk Hawk, renowned figures in the Mushroom Kingdom, formed a protective shield around the wounded, ensuring their safe passage.
The sight of their battered and bloodied comrades ignited a renewed sense of determination within this group of heroes. With the injured soldiers in their care, they vowed to seek justice for the fallen and to stop Mecha Mario's rampage. Guided by their unwavering resolve, they retreated to a secure location, tending to wounds and regrouping for a counterattack.
As the group made their way through the devastated battlefield, the echoes of battle continued to reverberate. The clash between Mecha Mario and the remaining soldiers waged on, but their hope was diminished, their morale shattered by the horrifying onslaught.
Math Notes: (filled with academic musings)
Finding the tangent
y=-x^3 + 8x^2 - 20x + 14 at (2,-2)
dy/dx or slope (m) = -3x^2 + 16x - 20 at (2,-2)
m= -3(2)^2 + 16(2) - 20
m=0 or y'=0
y=mx+b
y=(0)x+b
y=(0)2-2
b=-2
Answer: y=-2
...
y= 6 / x^2+2 at (3, 6/11)
dy/dx = (x^2 + 2) d/dx(6) - (6) d/dx(x^2+2) / (x^2 + 2)^2
dy/dx = (x^2 + 2) (0) - (6) (2x+0) / (x^2 + 2)^2
dy/dx = (3^2 + 2) (0) - (6)(2 times 3+0) / (3^2 + 2)^2
dy/dx: m=-36/121
y=mx+b
y=(-36/121)x+b
(3, 6/11)
6/11=(-36/121)(3)+b
174/121=b
y=-36/121x + 174/121
...
y=-2sin(x) at (0,0)
dy/dx=-2cos(x)
m=-2cos(0)
m=0
-2cos(0)
=-2(1)
m=-2
y=(-2)x+b
y=(-2)(0)+b
0=(-2)(0)+0
b=0
y=(-2)x+0
...
y=(2x-4)^1/3 at (-2,-2)
dy/dx=1/3 (2x-4)^1/3-1
f'(x)=f(g(x)) Chain Rule
1/3 (2x-4)^1/3 - 1 (2)
dy/dx=1/3 (2x-4)^-2/3 (2)
(-2,-2)
x=-2
dy/dx=1/3 (2(-2)-4)^-2/3 (2)
dy/dx=1/6
m=1/6
y=(1/6)x+b
(-2)=(1/6)(-2)+b
b=-7/3
y=(1/6)x - 7/3
...
y= -3sin(x) at (0,0)
dy/dx=-3sin(x)
m=-3cos(0)
m=0
cos(0)=1
So: -3cos(0)
=-3(1)
m=-3
y=(-3)x+b
y=(-3)(0)+b
b=0
y=(-3)x+0
...
y=(2x-4)^1/4 at (-2,-2)
y=(3x-5)^1/4 at (-3,-3)
dy/dx=1/4(3x-5)^1/4-1
f'(x)=f(g(x)) chain rule
dy/dx=1/4(3x-5)^1/4 - 1 (3)
(-3,-3)
dy/dx=1/4(3(-3)-5)^1/4 - 1 (3)
dy/dx=1/4(3(-3)-5)^-5/4 (3)
Domain Error
...
y=(3x-5)^1/2 at (-3,-3)
dy/dx = 1/2(3x-5)^1/2-1
f'(x)=f(g(x)) chain rule
dy/dx=1/2(3x-5)^-1/2 (3)
dy/dx=1/2(3(-3)-5)^-3/2(3)
Domain Error
...
Mean Value Theorm
Continious [a,b] vs (a,b)
f'(c)=f(b)-f(a) / b-a
mtan=msec
m=y2-y1 / x2-x1
f(x)=y
f(b)=y2
f(a)=y1
...
f'(c)= f(b)-f(a) / b-a
f(x)=2x^3 + x^2 + 7x - 1 on [1,6]
f(a)=2(1)^3 + (1)^2 + 7(1) - 1
f(a)=9
f(b)=2(6)^3 + (6)^2 + 7(6) - 1
f(b)=509
509-9 / 6-1 = 100
f'(c)=100
Derivative
dy/dx = f'(x) = 6x^2 + 2x + 7
100=6x^2 + 2x + 7
0=6x^2 + 2x - 93
-b+/- square root b^2 - 4(a)(c) / 2(a)
-2+/- square root 2^2 - 4(6)(-93) / 2(6)
=3.77... and -4.11...
x=3.77...
Plug in the x to the f(x)'s x
=(3.77... , 147.15...)
...
f(z)=4z^3 - 8z^2 + 7z - 2 on [2,5]
f(a)=4(2)^3 - 8(2)^2 + 7(2)-2
f(a)=12
f(b)=4(5)^3 - 8(5)^2 + 7(5) - 2
f(b)=333
333-12 / 5-2 = 321/3 = 107
f'(c)=107
dy/dx=f'(z)=12z^2 - 16z + 7
107=12z^2 - 16z + 7
0=12z^2 -16z - 100
Since difficult to factor: -b+/- sqaure root b^2 - 4(a)(c) / 2(a)
-(-16)+/- square root (-16)^2 - 4(12)(-100) / 2(12)
x= 3.63...
x=3.63...
y=4(3.63...)^3 - 8(3.63...)^2 + 7(3.63...) - 2
y=109.26...
(3.63... , 109.26...)
...
1) Find the derivative set = 0. Solve for x
2) Create a number line and plug in points close to the extremas to determined if the extremas are relative with minimums and maximums.
3) Plug in the endpoints as well as the relative extremas.
4) Determine the absolute extrema by carrying y-values
f(x)=x^2 - 4x + 9; [1,4]
x=2
f(1)=1^2 - 4(1) + 9
1-4+9=6
f(2)=2^2 - 4(2) + 9
4-8+9=5
f(4)=4^2 - 4(4) + 9
16-16+9=9
f(x)=x^2-4x+9
f'(x)=2x-4=0
0=2(x-2)
x=2
Number line: Before 2 has negative range, after 2 has positive range
(1,6)
(2,5): absolute min
(4,9): absolute max
...
y=-x^3 + x^2 - 3; [-1,1]
y'=-3x^2 + 2x
0=-3x^2 + 2x
x(-3x+2)
x=0, x=2/3
f(-1/3)=-3(-1/3)^2 + 2(-1/3)
f(-1/3)=-1
f(1)=-1
f(1/3)= -3(1/3)^2 + 2(1/3) = 1/3
Max: (-1,-1)
(2/3, -2.85...)
Min: (0,3), (1-3)