June 11, 20XX
Today marked a significant turn of events in the ongoing clash between heroes and villains. Turbo Mecha Sonic, an incredibly powerful robotic creation, underwent a formidable upgrade (was informed that this foe went on transforming into the fearsome entity known as Metallix). This upgrade was executed with a singular purpose in mind: to challenge and defeat the current belligerents Mario, Sonic, Luigi, Shadow, and Yoshi.
The imminent clash between Turbo Mecha Sonic and the combined might of the heroic team had been circulating for weeks. The anticipation hung heavy in the air, as both sides prepared for the ultimate showdown.
Dr. Robotnik, the brilliant yet sinister mastermind behind Turbo Mecha Sonic and Mecha Mario, had been working tirelessly in his secluded laboratory. His dedication and knowledge of robotics were unmatched, and he had spared no expense in upgrading his creation to its fullest potential. Metallix was the result of countless experiments and enhancements, boasting superior speed, strength, and intelligence compared to its previous incarnation.
As the sun reached its zenith, the heroes assembled in a vast open field, brimming with confidence and determination. Mario, the iconic plumber from the Mushroom Kingdom, stood at the forefront, rallying his companions with words of encouragement. Sonic the Hedgehog, his blue quills glistening in the sunlight, radiated an aura of unwavering resolve. Luigi, Shadow the Hedgehog, and Yoshi, each with their unique abilities, stood shoulder to shoulder, united in their quest to protect their respective worlds.
The tension grew palpable as Turbo Mecha Sonic emerged from the shadows, its metallic frame gleaming ominously. The transformation into Metallix had elevated it to a whole new level, with sleek and deadly features. Its crimson eyes burned with a cold, calculated intensity, betraying an upgraded artificial intelligence capable of analyzing every movement and anticipating the heroes' actions.
The battle began with a fierce clash of powers and abilities. Mario leaped into the air, launching fireballs towards Metallix, but the upgraded robot dodged them effortlessly, exhibiting newfound agility. Sonic, with his incredible speed, attempted to outmaneuver the mechanical monstrosity, but Metallix matched his every move, its reflexes operating at an unparalleled level.
Luigi and Shadow, utilizing their unique skills, launched a coordinated attack, combining the forces of darkness and ghostly prowess. However, Metallix proved to be an impenetrable fortress, deflecting their combined assault with ease. Yoshi, known for his versatility, tried to unleash his egg-throwing ability, but Metallix swiftly countered, utilizing an energy shield to nullify the attack.
As the battle raged on, the heroes realized the immense challenge they faced. Metallix was a formidable opponent, exhibiting a level of power they had never encountered before. The heroes needed to strategize quickly and find a weakness in their metallic adversary's defenses.
With their combined efforts, the heroes managed to exploit a split-second vulnerability in Metallix's defense. Sonic's lightning-fast reflexes allowed him to deliver a precise strike, momentarily disabling one of Metallix's systems. Seizing the opportunity, Mario and Luigi unleashed a devastating combination attack, exploiting the exposed weakness and dealing a significant blow to the upgraded robot.
However, Metallix's durability was astounding. It quickly recovered, retaliating with devastating force, sending shockwaves through the battlefield. The heroes found themselves on the back foot, forced to regroup and reevaluate their strategies.
The battle between good and evil raged on, neither side relenting. The heroes pushed themselves to their limits, leveraging every ounce of power and skill they possessed. Their determination was unwavering, knowing that the fate of their worlds hung in the balance.
As the sun began to set, the clash reached its climax. Fatigue began to take its toll on the heroes, while Metallix, fueled by its relentless pursuit of victory, appeared undeterred. The outcome of this titanic struggle remained uncertain, with both sides refusing to yield.
Today's events served as a stark reminder of the enduring power of evil and the unyielding resolve of heroes. Metallix, the upgraded form of Turbo Mecha Sonic, proved to be an adversary unlike any the heroes had faced before. The battle against the combined might of Mario, Sonic, Luigi, Shadow, and Yoshi was far from over, and the outcome remained uncertain.
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)