June 8, 20XX
Today was a monumental day in the ongoing battle between the forces of good and the menacing Koopa army led by the fearsome Basilisx. The fate of the Mushroom Kingdom and Mobius hung in the balance as Mario, Sonic, Luigi, and Yoshi prepared to confront the Koopa unit leaders and put an end to their reign of terror. The Battle of Mushroom Raceways now reached day six.
The day began with Shadow valiantly holding his ground against the relentless onslaught of the blue spikey robot, ensuring that no further reinforcements from the Koopa army could harm the Mushroom Kingdom and Mobius. His determination and skill in battle were truly remarkable, and the combined efforts of the futuristic intergalactic Earth's military forces only strengthened their defensive position.
As the battle raged on, more information about this battle's inhabitants approached my curiosity, accompanied by another military official and Mario's trusty friend, Stuffwell. They shared the exciting news that King Bowser's Koopalings, each equipped with their formidable attacking military vehicles, were ready to join the fight. It was a glimmer of uncertainty amidst the chaos, as the Koopalings had proven themselves to be a force to be reckoned with in the past.
Suddenly, a powerful robot, bearing an uncanny resemblance to Mario, appeared on the battlefield. Its imposing presence sent shockwaves through the ranks of both the heroes and the Koopa army. The robot was an enigma, its origins and intentions shrouded in mystery. It stood as a formidable obstacle between our heroes and victory.
I was informed that the blue spikey robot that Shadow was facing recently and continuously was known as Mecha Sonic (the creator was Dr. Eggman, who often creates copied robotic clones of his nemesis, Sonic) and the new robotic foe was called as Mecha Mario.
With renewed determination and a firm belief in the power of teamwork, Mario, Sonic, Luigi, and Yoshi rallied their strength and faced the remaining Koopa unit leaders head-on. Despite the overwhelming odds, they fought with unwavering courage and an unwavering sense of purpose.
In a display of incredible coordination, each member of the heroic quartet utilized their unique abilities to maximum effect. Mario's nimble agility and skilled jumps, Sonic's blazing speed and lightning-fast attacks, Luigi's uncanny precision and strategic thinking, and Yoshi's relentless ferocity all combined to create a formidable force that the Koopa army simply could not withstand.
One by one, the Koopa unit leaders fell, their defeat at the hands of our heroes signaling a turning point in the battle. The once-dreaded Basilisx was no more, and the remaining Koopa forces were left in disarray. The combined might of Mario, Sonic, Luigi, and Yoshi had proven to be an unstoppable force, united by a common goal and an unyielding spirit.
As the dust settled, our heroes regrouped, their eyes now fixed upon the powerful robot that stood in their path. With King Bowser's Koopalings at their side and the support of the futuristic intergalactic Earth's military forces, they prepared for their final confrontation.
The outcome of this battle remained uncertain, but the resolve of Mario, Sonic, Luigi, and Yoshi remained unshaken. Together, they would face any challenge that stood in their way, fighting with everything they had to ensure the safety and peace of the Mushroom Kingdom and Mobius.
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