June 18, 20XX
Today marked the conclusion of the epic Battle of the Mushroom Raceways, an event that will forever be etched in history. The day began with a sense of anticipation and excitement, as forces from different dimensions and universes clashed in a battle of epic proportions. Little did we know that it would end with a mysterious explosion on King Bowser's and Dr. Eggman's main doomship.
As the battle raged on, the combined might of Mario, Sonic, Luigi, Shadow, and Yoshi led the charge against the formidable alliance of Metallix and Mecha Mario. Their forces, comprising of Koopas and Badniks, were relentless and threatened to overrun our defenses. The air crackled with energy as projectiles flew and explosions rocked the landscape.
Amidst the chaos, I found myself among a group of military soldiers and civilians, desperately been ordered them to retreat to a safe distance. The devastation that unfolded before our eyes was unprecedented. The Mushroom Raceways, once a vibrant and lively place, now lay in ruins. It was a sight that tore at my heart, witnessing the destruction of a place that held so many cherished memories for the inhabitants here.
In the midst of many military leaders giving orders, I searched frantically for my two friends, a woman with a British accent and an older-looking benefactor with grey hair. They were the ones who had brought themselves with me to this futuristic era using their incredible time machine/spaceship. Their presence had been invaluable in preparing us for this battle, and now, I feared for their safety.
The explosion on King Bowser's and Dr. Eggman's doomship caught everyone off guard. A massive ball of fire and debris erupted into the sky, casting an eerie glow over the battlefield. The shockwave rippled through the air, knocking many of us off our feet. The scene was surreal, as if time had momentarily stood still.
As the smoke and dust settled, I scanned the area for any signs of my friends, hoping against hope that they had managed to escape the devastation. Yet, they were nowhere to be found. Questions swirled in my mind—had they been caught in the explosion? Did they make it out in time? The uncertainty gnawed at my heart, and I felt an overwhelming sense of loss.
In the aftermath of the explosion, it became clear that Metallix and Mecha Mario were retreating, their armies of Koopas and Badniks scattering in disarray. The battle might have been won, but at a great cost. The Mushroom Raceways lay in ruins, and the fate of my friends remained unknown.
After the long and arduous Battle of the Mushroom Raceways, where chaos and destruction reigned, I find myself consumed by the need to find Mario and Sonic. These two legendary heroes have been through countless adventures and battles (so I have been told), and I believe they hold the key to discovering the fate of my dear friends.
Mario and Sonic have crossed paths with my friends on numerous occasions, and their shared experiences may hold vital information or clues that could lead me to them. Their bond, forged through their joint efforts against formidable adversaries, gives me hope that they might have some knowledge or insight into the whereabouts of my friends.
Despite the fatigue and devastation surrounding us, I am fueled by determination. I know that the path to finding Mario and Sonic will not be easy, especially in the aftermath of such a cataclysmic event. However, the resilience and perseverance displayed by these heroes time and time again inspires me to press forward.
I gather a small group of trusted allies (other survivors from this battle that have been caught off guard from the recent destruction) and share my plan. Together, we will embark on a quest to locate the now-proclaimed heroes at this time, Mario and Sonic, by seeking their aid in our search for my missing mentoring friends. We are aware that they may be preoccupied with the aftermath of the battle, aiding in the recovery efforts and attending to their own wounded allies. Nonetheless, we will approach them with the utmost respect and urgency, hoping they will understand the gravity of our situation.
As we traverse the ruined landscape of the Mushroom Raceways, the scope of the destruction weighs heavily upon us. The once vibrant and bustling raceways now lie in ruins, serving as a stark reminder of the fierce struggle that took place here. We navigate through the debris, ever vigilant for any sign of Mario and Sonic.
Hours turn into days as we venture forth, encountering remnants of defeated enemies and signs of the battle that ensued. We come across wounded soldiers and civilians, offering assistance where we can, but our primary focus remains on finding our missing friends.
I have come across a surprising sight that has left me both intrigued and concerned. Among the group of heroes, I notice Luigi wearing the unmistakable attire of a black jumpsuit that strangely retains his former overalls' large buttons. Also wearing a green bandana. The sight of him in this outfit raises questions, as I wonder about the circumstances that led to his adoption of this guise during the last hours of this battle.
Accompanying Luigi is a figure I was introduced later named Rosalina, a powerful and enigmatic being from the cosmos who lives in the Mushroom Kingdom. Her presence in this critical juncture piques my curiosity further, as I try to comprehend the extent of her involvement in the events that have transpired. With her cosmic abilities and wisdom, Rosalina's inclusion in our search gives me hope that she may hold valuable insights or even possess the means to aid us in finding my friends.
Finally, after what feels like an eternity, we catch sight of familiar figures in the distance. Mario and Sonic, standing amidst the rubble, their faces etched with exhaustion but determination.
Math Notes: (filled with academic musings)
y=-x^3 + x^2 - 3
y'=-3x^2 + 2x
= x(-3x+2)
Slope=0 at 0 and 2/3
-1000(-3(-1000)+2)
f(-1)=-1 [ab max], f(0)=-3 [ab min], f(2/3)=-2.85, f(-1)=-3 [ab min]
y=-3 / x^2 - 4; [-1,4]
f(x): -3
g(x): x^2 - 4
y' = g(x) d/dx(f(x)) - f(x) d/dx(g(x)) / g(x)^2
y' = (x^2 - 4) d/dx(-3) - (-3) d/dx(x^2 - 4) / (x^2 - 4)^2
y' = (x^2 - 4)(0) - (-3)(2x) / (x^2 - 4)^2
0=(x^2)(0) - (-3)(2x) / (x^2 - 4)^2
0=(x^2)(0) - (-3)(2x) / (x^2 - 4)^2
0=6x/(x^2 - 4)^2
(x^2 - 4)^2
(x^2 - 4)(x^2 - 4)
6y=0
y=0
x=0
[-1,4]
Plugging it in -3 / x^2 - 4
f(-1)=1
f(0)=3/4
f(2)=DNE
f(4)=-1/4
No Ab Min. No Ab Max
f(x) = 3x+6 / x-1
Graph is increasing so its derivative is positive
Graph is decreasing so its derivative is negative
f '(x)=(x-1)(3) - (3x+6)(1) / (x-1)^2
0=3x - 3 - 3x - 6 / (x-1)^2
0= -9 / (x-1)^2
0= -9 / (x-1)^2
No interval that is making it increasing
Decreasing: (-infinity, infinity)
(-infinity, 1) U (1, infinity)
f '(x)=-9 / (x-1)^2
Finding the local extremas; f'(x)=0
If no x-value, no local extrema
(Derivative=0 then there is a local extrema)
Concave up and concave down
Concave up: slope is negative (-,+)
Concave down: slope is positive (+,-)
No local extrema, No concave up or down
(Concave example; f(x)=x^2 + 14x + 49)
Inflection Points
No concave up or down, No inflection points
Find all asymptotes
You cannot make denominator equal to zero
3x+6 / x-1 = 0
x should not equal to 1, Vertical asymptote at x=1
Inflection points are different than extremes
f "(x)=0
f '(x)= -9 / (x-1)^2 = -9(x-1)^-2
f "(x)= 18(x-1)^-3 = 18 / (x-1)^3
No Inflection points
f(x)= x^5 - 5x^4
f '(x)= 5x^4 - 20x^3
0=5x^4 - 20x^3
0=5x^3 (x-4)
Pull out the greater exponent
x=0, x=4
x=-1; +
x=1; -
x=5; +
f '(x)=5x^4 - 20x^3, x=-1
=5(-1)^4 - 20(-1)^3
=25; +
f '(x)=5x^4 - 20x^3, x=1
=-15; -
x=5
=625; +
Increasing: (infinity, 0) U (4, infinity)
(infinity, 0): +
(0,4): -
(4, infinity): +
Decreasing: (0, 4)
f(x)= x^5 - 5x^4
f '(x)=5x^4 - 20x^3
f "(x)=20x^3 - 60x^2
Max: (0,0)
Min: x^4 (x-5) = 4^4 (4-5) = -1
f(x)=x^5 - 5x^4
f '(x)=5x^4 - 20x^3 = 5x^3 (x-4) 0,4
f "(x)=20x^3 - 60x^2 = 20x^2 (x-3) 0,3
f(-1)= 5(-1)^3 (-1-4) = +
f(1)=5(1)(1-4)= -
f(10)=5(10^2)(10-4)= +
Max: (0,4)
f(0)=0
Min: (4, -256)
4^3 - 54 = -256
f "(x) = 20x^3 - 60x^2 = 20x^2 (x-3)
f "(-1)= 20(-1)^2 (-6-3)= -
f "(1)= 20(1)^2 (1-3)= -
f "(4)= 20(4)^2 (4-3)= +
f(3)=3^5 - 5(3)^4
Inflection Points: (3, -162)
f(3)= 3^5 - 5(3)^4
=3^4 (3-5) = -162
No Fraction, No Asymptotes
-Local extremas are first derivatives
-Inflection points are second derivatives