June 10, 20XX
I am writing this entry as I witness the evacuation of civilians from the raging battlefield where Mario, Sonic, Luigi, Shadow, Yoshi, and military forces are pitted against the formidable alliance of King Bowser and Dr. Eggman's military forces. During this siege, allied military forces are provided reinforcements by a nearby establishment known as Diamond City and a country's monarch named Ōsama with his commander.
As sirens blared and emergency announcements echoed through the streets, the residents quickly realized the urgency of the situation. Panic began to set in, but the city's evacuation plan swung into action promptly, orchestrated by the brave men and women of our military forces. Their unwavering commitment to protect and serve the people was evident as they coordinated the safe passage of civilians to designated evacuation points.
The sight was both surreal and awe-inspiring. With each passing moment, the skyline was transformed into a backdrop of chaotic clashes and spectacular displays of power. Buildings shook from the force of Mario's mighty jumps and Sonic's lightning-fast movements, while Luigi's thunderous thunderbolt strikes sent shockwaves rippling through the air. Shadow, with his mysterious demeanor, fought with incredible speed and agility, matching the formidable foes blow for blow, and Yoshi's long tongue lashed out, engulfing enemies whole. Their combined might was a sight to behold.
However, amidst the fierce battle, the military forces and emergency personnel worked tirelessly to ensure the safety of the fleeing civilians. They directed and guided us through the chaos, their reassuring presence calming our frayed nerves. The streets were filled with people of all ages, clutching their belongings tightly, seeking solace in the arms of their loved ones. Families huddled together, children's eyes wide with a mix of awe and fear, as they sought refuge from the calamity that raged around them.
In the face of imminent danger, the heroes fought valiantly to protect us from the villainous onslaught. Bowser's massive frame dominated the battlefield, roaring with fury as he unleashed fireballs and commanded legions of his minions. Dr. Eggman's robotic army advanced relentlessly, their mechanical claws clashing with the heroes' unwavering determination. It was a battle of wills, each side giving no quarter and demanding nothing less than victory.
As the evacuation continued, the heroes' resilience and bravery became a beacon of hope for all who witnessed their extraordinary feats. The occasional glimpse of Mario's iconic red hat or Sonic's blur of blue brought forth a surge of inspiration, reminding us that even in the darkest times, heroes rise to defend the innocent. Their selfless acts of heroism echoed through the city, igniting a spark of courage within each evacuee.
As nightfall descends upon the city, I can still hear the distant echoes of the battle. The heroes and villains continue to clash, their struggle for supremacy reaching a crescendo. But we, the evacuated civilians, find solace in the knowledge that we are safe, sheltered from the turmoil that ravages the fields.
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)