The Journey of Big O Who is the largest here? It's me! No!! I'm larger than you! I'm the largest! Why? With the same perimeter ... 2 2 = Lx - x 2 Area (A) = x ( L - x ) 2 L Let’s start with rectangular How can we know that

A is max or min? 4 x = L max min A or A when 4 x = L Finding maximum of A 2 dx d (A) 2 L - 2x dx 2 d (Lx - x ) = 0 = 0 ( ) L 4 When x = L x = L x = L 2 4 6 A = 0 A = L 2 2 16 A = L 18 MAX lower lower so A will be the maximum when x = L 4 L n h ¶ n ¶ - ¶ 2 n Perimeter of Jen = L Equilateral polygon, with n sides, consist of n isosceles triangles angle around the point (center) = 360 or 2¶ 2¶ n Note! Equilateral polygon is a polygon which has all sides of the same length ... Note! To find area of Jen, we need to find area of a isosceles triangle Let's call her "Jen" Hi, I'm Jen To find area of a isosceles triangle, we need to know the height of the triangle To know the height of the triangle, we have to encounter Trigonometry!!!! tan ¶ - ¶ = h 2 n L 2n h = L tan ¶ - ¶ 2n 2 n Area of a triangle = 1 . base . height 2 = 1 . L . L tan ¶ - ¶ ( ) ( ) ( ) 2 n 2n 2 = L tan ¶ - ¶ 4n 2 n 2 2 ( ) 0 = So Jen is composed of n isosceles triangles Area of Jen = n . area of a isosceles triangle Area of a triangle = 1 . base . height 2 !@#$%%@!@##%$%^$#@& But don't worry, it's just basic x Opposite Hypotenuse Adjacent sin x = cos x = tan x = O H A H O A n area of a triangle = L tan ¶ - ¶ 4n 2 2 2 n ( ) And let ¶ = y or n = ¶ n y L y tan ¶ - y 4¶ 2 ( ) 2 L y sin x cos x = 4¶ cos ¶ - y sin ¶ - y 2 2 2 sin A = cos (90-A)

cos A = sin (90-A)

tan A = cot (90-A) sin y cos y Jen's area cos y sin y y 4¶ 2 L L y cos y 4¶ sin y 2 = We want to proof that circle has the largest area so let n ∞ If

Jen

has 5 corners Pentagon 10 corners Decagon 18 corners Octdecagon ∞ corners or x 0 ( from letting ¶ = x ) n L 4¶ cos y sin y y lim 2 y 0 1 1 = L 4¶ 2 L (Perimeter of a circle) = 2¶r Perimeter of a rectangle = L 4¶ r 2 2 4¶ = Note! = So complicated!! n = ∞ ¶r 2 Let us explan about this. Next, take look at equilateral polygon that has n sides (or n corners) 2 2 L - L - x = L - x x To take this journey, you must know

basic calculus

basic trigonometry

basic geometry The Journey of Big O L 2n cos 0 = 1 0 0 0 0 lim sinA A A 0 = 1 Note! so area of Jen is n . L tan ¶ - ¶ 4n 2 n 2 2 ( ) n = 5 n = 10 n = 18 Circle Area of a Circle Who has the largest area?

and You are WINNER, Big O !!

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# math

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