MarkFL ‎
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6 followers
MarkFL's posts
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Floor Function Problem...
Solve the following equation: $\displaystyle \left\lfloor x+\frac{7}{3} \right\rfloor^2-\left\lfloor x-\frac{9}{4} \right\rfloor=16$ Note: $\displaystyle \lfloor x \rfloor$ denotes the largest integer not greater than $x$. This function, referred to as the ...﻿
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Find The Area Of The Equilateral Triangle
Show that the curve $x^3+3xy+y^3=1$ has only one set of three distinct points, $P$, $Q$, and $R$ which are the vertices of an equilateral triangle, and find its area. My solution: The first thing I notice is that there is cyclic symmetry between $x$ and $y$...﻿
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Find The Sum Involving The Inverse Tangent Function
We are given to evaluate: $S_n=\sum_{k=0}^n\left[\tan^{-1}\left(\frac{1}{k^2+k+1} \right) \right]$ My solution: Using the identity: $\tan^{-1}(x)=\cot^{-1}\left(\frac{1}{x} \right)$ we may write: $S_n=\sum_{k=0}^n\left[\cot^{-1}\l...﻿ Post has attachment Analytic Geometry: Orthogonal Trajectories The problem is as follows: a) Find the family of circles centered on the y-axis, that pass through the points (\pm a,0), where [math]0<a\le r\in\mathbb{R}$. b) Find the family of curves orthogonal to the family of circles found in part a). Hint 1:...﻿
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Probability: Coin Tossing
A while back I helped a student with a probability problem, and I took the problem, generalized it a bit, and wish to post it here. Here is the problem: A coin has the probability $p$ of turning up heads when tossed. Suppose we toss the coin $2n$ times, whe...﻿
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Trigonometric Inequality
Show that : $\left( {\sin x + a\cos x} \right)\left( {\sin x + b\cos x} \right) \leq 1 + \left( \frac{a + b}{2} \right)^2$ Let: $A=\tan^{\small{-1}}(a)$ $B=\tan^{\small{-1}}(b)$ Using a linear combination, we may write the i...﻿
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Maximize The Area Of The Lune
In the $xy$ plane, there are two circles, a larger one of radius 1 unit and a smaller one of radius $r$. The two circle intersect such that the two points of intersection are on a diameter of the smaller circle. Find the value of $r$ which maximizes the are...﻿
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Maximizing The Trajectory Of A Projectile
Suppose we are asked to compute the launch angle which will maximize the arc-length of the trajectory for a projectile, assuming gravity is constant and is the only force on the projectile after the launch. Eliminating the parameter $t$, we find the object'...﻿
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Test Post: Embedding Desmos API
Here is an interactive graph Did it work?﻿
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Minimize The Crease
Consider a rectangular piece of paper of width $W$ laid on a flat surface. The lower left corner of the paper is bought over to the right edge of the paper, and the paper is smoothed flat creating a crease of length $L$, as in the diagram: What is the minim...﻿