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: [math]S_n=\sum_{k=0}^n\left[\tan^{-1}\left(\frac{1}{k^2+k+1} \right) \right][/math] My solution: Using the identity: [math]\tan^{-1}(x)=\cot^{-1}\left(\frac{1}{x} \right)[/math] we may write: [math]S_n=\sum_{k=0}^n\left[\cot^{-1}\l...

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**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}[/math]. 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 : [math]\left( {\sin x + a\cos x} \right)\left( {\sin x + b\cos x} \right) \leq 1 + \left( \frac{a + b}{2} \right)^2[/math] Let: [math]A=\tan^{\small{-1}}(a)[/math] [math]B=\tan^{\small{-1}}(b)[/math] 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...

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