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# Calculus Warmups

The concepts of limits, infinitesimal partitions, and continuously changing quantities paved the way to Calculus, the universal tool for modeling continuous systems from Physics to Economics.

All these points are taken **in given order** for constructing a **convex** quadrilateral, which has maximum possible area.

Find minimum possible value of \(a+b+c+2\alpha +4\beta.\)

A rectangle's bottom is at \(y=0\)

while its top corners are on the curve \(y=x{ (x-1) }^{ 2 }\) between \(x=0\) and \(x=1\). The maximum area of this rectangle can be expressed as

\[ \dfrac { a\sqrt { a } -b }{ c\sqrt { d } }\]

where \( a\) and \( d\) are prime numbers. What is the sum \(a+b+c+d?\)

(Don't count \( a\) twice)

Let

\[ f(x) = \sin^{2} x - \frac{1}{2} \sin2x \times \sin^{2}x + \frac{1}{3} \sin3x \times \sin^{3}x - \ldots\]

Then \( \displaystyle f \left( \frac{\pi}{12} \right) \) can be written as

\[ \tan^{-1} \left( \dfrac{ a - \sqrt{b}}{c} \right)\]

where \(b\) is square free & \(a,b,c \in\mathbb N\). Find the value of \(a+b+c\).

Enter the answer for \(r=10\), rounded to the nearest hundredth.

\[ \displaystyle \lim _{ n\rightarrow \infty }{ \frac { \displaystyle \sum _{ r=1 }^{ { 2 }^{ n-1 }-1 }{ \tan ^{ 2 }{ \left (\frac { r\pi }{ { 2 }^{ n } } \right ) } } }{ { 4 }^{ n } } } \]

For positive integer \(n\), the limit evaluates to \( \dfrac a b\) for coprime positive integers \(a,b\). What is the value of \(a+b\)?

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