Calculus
# Calculus Warmups

Consider a curve \(y=ax^2 +bx+c\) with \( a, b, c \in \mathbb{N} \) which passes through four points \(A(-2,3),B(-1,1),C(\alpha ,\beta ),D(2,7)\).

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 \(a+b+c\).

A cow is tied to a silo with radius \(r\) by a rope just long enough to reach the opposite end of the silo.

Find the area available for grazing by the cow.

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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