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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.

# Calculus Warmups: Level 5 Challenges

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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