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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. See more

Level 5

         

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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