Calculus Warmups

Calculus Warmups: Level 5 Challenges


Consider a curve y=ax2+bx+cy=ax^2 +bx+c with a,b,cN a, b, c \in \mathbb{N} which passes through four points A(2,3),B(1,1),C(α,β),D(2,7)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+2α+4β.a+b+c+2\alpha +4\beta.

A rectangle's bottom is at y=0y=0

while its top corners are on the curve y=x(x1)2y=x{ (x-1) }^{ 2 } between x=0x=0 and x=1x=1. The maximum area of this rectangle can be expressed as

aabcd \dfrac { a\sqrt { a } -b }{ c\sqrt { d } }

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

(Don't count a a twice)


f(x)=sin2x12sin2x×sin2x+13sin3x×sin3x f(x) = \sin^{2} x - \frac{1}{2} \sin2x \times \sin^{2}x + \frac{1}{3} \sin3x \times \sin^{3}x - \ldots

Then f(π12) \displaystyle f \left( \frac{\pi}{12} \right) can be written as

tan1(abc) \tan^{-1} \left( \dfrac{ a - \sqrt{b}}{c} \right)

where bb is square free & a,b,cNa,b,c \in\mathbb N. Find the value of a+b+ca+b+c.

A cow is tied to a silo with radius rr 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=10r=10, rounded to the nearest hundredth.

limnr=12n11tan2(rπ2n)4n \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 nn, the limit evaluates to ab \dfrac a b for coprime positive integers a,ba,b. What is the value of a+ba+b?


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