Calculus

Calculus Warmups

Calculus Warmups: Level 4 Challenges

         

The ends of a stiff bar AB\overline{AB} of length 4 slide freely inside a parabolic track (specifically, the parabola y=x2y=x^2). As they do, the midpoint MM of that bar traces a curve. Find the area of the region between the parabola and the curve traced by MM.

Assume that the bar slides infinitely in both directions.

Ln=limx((x+a1)(x+a2)(x+an))1nx,L_n=\lim_{x\to\infty}((x+a_1)(x+a_2)\cdots(x+a_n))^{\frac{1}{n}}-x,

where ai=12i \large a_i = \frac{1}{2^i} .

Find limn(nLn)\displaystyle{\lim_{n\to\infty} (nL_n)}.

n=01(4n)!\sum_{n=0}^{\infty} \frac{1}{(4n)!}

The above sum can be expressed in the form

1n(k=1mneik)\frac{1}{n} \left( \sum_{k=1}^{mn} e^{i^{k}} \right)

where ii is the imaginary unit and nn is some positive multiple of 4.

Find mm.

e2xexx(e2x+1)(ex+1)dx \large \int_{- \infty} ^ \infty \frac { e^{2x} - e^x } { x ( e^{2x}+1)( e^x+1) } \, dx

The integral above has a closed form. Find this closed form.

Give your answer to 3 decimal places.

When nn is a positive integer, what is the value of

limn(1)n1sin(πn2+0.5n+1)? \lim_{n \rightarrow \infty} ( -1 ) ^{n-1} \sin ( \pi \sqrt{ n^2 + 0.5 n + 1 } ) ?

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