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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 4

         

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

Assume that the bar slides infinitely in both directions.

\[L_n=\lim_{x\to\infty}((x+a_1)(x+a_2)\cdots(x+a_n))^{\frac{1}{n}}-x,\]

where \( \large a_i = \frac{1}{2^i} \).

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

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

The above sum can be expressed in the form

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

where \(i\) is the imaginary unit and \(n\) is some positive multiple of 4.

Find \(m\).

\[ \large \int_{- \infty} ^ \infty \frac { e^{2x} - e^x } { x ( e^{2x}+1)( e^x+1) } \, dx \]

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

Give your answer your to 3 decimal places.

When \(n\) is natural, what is the value of

\[ \lim_{n \rightarrow \infty} ( -1 ) ^{n-1} \sin ( \pi \sqrt{ n^2 + 0.5 n + 1 } ) ?\]

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