Calculus
# Calculus Warmups

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 to 3 decimal places.

When \(n\) is a positive integer, 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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