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.
by
Matt Enlow
\[L_n=\lim_{x\to\infty}((x+a_1)(x+a_2)\cdots(x+a_n))^{\frac{1}{n}}-x,\]
where \( \large a_i = \frac{1}{2^i} \).
Find \(\displaystyle{\lim_{n\to\infty} (nL_n)}\).
by
Sanjeet Raria
\[\sum_{n=0}^{\infty} \frac{1}{(4n)!}\]
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 \(m\).
by
Jake Lai
\[ \large \int_{- \infty} ^ \infty \frac { e^{2x} - e^x } { x ( e^{2x}+1)( e^x+1) } \, dx \]
The integral above has a closed form. Find this closed form.
Give your answer to 3 decimal places.
by
Calvin Lin
When \(n\) is a positive integer, what is the value of
\[ \lim_{n \rightarrow \infty} ( -1 ) ^{n-1} \sin ( \pi \sqrt{ n^2 + 0.5 n + 1 } ) ?\]
by
Sanjeet Raria