Calculus

Calculus Warmups: Level 4 Challenges

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.

When $$n$$ is a positive integer, what is the value of
$\lim_{n \rightarrow \infty} ( -1 ) ^{n-1} \sin ( \pi \sqrt{ n^2 + 0.5 n + 1 } ) ?$