Sign up to access problem solutions.

Already have an account? Log in here.

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

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.*

Sign up to access problem solutions.

Already have an account? Log in here.

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

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

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

Sign up to access problem solutions.

Already have an account? Log in here.

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

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 \(m\).

Sign up to access problem solutions.

Already have an account? Log in here.

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

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

Give your answer your to 3 decimal places.

Sign up to access problem solutions.

Already have an account? Log in here.

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

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

Sign up to access problem solutions.

Already have an account? Log in here.

×

Problem Loading...

Note Loading...

Set Loading...