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4 votes
Accepted

What is the Maximum Theoretical Angle a Grand Piano Could be Held At?

One strategy for solving problems like this is to solve a simpler form of the problem if you can. Note that $$ \tan B = \frac{b\sin C}{a - b\cos C}, $$ which is much simpler than the expression for $\...
David K's user avatar
  • 100k
2 votes

Computing the Limit $\lim_{n \to \infty} n\left[ \frac{a_{n+1}}{a_{n}} - \left(\frac{n}{n+1}\right)^{\frac{1}{3}} \right]$

The expression can be represented as $${n\over 2(n+1)}-\left ({n\over n+1}\right )^{4/3}\,[-(n+1)]\left [\left (1-{1\over n+1}\right )^{2/3}-1\right ]$$ The limit of the last two factors is equal by ...
Ryszard Szwarc's user avatar
1 vote

Laplace transform of $\sin(\omega t)$

There are a few ways to find $\int \sin(\omega t) e^{-s t} dt$. One way is via integration by parts, and another is by using Euler's identity $e^{i x} = \cos x + i \sin x$ to relate the integral to ...
ConMan's user avatar
  • 25.6k
1 vote
Accepted

Help understanding Spivak's solution, and a verification of my proof. Spivak Chapter 5, Question 3(vi)

OP's analysis is fine, but he makes a small mistake. On the one hand we have $1-\epsilon < \sqrt x < 1 + \epsilon$, which after squaring leads to $(1-\epsilon)^{2} < x < (1+\epsilon)^{2}$. ...
M. Wind's user avatar
  • 3,781
1 vote
Accepted

Proving existence of certain step functions implies integrability

One way to circumvent this problem would be to take the $s_1 = s$ and $s_2 = t$ (you went from one notation to another, that is something to look out for) corresponding to $\varepsilon/2$ instead of $\...
Bruno B's user avatar
  • 5,849
1 vote
Accepted

Proof a curve is a geodesic on a sphere S

I'll still post a solution because it could be handy in terms of the surfaces more complicated than a sphere. The basic idea was to somehow define the tangent plane, or, rather, separate it from the ...
Egor Larionov's user avatar

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