# Tag Info

1 vote

### Problem 237 "Mathematical Quickies:270 Stimulating Problems with Solutions" Particle Movement

Here’s another rigorous explanation that doesn’t require thinking about triangles. Assume that the acceleration is at less than $4$ always. Repeatedly integrating and using that $f’(0)=0$, we have ...
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1 vote

### Prove $\frac{d}{dx} \frac{f(x)-f(a)}{x-a} \geq 0$ if $f(x)$ is convex without twice differentiability.

I don't think you're proving what you were asked. You said that you were asked to show that $f'(x)$ is increasing but instead showed that $\frac{f(x)-f(a)}{x-a}$ is, and $f'(x)$ and the other ...
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### Determining if a series converges or or diverges

For $x = -4$, think about the alternating series test. You have to check that the terms are indeed decreasing to zero in absolute value. For $x = 4$, think about Stirling's approximation to find an ...

### usage of Leibniz notation for things like $\frac{d^2y}{dt^2}$ and $\frac{dy'}{dy}$

For higher-order differentials to work algebraically, you need to adopt a notation that is a little non-standard. The typical notation, $\frac{d^2y}{dx^2}$ does not allow for algebraic manipulations. ...
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1 vote

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### Must the $x$ and $y$ axes have the same units? (Coordinate Geomtry)

Take South St. Paul, Minnesota, USA. House numbers are 1600 per mile along one axis, and 800 per mile along the other, to fit their particular block structure. I.e.. nobody is forcing one to use the ...
• 107

### Is there such a thing as the "Second Passage Time"?

Based on the comments provided by @Zack Fisher, I tried to write an answer: Using the Law of Conditional Probability: If $X$ and $Y$ are two random variables, the joint probability density ...
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### Using Graph of $\frac{1}{x}$ to Find $\delta$

Hint: First observe that solving for $1/x = .35$ and subtracting from $4$ will work but not solving for $1/x = .15$ and so on. And also check on which side you have to round. Edit: you can also check ...
• 41

### Evaluating $\lim_{x\to\infty}\left[\cos\frac1x\right]^{h(x)}$, where $h(x)=\frac{x^4+x^2-1}{2x+1}\sin\frac1x$

$\textbf{Hint:}$ For large $x$, we have that $$\cos\left(\frac{1}{x}\right) \sim 1 - \frac{1}{2x^2}$$ and $$\frac{x^4+x^2-1}{2x}\sin\left(\frac{1}{x}\right) \sim \frac{x^2}{2}$$ therefore your limit ...
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Accepted

### Bump function with integral $1$ and value $1$ at zero

Consider $$g_0(x)=f\left(x-\frac12\right)f\left(\frac12-x\right)$$ This is a smooth bump function centred at the origin, but $g_0(0)\neq1$. We can fix that with $g_1(x)=\frac{g_0(x)}{g_0(0)}$. Now for ...
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### Problem 237 "Mathematical Quickies:270 Stimulating Problems with Solutions" Particle Movement

Here's a more mathematically rigorous explanation of what the solution is saying: Suppose there exists a path $f(t)$ such that $f'(t)$ does not exceed the bounds of this isosceles triangle (whose path ...
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1 vote

### Integration of $1/x$ as a Riemann sum

$$\lim_{t\to 0}\frac{a^t - 1}{t} = \ln{a}$$ so $$\lim_{n\to \infty}\frac{(b/a)^{1/n} - 1}{1/n} = \ln{b/a}$$
Accepted

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### Issue in numerical integration of $\frac{1}{4\pi^{3/2}i}\int_Ce^{-\frac{z^3}{12}-tz}z^{-1/2}dz$

$$\operatorname{Ai}^2(t)=\frac{1}{{4 \pi ^{3/2} i}}{\int_{\sigma-i \infty }^{\sigma+i \infty } e^ \left(\color{red}+\frac{z^3}{12}-t z\right)\cdot z^{-\frac{1}{2}} \, dz}$$ Use SciPy Gaussian ...
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• 319
1 vote

### Find a sequence function for combinatorial sequences

Given by Mathematica, the sequence is $$a_{n}=\frac{30 (n-3)\, a_{n-1} +\left(18370 n^2-82832 n+92261\right)}{30 n+17}\qquad \text{with} \qquad a_1=0$$.
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• 33.8k
1 vote
Accepted

### Why $\lim_{x \rightarrow \infty } \frac {P(x)}{Q(e^x)} = 0$ for polynomials $P(x)$ and $Q(x)$?

I'm not sure I understand your objection. In general, if the numerator of a fraction grows sufficiently slower than its denominator, regardless of which infinity that denominator goes to ($+\infty$ or ...
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### Why $\lim_{x \rightarrow \infty } \frac {P(x)}{Q(e^x)} = 0$ for polynomials $P(x)$ and $Q(x)$?

If $Q(x)=a_0+a_1x+\cdots+a_nx^n$, with $n\in\{1,2,3,\ldots\}$ and $a_n\ne0$, then\begin{align}\lim_{n\to\infty}\frac{Q(e^x)}{e^{nx}}&=\lim_{n\to\infty}a_0e^{-nx}+a_1e^{-(n-1)x}+\cdots+a_{n-1}e^{-x}...
1 vote
Accepted

• 19.6k
1 vote

### About solution to homogeneous ODE $u'' + u = 0$

Another approach, let $f(x) = u(x)^2+(u'(x))^2$. Then $f(0) = 0$ and $f'(x) = 0$ hence $f(x) = 0$.
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### Find solution of the IVP $y' = y+ \frac12 |\sin(y^2)|,\,\, x>0,\,\, y(0) = -1$
You should check first that the right hand side is locally Lipschitz in $y$. Picard-Lindelöf implies existence and uniqueness. Under these circumstances, solutions to first order autonomous ODEs (like ...