# Tag Info

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### How can I argue that $f(x) = (2-x)^3-x+\frac{3}{2}$ is decreasing with a polynomial?

$$f'(x)=-3x^2+12x-13 = -3(x^2-4x+4)-1 = -3(x-2)^2-1<0 \text{ for all } x\in\mathbb R$$
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### Spivak, Ch. 20, Problem *4(i): Write down a sum which equals $\sin{1}$ with an error of less than $10^{-10^{10}}$.

$\sin 1 = \sum_{n=0}^\infty (-1)^{2n+1} {1 \over (2n+1)!}$ which is alternating so $|\sin 1 - \sum_{n=0}^N (-1)^{2n+1} {1 \over (2n+1)!} | \le {1 \over (2N+2)!}$. You want $(2N+2)! > 10^{10^{10}}$ ...
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### Question on the natural logarithm laws

Alternative approach: Assume that $\log(x) = r \implies e^r = x.$ Then $\displaystyle e^{(3r)} = \left[e^r\right]^3 = x^3.$ Thus, $3r = \log(x^3).$ Thus $3 \times \log(x) = \log(x^3).$
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### Proving that the maximum of two convex functions is also convex

Here is another way of seeing this. First notice we can rewrite the maximum function as follows: $$\max(x,y) \,=\, \frac{x+y+|x-y|}{2}.$$ So if $f,g$ are convex functions, using the fact that the ...
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### Variational Calculus - Derivation of Lagrangian Equation

About the reason for imposing the condition that the added variation $\eta(x)$ should be continuous and differentiable: The way that Jacob Bernoulli solved the Brachistochrone problem also underlines ...
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### if $f(x)f(y) = y^h f(x/2) + x^k f(y/2)$ for all positive reals $x,y$, then $f$ must be identically zero

Partial Solution: First, note that $$f(0)^2 = 0^hf(0) + 0^kf(0) = 0.$$ The remainder of this answer assumes that $f(1) = 0$. Now with $x = y = 1$, $$0 = f(1/2)+f(1/2) \implies f(1/2) = 0.$$ Finally, ...
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### I need some help with a specific probability question. It's about radioactive decay.

Since the particle can only decay once, the probability of decay in a particular interval is equal to the expected number of decays in that interval. Because of this, you can use linearity of ...
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### uniform continuity on $(a, b]$ implies limit at $a^+$ exists and finite

Here's a way without Cauchy sequences. We prove first that $f$ is bounded on $(a,b]$. By uniform continuity, there is some $\eta \in (0,b-a)$ such that $|x-y|<\eta \implies |f(x)-f(y)|<1$. Since ...
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### Derivative of an integral with respect to a function

I just see this question posted 5 years ago because I got into a similar problem. I struggled for days but I think I got a reasonable answer that convince me. So your integration variable x is ...
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### Asymptotic expansion as $r\to 0^+$ for $(2re^r -3(e^r-1))^2$ using small o
Take your initial expansions out a little further: $2re^r = 2r+2r^2+r^3+O(r^4)$ and $3(e^r-1) = 3r+3r^2/2+r^3/2+O(r^4)$; you now get $2re^r-3(e^r-1) = -r+r^2/2-r^3/2+O(r^4)=-r+r^2/2+O(r^3)=-r+O(r^2)$....