# Tagged Questions

For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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I was reading and doing problems from Spivak's Calculus on Manifolds. Q1-6 (a) stumped me a little. Let $f$, $g$ be integrable on $[a,b]$. Prove that $$\left| \int_a^b f\cdot g \; \right | \leq \left(... 1answer 87 views ### Books and sources concerning the mathematics of Leibniz and the feud with Newton I am trying to find books and other sources concerning the mathematical history of Leibniz, including the controversy due to the independent discoveries of calculus by both Newton and Leibniz. I can't ... 3answers 52 views ### Question about limits \lim_{x\to\infty}\frac{x-2}{e^{1/x}\cdot x} How to calculate this:$$\lim_{x\to\infty}\frac{x-2}{e^{1/x}\cdot x}$$3answers 198 views ### Possible alternative way of expressing continuity of a function? In Calculus or Real Analysis the usual form of definition of continuity of a function is \epsilon- \delta def. From a rigorous point of view, is it possible to say this way? and if so, why?: f(\... 1answer 355 views ### How to find singular points of a function without knowing the graph? Problem: Let f(x) = (x-1)^{2/3} - (x+1)^{2/3}. Locate and classify all local extreme values of this function. Determine whether any of these extreme values are absolute. Attempt at solution: We ... 2answers 47 views ### Continuous derivative vs Continuous partial derivatives Firstly, suppose f:\mathbb R^n\to\mathbb R^m has all continuous partial derivative. I believe I have proved that this imply continuous derivative. Please tell me if this is actually true. For its ... 1answer 75 views ### Solving x - a \log(x)=b Let a>0 and b \in \mathbb{R}: Assume there exists an x >0  s.t.$$x - a\log(x) = b$$holds. How can it be determined in closed-form? 3answers 91 views ### Why should I use derivatives and calculus? I know that this question maybe sounds pretty generic, but it's a curiosity that I have and I didn't found any answer yet. I recently started studying calculus using this material where is said that "... 2answers 88 views ### Finding F(x) from F(kx), where F(x) is the antiderivative of the function f(x). I have that F(e^{x}x) = e^{x}x^{2} - e^{x}x + e^{x} - 1, and I would like to find F(x). Attempt Since F(e^{x}x) = e^{x}x^{2} - e^{x}x + e^{x} - 1, F(t) = \alpha_{1}t^{\beta_{1}} + \alpha_{2}t^... 3answers 107 views ### Derivation of formula for gradient in spherical coordinates If we have a function f=f(r, \theta, \phi), where (r, \theta, \phi) are spherical coordinates on \mathbb{R}^3, how do we compute the gradient \nabla f by using the formula$$\nabla f \cdot d\...
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Marcinkiewicz–Zygmund inequality gives gives relations between moments of a collection of independent random variables. The statement of this inequality can be seen in Wiki https://en.wikipedia.org/...
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### How to find this kind of function?

I am trying to find a function $f(x,y)$ with $f:\mathbb{R}^{2} \rightarrow [a, b]$ where $[a, b]$ is equal to $[-1, 1]$, $[0, 1]$, or some other small interval (open intervals are fine as well). The ...
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### Calculate in closed form $\sum_{n=1}^{\infty} \frac{\arctan(1/n) H_n}{n}$

Playing with Taylor series is not helpful enough. What else would you try out? $$\sum_{n=1}^{\infty} \frac{\arctan(1/n) H_n}{n}$$ $$\approx 2.1496160413898356727147400526167103602143301206321$$ It's ...
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### How do i find a closed form expression for $\sum_{k=0}^n \frac{(x-1)^k}{k+1}$?

How do I Find a closed form expression for : $$\sum_{k=0}^n \frac{(x-1)^k}{k+1}$$ Note :I have no idea how to do that, I am bad at evaluating series when we cannot use some standard series to do it. ...
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### Prove any slope is obtained at some point

We are given $f:$$R\rightarrow$$R$ differentiable in $[a,b]$ such that $f'(a)<f'(b)$. We need to prove that $\forall\beta\in$$[f'(a),f'(b)]$$\exists$$x\in$$[a,b]$ such that $f'(x)=\beta$. This ...
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### A limit of a sequence statistifying $S_{n} = \frac{1}{2}(a_{n}+\frac{1}{a_{n}})=a_1+a_2+…+a_n$

Sequence $\{a_n\}$ is a positive sequence and satisfies $S_{n} = \frac{1}{2}(a_{n}+\frac{1}{a_{n}})$ where $S_n = a_1+a_2+...+a_n$. Find $\lim_{n\to \infty} S_{n+1}*(S_{n}-S_{n-1})$
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### Proof of change in position vector in spherical coordinates

I have found it hard to proof that ${d\vec r=dr\hat r+rd\theta\hat \theta}$ in spherical coordinates. Also it would be great if somebody can explain what ${d\vec r}$ is because I read different things ...
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### A question about the differentiability conditions for a fractional power

Now consider the fractional power $$f(x)=z^{m/n}=|z|^{m/n}(\cos \frac{mArgz}{n}+i \sin \frac{mArgz}{n})$$ Since $f(x)$ satisfies the Cauchy-Riemann equations, and is therefore differentiable, we can ...
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### Tangent to the curve

What is the equation of the tangent to the curve $$y = x^{1/3}$$ at the point $(0,0)$ ? This is a homework question. I tried solving it. The derivative comes out to be infinite at the given point. So,...
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### log normalizer - exponential family

i am studying the exponential family- and read that, for $p(x|\mu)=h(x)exp(\eta^T t(x)-a(\eta))$, that $a(\eta)$ is the log normalizer, which ensures that the probability distribution integrates to ...
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### Convergent sequence rigorous definition

I know what a convergent sequence is and how it works and everything but whenever I look at the definition my mind cant suddenly make the link with my intuition part of the brain so I realise that I ...