# Tagged Questions

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

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### Calculus Paradox. I mean, what's wrong with what I think? [closed]

Is not calculus based on the paradox that the closest point to a point A is a distinct point B which is the point A itself? For example, if we consider the limit, $$\lim_{x\to2} \frac{x^2-4}{x-2}$$...
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### How to detect inflection point in a non-differentiable part of a curve

In a $W$ shaped curve, is the middle point a maximum or an inflection point? In that regard how to differentiate mathematically between a maximum (or minimum) and an inflection point at a non-...
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### The existence of such function satisfies the infinitum condition

Let $t\in [0,+\infty)$ be a real number, I am looking for two functions satisfies: $f$: $[0,1]\to[0,1]$ such that $f(0)=0$, $f(1)=1$, and $f(t)>0$ if $t>0$. $f$ is lower semicontinuous ...
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### Proving $\lim\limits_{x \to \infty} xf(x)=0$ if $\int_{0}^{\infty}f(x) dx$ converges.

Let $f(x)$ be a monotone non-increasing function such that $\int_{0}^{\infty}f(x) dx$ converges. Prove: $\lim\limits_{x \to \infty} xf(x)=0$. My question is, why can't I simply contradict any other ...
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### How to integrate : $\int \sqrt{\tan^2x +2}dx$ [closed]

How to integrate : $\int \sqrt{\tan^2x +2}dx$ Please guide what to substitute or any approach as I am not getting any clue on this , thanks .
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### Solve this question involving temperatures?

So I am given 2 formulas: $$\frac{dT}{dt}=-k(T_t-T_s)$$ Where $\frac{dT}{dt}$ rate at which the object's temperature is changing $T(t)$ is the temperature of the object at time $t$ $T(s)$ is the ...
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### How to find the MacLaurin series of $\frac{1}{1+e^x}$

Mathcad software gives me the answer as: $$\frac{1}{1+e^x} = \frac{1}{2} -\frac{x}{4} +\frac{x^3}{48} -\frac{x^5}{480} +\cdots$$ I have no idea how it found that and i don't understand. What i did is ...
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### Derivative of a logarithm from first principles

The usual example where learning about the derivative is obtaining it for $f(x)=x^2$ from first principles (see this for example). I am stumped on how use first principles to obtain the derivative of ...