# Questions tagged [calculus]

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

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### the approximation of $\log(266)$?

Consider the following exercise: Of the following, which is the best approximation of $\sqrt{1.5}(266)^{1.5}$? A 1,000 B 2,700 C 3,200 D 4,100 E 5,300 The direct idea is using the "...
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### How to convince a layperson that the $\pi = 4$ proof is wrong?

The infamous "$\pi = 4$" proof was already discussed here: Is value of $\pi = 4$? And I have read all the answers, yet I think that they will not be of much help to me if I try to explain this thing ...
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### How do you prove $| \int (f)| \leq \int (|f|)$?

For an integrable function $f$ on $(a,b)$, how would you prove $| \int (f)| \leq \int (|f|)?$
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### Limits of function compositions

Is it possible to evaluate limits involving sequences of function compositions? For example, given the expression $$g(x, n) = \sin(x)_1 \circ \sin(x)_2 \circ [...] \circ \sin(x)_n$$ is it possible ...
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### $p_n = 1- \left( 1-\frac{1}{365}\right)\left( 1-\frac{2}{365}\right)\cdots \left( 1-\frac{n-1}{365}\right)$ then $p_n>\frac{1}{2}$ for $n>?$

$$p_n = 1- \left( 1-\frac{1}{365}\right)\left( 1-\frac{2}{365}\right)\cdots \left( 1-\frac{n-1}{365}\right)$$ Then $p_n>\frac{1}{2}$ for $n>?$ This occured in a probability problem. The result ...
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### Expanded concept of elementary function?

After searching about why $\int e^{x^2}$ is not an elementary function, I was disappointed that I should understand about Galois theory, but then I started to think about a concept that treats ...
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### Can the derivative be defined game-theoretically?

It is often said, by way of intuitive explanation, that the derivative of a function at a point is the slope of the line that “best fits” the function through that point. Can this be pressed into a ...
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### Asking for general form of Integral Inequality of this kind

Let $f\in C^1[0,a]$ and $f(0)=0$. Is it true that $$\int_0^a \left(\sqrt{x}f(x)\right)^{\prime} \left(\frac{f(x)}{\sqrt{x}}\right)^{\prime}\, dx\geq 0\;\;?$$ What is the general form of this type ...
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### Proof of $f(x) = (e^x-1)/x = 1 \text{ as } x\to 0$ using epsilon-delta definition of a limit

I am in calc 1 and we have just learned the epsilon-delta definition of a limit and I (on my own) wanted to try and use this methodology in order to prove $(e^x-1)/x = 1$ (one of the equivalencies), ...
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### Convexity of a Set

Consider the following function, $$f(x, y) = e^{m e^{-y}+n e^{-x}-x-y} \left(a x e^y+b e^x y+c x y\right)$$ where $a, b, c, m$ and $n$ are positive constants. I want to show $f(x, y)$ is quasi-...
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### Area vs Volume Paradox [duplicate]

Possible Duplicate: How can a structure have infinite length and infinite surface area, but have finite volume? Hi, I have this question that I quite cant explain why. So the area under the ...
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### Sum of derivative of integrals

For all $x$ in $\mathbb R$ define $\displaystyle f(x)=\left(\int_0 ^{x} e^{-t^2}dt\right)^2$ and $\displaystyle g(x)=\int_{0}^{1}\frac{e^{-x^2(t^2+1)}}{t^2+1}dt$. Show that for all $x$ in $\mathbb R$ ...
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### Power series of $\ln(x+\sqrt{1+x^2})$ without Taylor

The answer is $$x-\frac{ 1}{2}\frac{ x^3}{3}+\frac{ 1\cdot 3}{2\cdot 4}\frac{ x^5}{5}-\frac{ 1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{ x^7}{7}+\cdots$$ But I can't see how. Unfortunately, "how" can't be ...
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### Is this condition sufficient to ensure monotonicity of a function?

Suppose $f:[a,b]\to\mathbb{R}$ is continuous and $$\limsup_{h\to0}\frac{f(x+h)-f(x)}{h}\geq0$$ for every $x\in(a,b)$. Does it follow that $f$ increases monotonically on $[a,b]$? It is a problem in ...
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### Derivative of $f(x) = (x+x)$

I'm trying to teach myself algebra and derivatives. I learned the derivative for $f(x) = x^2$ from a lesson, and now I thought I would see if I could figure out the derivative of $f(x) = x+x$ on my ...
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### questions about limits and derivatives

I am trying to solve a set of problems, this one is causing my some troubles. For the first one I tried to use the $\epsilon-\delta$ definition but I couldn't solve it, I would appreciate some hints ...
319 views