Tagged Questions

For questions about infinitesimals, both in an intuitive sense as well as more rigorous settings (see also [nonstandard-analysis]).

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Learning differential calculus through infinitesimals

In class, we've studied differential calculus and integral calculus through limits. We reconstructed the concepts from scratch beginning by the definition of limits, licit operations, derivatives and ...
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Density of real numbers and density function

In Quantum mechanics, given a certain material, it is possible to write the density of energy states $\rho (E)$ as a function of $E$. That is: let's consider all the real values contained in the ...
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How does differentiation work?

I am a physics student and my teacher told me, to find the instantaneous velocity of an object, reduce the time interval to a very small extent. May the time interval be very very very close to 0, ...
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What is the value of $\tan (\frac{\pi}{2} - \epsilon)$? [closed]

$$\tan(\frac{\pi}{2}-\epsilon)$$ By epsilon I mean an infinitesimal.
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Cauchy's contribution

Sometime, I believe perhaps 2 years, ago I asked a question about breakthroughs, such as those within mathematics and physics which may lead a whole discipline forwards in many ways. One example from ...
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How is the concept of the limit the foundation of calculus?

My casual study of mathematics and calculus introduced me to the notion that calculus didn't find a firm foundation until Cauchy, Weierstrauss (et al) developed set theory some ~100 years after Newton ...
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Proving a Function Continuous with Non-Standard Analysis

I am reading a text on non-standard analysis. I need to prove the following: Suppose that $f$ is non-decreasing on the real interval $[a,b]$ and that $f$ satisfies the intermediate value property. ...
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Why can't the reals be constructed from the infinitesimal?

If the infinitesimal gives an unlimited precision as 1/∞ --> 0 Which can be thought of as the decimal 0.000000.....00000... then Why can't the reals, which demands, simply, unlimited precision (this ...
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Do we have $\int f dxdy = \int fdydx$ or $\int f dxdy = -\int f dydx$?

If $f : \mathbb R^2 \to \mathbb R$ is an integrable function, then do we have $$\int f dxdy = \int f dydx$$ or $$\int f dxdy = -\int f dydx?$$ (I am leaving the domain of integration as it does ...
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proof that $\lim \limits_{n \to \infty} \sqrt[n]{n^5 -2n + 7} = \infty$

I started learning infinitesimally math and I have the following question: Is the following sentence true $\lim \limits_{n \to \infty} \sqrt[n]{n^5 -2n + 7} = 1$ I can see that it tends to $\infty$...
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What's an example of an infinitesimal?

If you want to use infinitesimals to teach calculus, what kind of example of an infinitesimal can you give them? What I am asking for are specific techniques for explaining infinitesimals to students, ...
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How is an infinitesimal $dx$ different from $\Delta x\,$? [duplicate]

When I learned calc, I was always taught $$\frac{df}{dx}= f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-x}$$ But I have heard $dx$ is called an infinitesimal and I don't know what this means....
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$1/\infty$ is zero or infinitesimal? [closed]

Since $\infty>0$, so $1/\infty >0$, thus I think $1/\infty$ should be infinitesimal, but the calculus book says $$\lim_{x \to \infty} \frac{1}{x}= 0$$ So is $1/\infty$ zero or infinitesimal ? ...
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Are there concepts in nonstandard analysis that are useful for an introductory calculus student to know?

Studying calculus I became aware that nonstandard analysis had some methods that that made the concept of infinitesimal concrete, so that $dx$ actually made sense. Can someone elaborate on this ...
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Infinitesimal calculus

I have been reading some non-standard analysis from Keisler's book and I think it is logically consistent till now but there are criticisms against it and why isn't non-standard analysis accepted more ...