# Tagged Questions

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### Single variable justification for the multivariate chain rule.

I $\def\d{\mathrm d}\def\p{\partial}$am going to ask everyone to switch their paradigms to that of the real line. I am looking for a "lowbrow" explanation of the following phenomena. I am talking ...
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### Does this integral variable change makes sense to you?

I was Reading a book about calculus when I've found this part about variable substitution in integrals: Consider $f$ defined in na interval $I$. Suppose that $x =\phi(u)$ is inversible, and its ...
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### Confused about differentiation

I'm new to calculus and have been taught that $\displaystyle \frac{dy}{dx}$ is the rate of change of y with respect to x. Does $\displaystyle \frac{dy}{dx}$ show how much the variable y changes as x ...
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### What is the meaning of infinitesimal?

I have read that an infinitesimal is very small, it is unthinkably small but I am not quite comfortable with with its applications. My first question is that is an infinitesimal a stationary value? It ...
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### What is the difference between a discrete function and a continuous function

Intuitively it seems that both concepts should be disjoint because if a function is discrete then it has some holes on it and if a function is continuous then it doesn't have holes. But now I'm not ...
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### Understanding Continuity of Functions

I know that graphically a function $f(x)$ is said to be continuous in $[a,b]$ if there are no breaks in the curve for $f(x)$ in the interval $[a,b]$ I also know that by definition, a function $f(x)$ ...
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### Integral and series convergence intuition

I have this problem I ran into during my studies to the upcoming exam: I don't feel I have the intuition of whether a series or an integral converges or not. What are the things I should look for when ...
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### Why does Newton's method work?

I find many sites explaining how to use Newton's method, but none explaining why it works. Could someone give me the intuition behind it? Thanks.
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### conditional convergence

This is an practice question from "Advanced Calculus, Folland" Chapter 6.3, Q.2 (not HW) I am not sure how to go about this question :: suppose $\sum { { a }_{ n } }$ is conditionally convergent. ...
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### Examples of convergence of series

These questions are practice questions from the text "Advanced Calculus, Folland" chapter 6.2 (not HW) I am working on some exercises on convergence of series and I feel that I understand it well but ...
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### The Meaning of the Fundamental Theorem of Calculus

I am currently taking an advanced Calculus class in college, and we are studying generalizations of the FTC. We just started on the version for Line Integrals, and one can see the explicit symmetry ...
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### Why does the power rule work?

If $$f(x)=x^u$$ then the derivative function will always be $$f'(x)=u*x^{u-1}$$ I've been trying to figure out why that makes sense and I can't quite get there. I know it can be proven with limits, ...
I am working on some exercises for Improper Integrals (not homework). The question is 1.c in Folland Advanced Calculus : $$\int_0^\infty x^2 e^{-x^2 } \, dx$$ It asks whether the above Improper ...
I am working on some practice exercises (not homework) on transformations and need some intuition and help. One of the questions is: $(u,v)=f(x,y)$ where \$ \quad u= { e }^{ x }\cos(y), \quad v = { e ...