# Questions tagged [calculus]

For basic questions about limits, continuity, derivatives, integrals, and their applications, mainly of one-variable functions. For questions about convergence of sequences and series, this tag can be use with more specialized tags.

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So I came across the integral $\displaystyle \int \frac{1}{1+x^2}\, dx$ Now the conventional way is to solve this via trigonometric substitution, and you get the answer as $tan^{-1} x + C$. However, ...
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### A question on the "intermediate value property" for disconnected sets!

We know that a continuous function on a connected space satisfies the Intermediate value property. Let $\mathbb{R}^+\times Z_{\alpha}=X_{\alpha}$, where $$Z_{\alpha}=\{\alpha+n: n\in \mathbb{N}\}$$ ...
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### Sum of binomial coefficients on $n$ instead of $k$

There are lots of results on sum of binomial coefficients over $k$. How do we show that $$\sum_{n=k+1}^s \binom{n}{k}=\binom{s}{k+1}$$? Thank you in advance!
1 vote
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### Flux through surface of revolution

I'm trying to solve the following problem Let $C$ be the curve in the $xy$ plane given in polar coordinates by $r = 2-\sin(\theta),\ 0 \leq \theta \leq \pi$ and let $S$ be the surface given by ...
1 vote
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### Concavity of a Derivative from a Graph

Is it possible to know the concavity of a derivative $f'$ given the graph of $f$? For example, I was given this graph here: How can I know the concavity of the function $f'$ over $(0,1)$ by simply ...
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### Calculus: Differentiating Related Functions

problem that uses the y-term with DX/DT and the x-term with DY/DT How do you decide to which variable term (x, y) you're going to derive with DX/DT or [problem that uses the y-term with DX/DT and the ...
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### Series Equaling Euler–Mascheroni Constant

Is there a known series equaling the Euler–Mascheroni Constant? And if there is this, wouldn't that imply that the Harmonic series plus this new series equal a series that is exactly $\ln(x)$? I have ...
1 vote
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### Two Power Series Identify on an Open interval not Containing Zero

I searched a lot for a convincing answer for this question but failed to find one (That is formally complete). I wonder if the following claim is true, and if so, for a formal proof. Claim: Let there ...
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### Spivak's Calculus, Ch. 11, **69b: $f$ increasing at every $a \in [0,1]$. Prove $f$ increasing on $[0,1]$.

A function $f$ is increasing at $a$ if there is some number $\delta>0$ such that $$f(x)>f(a) \text{ if } a<x<a+\delta$$ and $$f(x)<f(a) \text{ if } a-\delta<x<a$$ (a) Suppose ...
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### Uniform convergence of $\sum_{n=1}^{\infty}(\sum_{k=0}^{m}kn^k)x^n$ $m\in \mathbb{N}$ Some constant

How can I prove or disprove uniform convergence $\sum_{n=1}^{\infty}(\sum_{k=0}^{m}kn^k)x^n$ $m\in \mathbb{N}$ Some constant
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### wedge volume problem

Find the volume of the wedge cut from the first octant by the cylinder $z = 12 - 3y^2$ and the plane $x+y=2$. What I did- sketched parabola and repeated in all of x axis, drew the plane and found the ...
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### Understanding numerical integration

I need help, with the fourth line where they introduced $t$. What does a "dummy" variable mean and why are they taking the integral with respect to $y$? Secondly, why have they added the &...
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### The concept of Power series in Strang's Calculus Lectures : how to understand " matching at $x=0 \space f(0), f'(0), f''(0), f'''(0) ...$"?

Source : https://www.youtube.com/watch?v=N4ceWhmXxcs In his Calculus Course , Pr. Strang kindly takes the pain to explain to the layman the concept of Power series. The basic idea, he says, is to ...
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### Question about calculating flux with different coordinate systems

For a question, I am asked to find the flux of $F=\langle 3x,0,2\rangle$ across the surface of $x^2+y^2+z^2=4, x>0, \ y<0,\ z<0$. I tried solving this with cylindrical and polar coordinate ...
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### If $P$ is a partition of $[a,b]$, Is there always a choice of points $q$, for $P$ is such that $\sigma(f,P,q) = \underline{s}(f,P)$?

The title says it all. If $P$ is a partition of $[a,b]$, does there always exist a choice of points $q$, for the associate partition $P$ is such that $$\sigma(f,P,q) = \underline{s}(f,P)\;?$$ Here ...
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### Riemann Sum Problem Explanation f(x)=mx on left endpoints using xk

I am learning Riemann when I encountered this question and its solution. Question A curve f(x)=mx in closed interval [a,b] where m>0 and a>=0. Calculate riemann sum of f(x) using xk as left ...
I'm studying Spivak's Calculus on manifolds, chapter 5.1. I cannot understand the following theorem. Let $A \subset \mathbb{R^n}$ be an open and let $f : A \rightarrow \mathbb{R^p}$ be a ...
### How to prove the function $x^3 + 3x^2 + 4x + 1$ is increasing for all values of $x$?
If I have the function $x^3+3x^2+4x+1$, how can I prove it is increasing for all values of $x$? I tried setting it equal to zero after differentiating it $3x^2+6x+4$, but I got an imaginary number. ...