Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [calculus]

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

Calculus is the branch of mathematics studying the rate of change of quantities, which can be interpreted as slopes of curves, and the lengths, areas and volumes of objects.

Calculus is divided into differential and integral calculus, which are concerned with derivatives

$$\frac{\mathrm{d}y}{\mathrm{d}x}= \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$

and integrals

$$\int_a^b f(x)\,\mathrm{d}x = \lim_{\Delta x \to 0} \sum_{k=0}^n f(x_k)\ \Delta x_k,$$

respectively.

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Is this argument that $(0,0)$ is a local minimum for $f(x,y)=x^4+3xy^2+y^2$ correct?

I'm trying to find a local minimum for $f(x,y) = x^4 + 3xy^2 + y^2$. After finding that a critical point is at $f(x,y)$ is $(0,0)$, I am wondering if this is proper formal justification to prove that this is indeed the case? Let $\epsilon >0$ be…
Sam C
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To prove divergence of a series.

Let $0
Gobi
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Spivak Chapter 11 Appendix Problem 6 (c)

I'm having trouble on Part (c). I read the solution, which assumes that $f$ has a minimum on the interval $[a, b]$, but we have not proved that convexity has any implication of continuity or anything. How does one show that $f$ does indeed have a…
minimario
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Proving Mean Value Theorem with Rolle's Theorem?

How to prove the Mean Value Theorem using Rolle's Theorem? I am getting the impression that it is possible by adding a linear function to a function where Rolle's theorem applies to prove the MVT. However, I can't quite turn this idea into a…
hesson
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Differences between using partial fractions or completing the square to solve an integral?

I have this question $$\int \frac{1}{4x^2-4x-3}\, dx$$ I tried to solve it by using completing square method, and I got $$\frac{1}{4} \arctan\left(\frac{2x-1}{2}\right),$$ but when I saw the answers, I found that it should be solved by partial…
yara
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Having trouble of finding this integral in the standard integral list

I am facing difficult problem to deal with this integral $$\int_{0}^{\pi/2}\ln(9-4\cos^2\theta)\,\mathrm{d}\theta \tag*{(1)}$$ note $(3)^2-(2\cos{\theta})^2=(3-2\cos{\theta})(3+2\cos{\theta}) \tag*{(2)}$ also note that, $$\log(AB)=\log A+\log B…
lion
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$f'(x)>f(f(x))$ implies $f(f(f(x)))\leq0$ for nonnegative $x$

If $f\in C^1(\mathbb R)$ satisfies $f'(x)>f(f(x))$ for all $x\in\mathbb R$, then $f(f(f(x)))\leq0$ for all $x\geq0$. Could anybody provide a solution or some hints on this problem?
Keke Zhang
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Prove that $x^2\arctan x$ is not uniformly continuous in $\Bbb R$.

I have proven that (and we are required to use this) $y^2\arctan y-x^2\arctan x\geq(y^2-x^2)\arctan x$ (the proof was near-trivial). I now have to use this to show that $f(x)=x^2\arctan x$ is not uniformly continuous in $\Bbb R$. Here is what I have…
James
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Proving: If $|x-y|<\frac{1}{n}$ for every natural $n$ then $x=y$

It's really basic but I am trying to prove: $$ \forall x,y\in \mathbb{R}.(\forall n\in \mathbb{N}. |x-y| <\frac{1}{n} \Rightarrow x=y)$$ I tried to prove it by proving that that if $$A= \left\{ \frac{1}{n} \ \middle| \ n\in\mathbb{N}…
theshopen
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What is $\int_{[0,1]^2}\!dx\,dy\,\mathcal{P} \frac{\log(x)}{x-y}$?

What is the value of $$\int_{[0,1]^2} \!\!\!dx\,dy\,\mathcal{P} \frac{\log(x)}{x-y},$$ where $\mathcal{P}$ denotes Cauchy's principal value. I solved this once for a homework, but I can neither remember the answer nor reproduce it right now... (bad…
Fabian
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Show that $ f''(x)+g(x)f'(x)-f(x)=0 $ has at least one root

Let $f$ be continuous on $[a,b]$, with continuous first and second derivatives on $[a,b]$. Suppose $f$ has at least 3 distinct zeroes in $[a,b]$. Show that $f''(x)+g(x)f'(x)-f(x)=0$ has at least one root in $[a,b]$, where $g$ is any continuous…
Vulcan
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Applying Chain Rule to more than two functions

I understand how to apply the chain rule to two functions, but trip up when it comes to three or more. Given $f~(x~)= x~\sqrt{1-x^{2}}$, I can see there are three values: $a: 1-x^{2}$ $b: \sqrt{a}$ $c: x~\cdot b$ Now, $$f(b(a))=…
Jason
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Determine the Volume of Gabriel's horn with a lodged sphere

Just recently I was posed with the following interesting question: Say you took a sphere with an arbitrary, constant radius "$R$" and dropped it into Gabriel's Horn. Then you filled the horn perfectly with water such that the sphere acts like a plug…
Nadia Bianca
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Why must we use dummy variables when integrating?

Given the function: $x+ye^{-x} \frac {dy}{dx}=0$ where $y(0)=1$ and you were told to solve it, I know I must multiply by $e^x$ giving $xe^x+y \frac {dy}{dx}=0$. Then re-arranging to give $ydy=-xe^xdx$. After this I proceed to integrate both…
Lo-urc
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