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.

134529 questions
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Does $\int_{0}^{\infty}\frac{dx}{1+(x\sin5x)^2}$ converge?

I would like your help with deciding whether the following integral converges or not: $$\int_{0}^{\infty}\frac{dx}{1+(x\sin5x)^2}.$$ I tried to compare it to other functions and to change the variables, but it didn't work for me. Thanks a lot!
Jozef
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$\int dx/x =\cdots$ (pedantic nitpicking?)

It seems that "everybody knows" that $\displaystyle\int\frac{dx}{x}=\log|x|+C$ (or if one really must, then $\ln$ instead of they synonymous $\log$). Does any textbook or reference work say that $$ \int\frac{dx}{x} = \log |x| + \left.\begin{cases}…
Michael Hardy
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How to bound this integral?

I want to ask how a hint how to show this integral inequality: $$ \frac{1}{\pi}\int^{\infty}_{0}\frac{x}{y^{2}+x^{2}}\log\frac{1}{1-e^{-2\pi y}}dy< \frac{1}{12x} $$ This is from Ahlfors, Complex Analysis, page $206$. I tried to compute a rough…
Bombyx mori
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What is wrong with my contradiction?

Spivak says the following function does not have an integral: $$ F(x) = \left\{ \begin{array}{lr} 1 & : x \in \mathbb{Q}\\ 0 & : x \notin \mathbb{Q} \end{array} \right. $$ It makes sense, for any partition $P$ there…
Dair
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Prove that the sequence$ c_1 = 1$, $c_{n+1} = 4/(1 + 5c_n) $ , $ n \geq 1$ is convergent and find its limit

Prove that the sequence $c_{1} = 1$, $c_{(n+1)}= 4/(1 + 5c_{n})$ , $n \geq 1$ is convergent and find its limit. Ok so up to now I've worked out a couple of things. $c_1 = 1$ $c_2 = 2/3$ $c_3 = 12/13$ $c_4 = 52/73$ So the odd $c_n$ are…
Justin
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A sufficient condition for linearity?

If $f$ is a linear function (defined on $\mathbb{R}$), then for each $x$, $f(x) – xf’(x) = f(0)$. Is the converse true? That is, is it true that if $f$ is a differentiable function defined on $\mathbb{R}$ such that for each $x$, $f(x) – xf’(x) =…
Mike Jones
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Limit of $(\sin\circ\sin\circ\cdots\circ\sin)(x)$

I'm trying to find this limit: $$\lim_{n \to \infty} \underbrace{\sin \sin \ldots \sin }_{\text{$n$ times}}x$$ Thank you
Jozef
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Taking the second derivative of a parametric curve

I understand that for the parametric equations $$\begin{align*}x&=f(t)\\ y&=g(t)\end{align*}$$ If $F(x)$ is the function with parameter removed then $\displaystyle F'(x) = \frac{\text{d}y}{\text{d}t}\big/\frac{\text{d}x}{\text{d}t}$ But the…
Noteventhetutorknows
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Importance of Rolle's and Lagrange's theorem in daily life: for school children

I am school teacher and teaching math. I know the Rolle's theorem and Lagrange's theorem. I can solve the problems numerically then and there. However, I was completely failed to explain the significance or applications of these theorems by…
russila-amity
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$\lim_{n \to +\infty } \left \{ en! \right \}$

I'd love your help this time with the following limit: $\lim_{n \to +\infty } \left \{ en! \right \}$ when $\{ a \}=a-[a].$ Honestly, I don't have a clue. Thank you.
user6163
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If $\lim_{x \rightarrow \infty} f(x)$ is finite, is it true that $ \lim_{x \rightarrow \infty} f'(x) = 0$?

Does finite $\lim_{x \rightarrow \infty} f(x)$ imply that $\lim_{x \rightarrow \infty} f'(x) = 0$? If not, could you provide a counterexample? It's obvious for constant function. But what about others?
Igrek
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How is calculus useful in economics; a subject where the quantities studied are often discrete?

Problem statement: It costs: $$c(x)=x^{3}-6x^{2}+15x$$ dollars to produce x toys when 8 to 30 toys are produced and that $$r(x)=x^{3}-3x^{2}+12x$$ gives the dollar revenue from selling $x$ toys. Your toy shop currently produces $10$ toys a day.…
charlietan84
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Show that $d/dx (a^x) = a^x\ln a$.

Show that $$ \frac{d}{dx} a^x = a^x \ln a. $$ How would I do a proof for this. I can't seem to get it to work anyway I try. I know that $$ \frac{d}{dx} e^x = e^x. $$ Does that help me here?
1ftw1
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What needs to be true of $f$ for $f_{xy}=f_{yx}$?

What conditions does a function $f$ have to fulfill in order that: $\frac{\partial}{\partial x}(\frac{\partial f}{\partial y})=\frac{\partial}{\partial y}(\frac{\partial f}{\partial x}) $? I am trying to prove something else, and I have just got it…
user71854
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Help figuring a formula for my job

I'm a metal worker, I cut, weld, whatever. I'm trying to figure out a formula where I could take my cutting list And figure out the most efficient way to cut it with the materials I have. For example I have 2 20ft lengths of tubes I want 5 peices…
John Don
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