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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Can't see why one of these functions is conservative, and the other isn't.

I am really confused here: Why one of these functions is conservative, while the other not? $F_{1} = \frac{-y \hat i + x \hat j}{x^2+y^2}$ $F_{2} = \frac{x \hat i + y \hat j}{x^2+y^2}$ Suppose both these vector functions are applied on a region…
Lac
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Determine if the sequence functions $f_n(x)=\frac{\sin(nx)+\cos(nx)}{\ln(n)}$ uniformly converges by Cauchy's criterion in $x\in[e,\infty)$

Determine if the sequence functions $f_n(x)=\frac{\sin(nx)+\cos(nx)}{\ln(n)}$ uniformly converges by Cauchy's criterion in $x\in[e,\infty)$ Attempt: Let $0<\varepsilon $ we chose $\displaystyle n^{*} =\left\lceil \frac{1}{\varepsilon }\right\rceil…
Esty.R
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Rolle's theorem for limits

Given a function defined by $a_n(x) = \frac{d^n}{dx^n} e^{-x^2}$. Every function can be written as $a_n(x) = h_n(x) e^{-x^2}$ where $h_n(x)$ is a polynomial (the Hermite-Chebyshev polynomial of degree $n$ to be precise). I want to prove that every…
iblue
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Why does $a=\prod_{n=1}^{∞}\left(a^{\frac{1}{2^{n}}}\right)$

So before I start, disclaimer that I'm just a high school student, so apologies if I'm missing something obvious here. A while back I found the expression $$\frac{\sqrt{n\sqrt{n\sqrt{n\sqrt{n\sqrt{n\sqrt{n}}}}}}}{n}$$ which led me to start…
LANCE
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Unsure about changing order of integration

I want to change the order of integration of the following $$\int_{1}^{+ \infty} \left (\int_{1}^{\sqrt{y}} x^3e^{-xy} dx \right) dy.$$ I get the bounds $1 \le y \le x^2$ and $1 \le x \le \infty$, and so the integration becomes $$\int_{1}^{+…
user4167
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How to show $\int_{0}^\infty e^{x^2}dx$ does not converge?

To show that $\int_0^\infty e^{x^2} \, dx$ does not converge, I can use polar coordinates, and say $$\int_0^\infty \int_0^\infty e^{x^2+ y^2} \, dx \, dy = \int_0^\infty\int_0^{\pi/2} e^{r^2}r d\theta \, dr$$ $$ = \frac{\pi}{2}\int_0^\infty e^{r^2}r…
user80456
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Exercise on true or false.

If $ f $ is continuous over $ [1,3] $ and $ \int_1 ^ 3f (x) dx = 0 $, then $ f (x) = 0 $ for $ 0 \leq x \leq 1 $. I think that is false. I'm right?
asd asd
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show $\lim_{x\rightarrow\infty} x^n\exp(-x^2)=0$

I want to show for all $n\in\mathbb N$ $$\lim_{x\rightarrow\infty}\frac{x^{n}}{\exp(x^2)}=0$$ I am pretty sure that I have to use L'Hospital. I've tried induction: $n=1$: $$\lim_{x\rightarrow\infty}\frac…
sheldoor
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Integral of $\int_1^N\frac{\{x\}}xdx$?

For $N\in\mathbb{N}$ how can the following integral be computed? $$ \int_1^N\frac{\{x\}}{x}dx $$ The notation $\{x\}$ is the fractional part of $x$, so $\{x\}=x-\lfloor x\rfloor$. Apparently, the integral evaluates to…
Hikawa
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How does integration find average?

I am slightly new to calculus. I haven't been able to get the concept that how plotting the area under the line gives us average value, not the instantaneous value. Then, how do you find the instantaneous value? I have seen few answers but they are…
Blank
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Best tools to teach animations of solids of revolution in a Calculus II class?

I would like to show animations of solids of revolutions particularly to those students who are not visualizers. Is there a tool that I (and preferably my students also) can use to create animations? It would be great to know what you use in…
rose
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If $\lim_{x \to \infty} \frac{f'(x)}{x}=2$ does it follow that $\lim_{x \to \infty} \frac{f(x)}{x^2}=1$?

I need to show that the following statement is true or false. $$\displaystyle\lim_{x \to \infty} \frac{f'(x)}{x}=2 \Rightarrow \displaystyle\lim_{x \to \infty} \frac{f(x)}{x^2}=1$$ I considered $f(x)=x^2$, and it seems that the above statement is…
mathmo
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Integrate function $\int_{0}^{\infty}\lambda^{x}e^{-2\lambda}d\lambda$

How can I integrate this function? It's originated by an exponential prior and a poisson likelihood. $\int_{0}^{\infty}\lambda^{x}e^{-2\lambda}d\lambda$
Filipe Ferminiano
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Limit of a sequence defined by a recurrence

Let $x_1$ and $x_2$ be positive real numbers and define, for $n>2$:$$\displaystyle x_{n+1}=\sum_{k=1}^{n}\sqrt[n]{x_k}$$ Evaluate $\displaystyle \lim_{n\to\infty}\frac{x_n-n}{\ln n}$
rezvane
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How to prove $\int_0^{2r} 2h \sqrt{r^2 -(x-r)^2}dx = \pi r^2 h$ without geometry?

I wanted to find the volume of a cylinder, radius r, height h by slicing it in to rectangles: I placed the cylinder on the x-axis, one corner of the base diameter at (0,0) the opposite at (2r, 0). I have found that an area of a cross-section…
tutorscomputer
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