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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If a particle travels $30$ meters every $3$ seconds, does it necessarily travel $20$ meters every $2$ seconds?

Is it possible if its motion were discontinuous? I'm trying to understand if there is a function that has this property, but I chose to say it in terms of motion because it's easier to explain.
Mostafizur Rahman
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Evaluate $\int_{1}^{\infty}e^{-x}\ln^{2}\left(x\right)dx$

Evaluate :$$\int_{1}^{\infty}e^{-x}\ln^{2}\left(x\right)\mathrm{d}x$$ I've tried to solve this with some elegant substitutions like $t=e^x$ or $t=\ln\left(x\right)$ . I've also tried to integrate by parts without any success. any help would be good.
Tomer Shetah
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Mean value theorem with integrals?

Here is the question: Let $f:[0, 1]\rightarrow \mathbb{R}$ be a continuous function satisfying $$\int_0^1 (1-x)f(x) \,dx = 0$$ Show that there exists $c\in (0, 1)$ such that $$\int_0^c xf(x)\,dx = cf(c)$$ I'm pretty sure that the problem…
zxcvber
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Proving $-\frac{1}{a}<\int_a^b \sin(x^2) dx<\frac{1}{a}$

I have encountered a question: Prove $$-\frac{1}{a}<\int_a^b \sin(x^2) dx<\frac{1}{a}$$ There are plenty of solutions to $\int_0^{\infty} \sin(x^2) dx$ online, but there seems to be no solution to the boundary of $\int_a^b \sin(x^2)…
Xiaowei Tang
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Prove that $\int_{0}^{1}{f^{2}(x)dx}\leq \frac{4}{3}\left(\int_{0}^{1}{f(x)dx}\right)^2$

Let $f(x)$ be a concave nonnegative function on $[0,1]$ Prove that $$\displaystyle \int\limits_{0}^{1}{f^{2}(x)dx}\leq \frac{4}{3}\left(\int\limits_{0}^{1}{f(x)dx}\right)^2$$ My friend tian_275461 told me we even have the general result Let $f(x)$…
pxchg1200
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How prove this identity wih the sum equal to other sum

Question: Given $l\in \mathbb{N^+}$. $a_1,\cdots,a_l,b_1,\cdots,b_l$ are real numbers.$a_0=b_0=a_{l+1}=b_{l+1}=0$.Define $$g(m,l)=-\dfrac{\displaystyle\prod _{r=0} ^{l}{(a_m-a_r+b_{r+1}-b_m)}}{\displaystyle\prod _{r=1,r\ne m}…
math110
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Prove there exists $\xi \in (a,b)$ such that $f(\xi)=f^{(n+1)}(\xi)$.

Problem Let $f(x)$ be $n$-times differentiable over $[a,b]$ and $n+1$-times differentiable over $(a,b)$. $f^{(k)}(a)=f^{(k)}(b)=0$, where $k=0,1,2,\cdots,n$. Prove there exists $\xi \in (a,b)$ such that $f(\xi)=f^{(n+1)}(\xi)$. Attempt Consider…
mengdie1982
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Prove that the unit circle is path-connected?

I need to show that the unit circle is path connected and connected. I was able to show that it is connected, by $f:[0,2\pi] \to \mathbb{R}^2$, $f(x)=(r\cos x,r \sin x)$ which is a continuous function. The interval $[0,2\pi]$ is connected and the…
hawaii99
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Are all functions that have a primitive differentiable?

Are all functions that have a primitive differentiable? For some background, I know that not all functions that are integrable are differentiable. For example: $$ f = \begin{cases} 0 & x \neq 0 \\ 1 & x = 0 \end{cases} $$ is integrable over…
ninivert
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Prove that there exists no differentiable real function $g(x)$ such that $g(g(x))=-x^3+x+1$.

Prove that there exists no differentiable real function $g(x)$ such that $g(g(x))=-x^3+x+1$. I have googled it but find nothing useful. Now I know it's a Iterated function problem. It's an exercise problem after the chapter DERIVATIVE, so I guess…
闫嘉琦
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Point of discontinuity

I have a function: $$f(x) = x$$ Defined over the domain $\mathbb{R} \backslash 0$. Is it correct to say that: The function is continuous, but it has a point of discontinuity at $x=0$?
TomDavies92
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Confused about an integral from MIT integration bee 2012

One of the integrals is: $$\int \frac{\mathrm{d}x}{2+2\sin x + \cos x}\, \mathrm{d}x $$ How can there be two $\mathrm{d}x$? MIT's Integration Bee
yiyi
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Can distance between two closed sets be zero?

Is given metric space $(M, d)$. Let $A\cap B = \emptyset; \,\,\text{dist}(A,B):=\inf\{d(x,y):x\in A, y\in B\}$. $A, B$ are both closed sets. Is it possible that $\text{dist}(A,B)=0$? The first thought comes into mind is that obviously…
nakajuice
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Prove that $ f(x) $ has at least two real roots in $ (0,\pi) $

Let $ f $ be a continuous function defined on $ [0,\pi] $. Suppose that $$ \int_{0}^{\pi}f(x)\sin {x} dx=0, \int_{0}^{\pi}f(x)\cos {x} dx=0 $$ Prove that $ f(x) $ has at least two real roots in $ (0,\pi) $
drawar
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Convex and bounded function is constant

Let f be a convex and bounded function, meaning there is a constant $C$, such that $f(x) < C$ for every $x$. I need to prove that $f$ is a constant function. Thanks!
user6163
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