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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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Prove that $f(x)=O(x)$ as $x\to 0$

Let $f$ be a function defined on $R$. for any absoulutely convergent series $\sum_{n=1}^{\infty}a_{n}$,the series $\sum_{n=1}^{\infty}f(a_{n})$ converges,Prove that $$ f(x)=O(x)\qquad (x\to 0) $$. I have tried to prove the contradiction side,assume…
pxchg1200
  • 2,050
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Why does $u$-substitution not work here?

$$ \int{\frac{1}{2y}dy} $$ Method 1: $$\int{\frac{1}{2y}dy} = \frac{1}{2}\int{\frac{1}{y}dy} = \frac{1}{2}\ln|y|+C$$ Method 2 ($u$-substitution): $$\int{\frac{1}{2y}dy} = \int{\frac{1}{u}dy} = \frac{1}{2}\int{\frac{1}{u}(2)dy}= …
JackOfAll
  • 4,701
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How to solve this integral: $ \int_{-1}^{1} \frac{x^4}{a^x+1}dx $?

I got this at my final calculus 1 exam. Any help with the solution? $$ \int_{-1}^{1} \frac{x^4}{a^x+1}dx $$ Thank you!
user83081
  • 777
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Continuous $f: \mathbb R \to \mathbb R$ with infinitely many zeroes in every interval

Is there a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for any two real numbers $a>b$, $f(x)=0$ has exactly a countable infinite many solutions with $a>x>b$?
halelulja
  • 71
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How to compute the volume of this object via integration?

What is the volume of intersection of the three cylinders with axes of length $1$ in $x, y, z$ directions starting from the origin, and with radius $1$?
user218
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Calculus Riemann Sums - why do the partitions have to be of the same size?

To set up Riemann sums for integration, my calculus text will say that the intervals of partition are all of the same size. Isn't it rather the case that they could be any size, as long as they are bounded by a largest partition which goes to zero…
hefalump
  • 305
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How to prove this to be an Irrational number?

$$ \int^1_0 e^{-x^2} \, \mathrm{d} x $$ It seems that needs more than 30 word to make a discription of this problem,but actually that all included in the title. Thank you for your answer.
user148210
  • 81
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Tauber theorem?

Let $f(x) = \sum\limits_{k = 0}^\infty {{a_k}{x^k}} $, where $a_k\ge0 $, and when $0 \le x < 1$ , $f(x)$ converges. Then please show that $\mathop {\lim }\limits_{n \to \infty } \frac{1}{{n + 1}}\sum\limits_{k = 0}^n {{a_k}} = \mathop {\lim…
Summer
  • 6,893
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Prove that $\sqrt{5}$ exists

Prove that $\sqrt{5}$ exists; in other words prove that there exists a positive number $x\in \mathbb R$ satisfying $x^2=5$ Here's what I've done: I let $A= \{x>0:x^2\leq 5\}$ We know that $A$ is not empty because clearly $2$ is in it: $2^2<5$ and we…
user3412839
  • 475
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Optimization, getting close to origin

I have no idea how to do this at all. "Find the point on the line $y=2x+3$ that is closest to the origin" I am just not very smart or creative so I have no idea how to do this. I graphed it and I think I can assume that it will be closest to the…
user138246
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Calculation of $\lim_{x\rightarrow 0}\frac{\sin (\pi\cos^2 x)}{x^2}$

Calculation of $\displaystyle \lim_{x\rightarrow 0}\frac{\sin (\pi\cos^2 x)}{x^2}$ $\bf{My\; Try::}$ Given $\displaystyle \lim_{x\rightarrow 0}\frac{\sin (\pi\cos^2 x)}{x^2} = \lim_{x\rightarrow 0}\frac{\sin (\pi (1-\sin^2 x))}{x^2}$ $\displaystyle…
juantheron
  • 53,015
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Functions of the form $\int_a^x f(t) dt$ that are commonly used.

I am a graduate student and teaching assistant, and I am teaching Calc 1 for the first time. In a few weeks I will be covering the Fundamental Theorem of Calculus. I'm using James Stewart's Calculus textbook, and I was hoping to give students…
user122916
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how to calculate the limit of an integral?

Could you please tell me how to calculate the limit: $$\lim_{x\rightarrow+\infty}\left(\int_0^1\sup_{s>x}\frac{s}{e^{(s\log s)t}}dt\right)$$ Thank you so much!
jenny
  • 69
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Volume of n dimensional ball

The open ball of radius $r$ in $\mathbb{R}^{N}$ is the set $\left\{\left(x_{1}, x_{2},\ldots,x_{n}\right)\in \mathbb{R}^{N} \mid \sum_{i = 1}^{N} x_{i}^{2} < r^{2}\right\}$. By definition its volume $V_N\left(r\right) = \int\int\cdots\int…
clarkson
  • 1,907
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Prove that there exists infinitely many pairs of positive real numbers $\alpha$ and $\beta$ such that $\alpha^\alpha = \beta^\beta$

I was searching online for some difficult Calculus problems and this one stumped me. An example of a pair of numbers would be $\left(\frac{1}{2}, \frac{1}{4}\right)$ because $$\left( \frac{1}{2} \right)^{\frac{1}{2}} = \left( \frac{1}{4}…
codemonkey800
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