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
7
votes
3 answers

If $\int_0^\infty f\text{d}x$ exists, does $\lim_{x\to\infty}f(x)=0$?

Are there examples of functions $f$ such that $\int_0^\infty f\text{d}x$ exists, but $\lim_{x\to\infty}f(x)\neq 0$? I curious because I know for infinite series, if $a_n\not\to 0$, then $\sum a_n$ diverges. I'm wondering if there is something…
de cyclinders
  • 71
7
votes
3 answers

Proving that $\int_{-\infty}^{\infty} f(x - \frac{1}{x}) dx = \int_{-\infty}^{\infty} f(x) dx.$

Consider $f : \mathbb{R} \rightarrow \mathbb{R}$ to be a continuous function such that $\int_{-\infty}^{\infty} f(x) dx$ exists. I would like to prove that $$\int_{-\infty}^{\infty} f(x - \frac{1}{x}) dx = \int_{-\infty}^{\infty} f(x) dx.$$ I…
Elissa
  • 238
7
votes
3 answers

$\int_0^4\frac{\log x}{\sqrt{4x-x^2}} dx=0$

I am having trouble proving that it is equal to zero analytically. I have tried plotting and know that for $0
kantorovich
  • 157
7
votes
1 answer

$f'+\lambda f$ acts as $f'$ without integral

Problem Suppose that $f(x)$ is a differentiable function and $f'(x)$ is the derivative of $f(x)$. Without aids of integral, can we prove that $f'(x)+\lambda f(x)$ has intermediate property? Intermediate property A (real) function $f(x)$ having…
Yai0Phah
  • 9,733
7
votes
4 answers

$f(x)=\sin x$, $f$ polynomial

$f$ is a polynomial and suppose that $f(x)=\sin x$ has infinitely many solutions. Prove that $f$ is a constant between -1 and 1. I get the problem intuitively, but how would I prove this formally? Can anyone give me any hints?
zxcvber
  • 2,081
7
votes
4 answers

Is the function differentiable at $0$?

Let $$f(x) = \begin{cases}\begin{align*}&\cos{\dfrac{1}{x}}, &x \neq0 \\ &0, &x=0. \end{align*}\end{cases}$$ Is the function $F(x) = \displaystyle \int_{0}^x f dx$ differentiable at $0$? We can see that the function $f(x)$ is continuous everywhere…
user19405892
  • 15,592
7
votes
3 answers

When are differentials actually useful?

I think of differentials as a way to approximate $\Delta y$ in a function $y = f(x)$ for a certain $\Delta x$. The way I understood it, the derivative itself is not a ratio because you can't get $\frac{dy}{dx}$ by taking the ratio of the limits of…
jeremy radcliff
  • 4,805
7
votes
2 answers

Is this series absolutely convergent (doesn't look like an easy problem)?

Is the series $$ \sum_{n=1}^{\infty} \frac{\cos n}{n} $$ absolutely convergent? (I've got a feeling that most probably it isn't due to the fact that for given $\varepsilon>0$ we can find infinitely many $n$ such that $|\cos n|>1-\varepsilon$, the…
Ievgen
  • 477
  • 3
  • 12
7
votes
3 answers

Prove that $(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)$

Assume $f$ and $g$ are differentiable at $x$. Prove that $(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)$ I am assuming here $fg = f(x) g(x)$. Then we can prove this via induction. If $n = 0$ we have $1 = 1$ which is true. Now assume…
user19405892
  • 15,592
7
votes
2 answers

Spivak's Calculus exercise. Chapter 10, Problem 27

Suppose that $f$ is differentiable at 0, and that $f(0) = 0$. Prove that $f(x) = xg(x)$ for some function $g$ which is continuous at 0. This is a problem from Spivak's Calculus, namely problem 27 of Chapter 10. (This is not homework, but…
Arpon
  • 1,246
7
votes
1 answer

Is this limit evaluation correct?

In trying to give the OP an elementary answer to this question, I made some rather stupid mistakes. I feel terrible about giving a wrong answer (in lieu of a complicated but correct one). I devised a new proof, and wanted to check it before editing…
Chris
  • 4,865
7
votes
3 answers

Stirling approximation note

During my study to Stirling approximation I find this formula $n! \approx \sqrt{2\pi n} n^{n}e^{-n} $ but we know that $ 0! =1 $ And in this formula if we replace every $ n $ with $ 0$ we will have $ 0^0$ which is undefined. My question is…
user257567
  • 511
7
votes
2 answers

Proving a Lipschitz constant does not exist.

So I wish to find for each of these functions a Lipschitz constant or prove that none exists. So my definition for a function to be Lipschitz is: A function $f:[a,b] \rightarrow \mathbb{R}$ is Lipschitz if there exists a $L$ such that $|f(x) - f(y)|…
David
  • 79
7
votes
7 answers

showing an inequality not using stirling formula

I don't know how to show that $$ \frac{k^k}{k!}\leq e^{k} $$ without using Stirling's approximation. I want to show it directly. I guess I need some inequality to achieve this but I don't know.
cali
  • 315
7
votes
1 answer

Integrable monotonic functions

Suppose that $f \in L^1(0,+\infty)$ is a monotonic function. Prove that $\lim_{x \to +\infty} x f(x)=0$.
Siminore
  • 35,136
1 2 3
…
99 100