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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Finding when $\frac{1}{\pi}\int_0^{j\pi} \frac{\sin t}{t}\,dt - \frac{1}{2}$ is positive / negative

Could you help me with the following question? Show that the following numbers are positive for $j$ odd and negative otherwise: $$\frac{1}{\pi}\int_0^{j\pi} \frac{\sin t}{t}\,dt - \frac{1}{2}$$
User4356
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$\lim_{n \rightarrow \infty} \frac{(2n)!}{n^{2n}}$

I am trying to show $$\lim_{n \rightarrow \infty} \frac{(2n)!}{n^{2n}}$$ I tried breaking it down, and got stuck when trying to $\left( \frac{2^{n}n!}{n^{n}} \right)$ goes to 0.
Jon
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Why is this not correct？

We are assigned to deal with the following task. Assume that $f(x)$ is derivable for any $x \in \mathbb{R}$. We want to research $\lim\limits_{x \to x_0}f'(x)$ where $x_0 \in \mathbb{R}$. Notice that \begin{align*} f'(x_0)=\lim_{x \to…
mengdie1982
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Is the following series convergent?

Let $A=\sum_{1}^{\infty}a_{n}$ be a convergent series. For every $n$, define $$b_{n}=\frac{a_{1}+2a_{2}+...+na_{n}}{n(n+1)}.$$ Is $\sum_{1}^{\infty}b_{n}$ convergent?
Aliakbar
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The intuitive meaning of integrals

I am an engineering student and i always encounter problems that needs integrals I know that integral is area under the curve , etc.... but till now i could not develop and intuitive meaning for integration. does integration rely only on the idea of…
A.ELAK
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When $dy/dx =0$ for all $x$ in the domain, is $dx/dy$ also zero?

If $dy/dx = 0$ for all $x$ in the domain, is $dx/dy$ also zero? This seems problematic because $dy/dx$ can be thought as $0/1 = 0$ but when you reverse the upper and lower part of the fraction, the fraction is an invalid number.
hues
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The $43$rd derivative of $\sin(x^{13}+x^{15})$ | Calculus 1

A question from a test of my professor on calculus 1: Find the $43$rd derivative of $\sin(x^{13}+x^{15})$ at $x=0$. Any idea I had didn't work, BTW, good luck to me on the test next week :X
ORBOT Inc.
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Composition of functions and number of solutions

If $f(x)= 4x(1-x)$, if $x \in [0,1]$. find number of solutions of equation $$f(f(f(x)))={x\over 3}$$ method: i tried finding the compostion but it is too lenghty. also i tried put $x= \sin ^2 a$ but no success.
maveric
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A convergence of series when ratio test doesn't work

Let $u_n$ be a sequence of positive numbers, for each n: ${u_{n+1}\over{u_n}}\le(\frac{n}{{n+1}})^\alpha$ when $\alpha>1$. Prove that $\sum_{n=1}^\infty {u_n}$ converges. I would like to get a hint.
Gyt
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How to find an analytic solution to $\lim_{x \to +\infty} \frac{x+\sin(x^2)}{\sqrt{x^2+1}}$

To solve this limit: $$\lim_{x \to +\infty} \space \frac{x+\sin(x^2)}{\sqrt{x^2+1}}$$ At the beginning I didn't know how to start. Then I thought, no matter the value that $x$ takes, $sin(x^2)$ will always be between $-1$ and $1$. So for large…
user24047
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Suppose $\lim_{n\to \infty} \frac{a_n}{n}=0 $

Suppose $$\lim_{n \to \infty} \frac{a_n}{n}=0. $$ Prove $$\lim_{n\to \infty} \frac{\max\{a_1,a_2,\ldots, a_n \}}{n}=0. $$ Below is what I tried , but I am not sure about my proof. Denotes…
Jaqen Chou
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Tips on proving this convergence.

We have an inductively defined sequence $x_n=x_{n-1}+2y_{n-1}$ and $y_n=x_{n-1}+y_{n-1}$ where $x_n^2-2y_n^2=\pm 1$, where $x_0=1$ and $y_0=0$. I need to prove that the sequence $\left(\frac{x_n}{y_n}\right)_{n=1}^\infty$ converges to $\sqrt2$. Now…
duncster94
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Looking to evaluate $\int_{0}^{1}\mathrm dx\ln^2(1+\sqrt{x})\ln(1-\sqrt{x})$

How do I evaluate this integral $$\int_{0}^{1}\mathrm dx\ln^2(1+\sqrt{x})\ln(1-\sqrt{x})?$$ Enforcing $x=\tan^2(y)$ $$\int_{0}^{\pi/4}\mathrm dy\sec^2y\tan y\ln^2(1+\tan y)\ln(1-\tan y)$$ Enforcing $v=1+\tan y$ $$\int_{1}^{1+\pi/4}\mathrm dv…
user550936
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How can I tell if one polynomial is greater than another over an interval?

$y_1=ax^2 + bx + c$ $y_2=dx^2 + ex + f$ Question: what is the proper way for me to test that $y_1 > y_2$ over an interval (say 0 to 10, inclusive) for all numbers in that interval? I assume it has something to do with testing the integral of y1 -…
MikeRand
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Some Fundamental theory of calc questions

I just wanted to sanity check these questions: Find the derivative of these functions $$g(s) = \int_{5}^s (t - t^2) ^8 dt$$ $$ g'(s) = (t - t^2)^8$$ $$h(x) = \int_{1}^\sqrt{x} \frac{z^2}{z^4 +1} dz$$ $$h'(x) = \frac{1}{2\sqrt{x}} \cdot…
Jwan622
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