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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What does $\frac{dy}{du}$ mean?

I understand that $dy/dx$ is the rate of change, and that it means "the rate of change of $y$ with respect to $x$", but when I see people use $dy/du$ I get confused ($u$ of course being any variable). What is meant by using $du$ or anything else but…
user1534664
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Teacher needs help with an integral involving trig identities that a student found online

$$\int\frac{\sqrt{\sec^3(x)+\tan^3(x)+\tan(x)}}{1+\sec(x)-\tan(x)}\,dx\quad\text{for $0
KitKatCatHat
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Find number of roots of an equation

I am stuck on the following problem: Given $ f(x)=\displaystyle\frac{4x+3}{x^2+1}$, find how many roots the equation $$f(f(x))=\int_3^4{f(x)\mathrm{d}x}$$ has in the interval $[1, 4]$. Consulting GeoGebra for the graphs, there is a root in that…
Peter Allen
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Why does $ \lim_{x\to 0}\ x^{x^{x}}=0?$

I'm having a hard time understanding why this limits equals to 0. By simply using logarithm identity I get that this limit equals $$ \lim_{x\to 0} \ e^{x\ln x^{x}}=\lim_{x\to 0}\ e^{x^{2}\ln x}$$ and by L'Hopital we get that the limit of the…
user6163
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Symmetry functions and integration

I have this: Case 1) If f is a pair function $f(-x)=f(x)$ then $\int_{-a}^a f(x)dx=2\int_0^af(x)dx$ Case 2) If $f$ is a inpair function $f(-x)=-f(x)$ then $\int_{-a}^a f(x)dx=0$ I understand the reasons for the case 1 to be double area and for…
Haizum Skallah
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The taxonomy of real functions of one real variable

In treatments like Spivak's Calculus, and in Pete Clark's notes here, the motivation is to describe a set of real functions of one real variable with "sufficiently nice properties" -- that is to say, a taxonomy of such functions. It seems to me that…
EE18
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$f$ is differentiable and $\lim_{x\to \infty }f(x)=\lim_{x\to -\infty }f(x)=0$

This is quiet a simple question, but still I'm not sure that I am correct. Let $f$ be a differentiable function in $\mathbb{R}$ such that: $\lim_{x\to \infty }f(x)=\lim_{x\to -\infty }f(x)=0.$ We need to prove that there is $a\in \mathbb{R}$ such…
user6163
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Evaluating a limit of an integral

I am trying to solve the following problem. Let $f:[0,\infty)\rightarrow\mathbb R$ be a continuous function and $b>a>0$ be real numbers. Prove that $$ \lim_{\epsilon\rightarrow+0}\int_{a\epsilon}^{b\epsilon}\frac{f(x)}{x}dx =…
Pteromys
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Bringing limits inside functions when limits go to infinity

A standard result says that under suitable conditions to make sure the functions are defined where they need to be, we can write $$\lim_{x \to c} f(g(x)) = f \left( \lim_{x \to c} g(x) \right)$$ as long as $f$ is continuous at $\lim_{x \to c}…
Barry Smith
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How to find this indefinite integral? $\int\frac{1+x^4}{(1-x^4)\cdot \sqrt{1+x^4}}dx$

I am thinking of a trig sub of $x^2 = \tan{t}$ but its not leading to a nice trigonmetric form, which i can integrate. Our teacher said that it can be computed using elementary methods, but I'm unable to think of the manipualtion.
q123LsaB
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Compute $\lim_{n\to\infty} nx_n$

Let $(x_n)_{n\ge2}$, $x_2>0$, that satisfies recurrence $x_{n+1}=\sqrt[n]{1+n x_n}-1, n\ge 2$. Compute $\lim_{n\to\infty} nx_n$. It's clear that $x_n\to 0$, and probably Stolz theorem would be helpful. Is it really necessary to use this theorem?
A brilliant crazy girl
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Calculating an improper integral as a limit of a sum.

This question arose from a solution I saw yesterday: Suppose f is continuous on (0,1] and has an infinite discontinuity at 0. If the improper integral $\int_0^1 f(x) dx$ converges, is it always the case that: $$\int_0^1 f(x) dx =…
user84413
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Find a point on a parabola that's closest to another point.

Find the point on the parabola $3x^2+4x-8$ that is closest to the point $(-2,-3)$. My plan for this problem was to use the distance formula and then that the derivative to get my answer. I'm having a little trouble along the way. $$ d =…
Revan
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Evaluating $\sum_{n=1}^{\infty} {(-1)^n \cdot \frac{2^{2n-1}}{(2n+1)\cdot 3^{2n-1}}}$

Calculate the summation $\sum_{n=1}^{\infty} {(-1)^n \cdot \frac{2^{2n-1}}{(2n+1)\cdot 3^{2n-1}}}$. So I said: Mark $x = \frac{2}{3}$. Therefore our summation is $\sum_{n=1}^{\infty} {(-1)^n \cdot \frac{x^{2n-1}}{(2n+1)}}$. But how do I exactly…
TheNotMe
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How do we take second order of total differential?

This is the total differential $$df=dx\frac {\partial f}{\partial x}+dy\frac {\partial f}{\partial y}.$$ How do we take higher orders of total differential, $d^2 f=$? Suppose I have $f(x,y)$ and I want the second order total differential $d^2f$?
dan
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