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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Limit composition

If I am given the following graph, how do I compute the limit of x->2 of f(f(x))? I tried to "compose" the limits, but lim x->2 f(x) is 1, but then f is not continuous at 1, so the limit DNE? Is that right?
MikeS
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Find the Area of the ellipse

Given $$\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$$ where $a>0$, $b>0$ I tried to make $y$ the subject from the equation of the ellipse and integrate from $0$ to $a$. Then multiply by $4$ since there are $4$…
Fred
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Differentiating under integral sign -- trig counterexample

I really hate integration by parts, so when faced with $\int_{-\pi}^\pi x^2 \cos n x \, dx$ I tried writing it as $$\int_{-\pi}^\pi x^2 \cos n x \, dx = \frac{d}{dn} \int_{-\pi}^\pi x \sin n x \, dx = \frac{d^2}{dn^2} \int_{-\pi}^\pi \cos n x \,…
cactus314
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How find this equation $\prod\left(x+\frac{1}{2x}-1\right)=\prod\left(1-\frac{zx}{y}\right)$

let $x,y,z\in(0,1)$, find the pairs of $(x,y,z)$ such $$\left(x+\dfrac{1}{2x}-1\right)\left(y+\dfrac{1}{2y}-1\right)\left(z+\dfrac{1}{2z}-1\right)=\left(1-\dfrac{xy}{z}\right)\left(1-\dfrac{yz}{x}\right)\left(1-\dfrac{zx}{y}\right)$$ my…
math110
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Limit exists v. Differentiable

Does existence of a limit at a point not necessarily mean that it's differentiable at that point? Take this function: $$f(x) = \frac{(x - 1)^{2}}{x - 1}$$ The function is not defined at x = 1, but as x approaches 1, f(x) goes to 0; i.e., a…
calc_student
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Showing Intermediate Value Property Holds

Let $$f(x)=\begin{cases} \sin \tfrac 1 x &\text{if $x\ne 0$}\\ 0 & \text{if $x = 0$.}\end{cases}$$ I have to show that $f$ has the intermediate value property. That is, for any $a < b$, if $y$ is any real number such that $f(a) < y< f(b)$ or…
user72195
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A “general definition” of Riemann sum

Suppose $f$ is Riemann integerable in $[0,1]$.Prove that $$\lim_{n\rightarrow \infty}\frac{1}{\phi(n)}\sum_{1\leq k\leq n,(k,n)=1}f\left(\frac{k}{n}\right)=\int_0^1f(x)dx$$Here $\phi(n)$ is Euler's function My attempt: Let $\mu$ be the Möbius…
math
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Proving the existence of $ξ$ and $η$ such that $f'(\xi)(\xi-a)+f'(\eta)(\eta-b)+f(a)+f(b)=0$

Let $f$ be continuous on $[a,b]$, and differentiable on $(a,b)$, $\int_a^b f(x)dx=0$. Show that there exists two distinct $\xi,\eta\in(a,b)$, such that $$f'(\xi)(\xi-a)+f'(\eta)(\eta-b)+f(a)+f(b)=0$$. My attempt: Let…
xldd
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How do I evaluate $\int_{1/3}^3 \frac{\arctan x}{x^2 - x + 1} \; dx$?

I need to calculate the following definite integral: $$\int_{1/3}^3 \frac{\arctan x}{x^2 - x + 1} \; dx.$$ The only thing that I've found is: $$\int_{1/3}^3 \frac{\arctan x}{x^2 - x + 1} \; dx = \int_{1/3}^3 \frac{\arctan \frac{1}{x}}{x^2 - x + 1}…
Dave Johnson
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Does the series $\sum\frac{\ln n}{n^{2}}$ Converge?

Does the series $$\sum_{n=2}^{\infty}\frac{\ln n}{n^2}$$ converge? I'm searching for a solution that does not use the Integral test, Stirling, L’Hôpital or functions theorems. I tried ratio test, and also comparing it to another series. Thank you…
user6163
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evaluation of limit $\lim_{n\rightarrow \infty}\left(\frac{n!}{n^n}\right)^{\frac{1}{n}}$

Calculate the value of the limit $$ \lim_{n\rightarrow \infty}\left(\frac{n!}{n^n}\right)^{1/n} $$ Can we solve this without using a Riemann sum method? If so, how?
juantheron
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Can infinity be divided by anything?

I'm in ninth grade and I've been thinking about this for a while. It's related to a question that came to my mind, namely which is the highest number you can divide 11 by. Is "infinite" a number and is it the highest number by which 11 can divide…
Ilearnmathnotknowit
  • 99
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Question on conservative fields

I'm hoping to really knock out several questions I have in my mind with just this one. I've been doing a lot of practice problems on this topic, and although I get the right answers, I really don't know what the answers mean. So there's a theorem…
Snowman
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What is the infinite product series for $\exp(\sin(x))-1$?

$e^{\sin(x)}-1$ has the same roots as $\sin(x)$. What is the difference between infinite product series expansions of $\sin(x)$ and $e^{\sin(x)}-1$ if they both have same infinite roots ?
Omid Motahed
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Non-equivalence of D'Alembert's and Cauchy's criterion?

Is there a simple example where D'Alembert's and Cauchy's criterion (the root test) for convergence of infinite series don't agree, i.e. one of them proves inconclusive? Can you explain why that happens? Intuitive explanations along with rigorous…
lel
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