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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Property of a Cissoid?

I didn't think it was possible to have a finite area circumscribing an infinite volume but on page 89 of Nonplussed! by Havil (accessible for me at Google Books) it is claimed that such is the goblet-shaped solid generated by revolving the cissoid…
ThudanBlunder
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Prove: the function $g$ has a global minimum in $\mathbb{R}$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ a polynomial of a even $n$ degree, such that $0\leq f(x)$ let $g=f+f'+f''+\cdots+f^{(k)}$, prove $g$ has a global minimum in $\mathbb{R}$ when $k$ is the $k$-th derivative How should I approach this?
gbox
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Why do $r=\cos[2\theta]$ and $r=\frac{1}{2}$ have 8 intersections?

I am not sure if the question is a duplicate. The conclusion from the title is clear, for example by viewing the wolframalpha graph at here. But on the other hand I feel intuitively there should be only 4, since a line intersect with…
Bombyx mori
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Find Area Enclosed by Curve

I want to find the area enclosed by the plane curve $x^{2/3}+y^{2/3}=1$. My attempt was to set $x=\cos^3t, \ y=\sin^3t$ so:$$x^{2/3}+y^{2/3}=\cos^2t+\sin^2t=1$$ Then the area is…
user124910
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Find all $a>0$, so that the equation $x=a\sin x+b$ has exactly one solution in $[0,a+b]$, ($b>0$).

For what $a>0$ is there exactly one solution in $[0,a+b]$ to the equation $$x=a\sin x+b$$ for all $b>0$. Thank you, Eitan.
Eitan Porat
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There exists a constant $C$ such that $|\sin(x)-x|\le C|x^3|$ for all real $x$

Problem: Show that there exists a constant $C$ such that $|\sin(x)-x|\le C|x^3|$ for all real $x$. I'm stuck on this question. First, I noticed that $|\sin(x)-x|\le 1+|x|$. But I'm not sure how to deal with $|x|\le 1$. Let $f(x)=|\sin(x)-x|$ and…
3x89g2
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Calculate limit for, $\lim\limits_{x\to 0}\frac{1-cos(x^6)}{x^{12}}$, but in there have suprize.

Let's think about this function, $\quad \to f(x)=\dfrac{x^2-1}{x-1}$, $\lim\limits_{x\to 1}\dfrac{x^2-1}{x-1}=0/0$ , First Solution : $\lim\limits_{x\to 1}\dfrac{x^2-1}{x-1}=\lim\limits_{x\to 1}\dfrac{x+1}{1}$ $=\lim\limits_{x\to…
Micheal Brain Hurts
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Evaluate $\lim_{x\to 0^{+}}\frac{\ln(x)}{x}$

$$\lim_{x\to 0^{+}}\frac{\ln(x)}{x}$$ I know that $\lim_{x\to 0^{+}}\ln(x)=-\infty$ and $\lim_{x\to 0^{+}}\frac{1}{x}=\infty$ can I say anything about $-\infty\cdot\infty$ or it is intermediate of L'hopital?
gbox
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Is there a non-continuous Riemann integrable function with an anti-derivative?

Is there a function $f:[a,b]\rightarrow\mathbb{R}$ (for real $a
Bel
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Applying mean value theorem

My question is about this question. We know that $f$ is differentiable on $(0,\infty)$ and $f(x)$ has a finite limit (let's say $L$) as $x\to\infty.$ My exact question is if we really need the assumption that $f'(x)$ has a finite limit as…
thanasissdr
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A function who's image is $\mathbb{R}$ on every interval

I was wondering if there exists a function $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfies the property that for every $a
35T41
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Is $b|x|(\sin|x^2+x|)$ differentiable? $b$ can have any real value

So I get that if only $\sin|x^2+x|$ was given it is not differentiable at $x=0$, but why does it become differentiable at $0$ when a factor of $b|x|$ is introduced? And if it does, then is the statement "A non differentiable function times a factor…
James
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Prove that if $f(x) = \int_{0}^x f(t)\,dt$, then $f(x) = 0$

Prove that if $f(x) = \displaystyle\int_{0}^x f(t)\, dt$ for all $x$, then $f(x) = 0$. I first differentiated to get $f'(x) = f(x) - f(0)$. Then by the mean value theorem there exists a $c$ in $(0,x)$ such that $f'(c)=\dfrac{f(x)-f(0)}{x}$. Thus,…
Puzzled417
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Why this limit of integration?

I've been solving problems from the book by DeGroot and Schervish and I can't understand why m is the upper limit of integration in the solution to this problem. Why not the lower one? Here is the problem: Suppose that a random variable X has a…
Alan
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What angle results in the shortest route across a river?

Take a river $100m$ wide flowing at $10m/s$ ($x$-axis). Take a boat which can go at $8.0m/s$. Clearly this boat cannot go straight across the river without being dragged some distance downstream. What angle $\alpha$ ( relative to the negative…
Kantura
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