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.

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I don't understand why the level of rigor used here is necessary

These are the course notes for MIT's calculus class, 18.014. Beginning at the bottom of page 4, the professor launches into a long proof that the set of all integers $\mathbb Z$ is closed under addition. I don't see why I can't demonstrate it by…
terriblestudent
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What is $\mathop {\lim }\limits_{x \to 0} \frac{{\tan 2x + \tan 4x - \tan 6x}}{{{x^3}}}$? .

What is $\mathop {\lim }\limits_{x \to 0} \frac{{\tan 2x + \tan 4x - \tan 6x}}{{{x^3}}}$?
Under sky
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Range of $xyz\;,$ If $x+y+z=4$ and $x^2+y^2+z^2=6$

If $x,y,z\in \mathbb{R}$ and $x+y+z=4$ and $x^2+y^2+z^2=6\;,$ Then range of $xyz$ $\bf{My\; Try::}$Using $$(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)$$ So we get $$16=6+2(xy+yz+zx)\Rightarrow xy+yz+zx = -5$$ and given $x+y+z=4$ Now let $xyz=c\;,$ Now leyt…
juantheron
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Square root of both sides

If you have the equation: $x^2=2$ You get: $x=\pm \sqrt{2}$ But what do you do actually do? What do you multiply both sides with to get this answer? You take the square root of both sides, but the square root of what? If you understand what i mean?
euroshopper
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Find all continuous functions $f$ satisfying $\int_{0}^x f = (f(x))^2+C$

Find all continuous functions $f$ satisfying $$\int_{0}^x f = (f(x))^2+C.$$ Edit: You can post your own solution to this question if this is wrong. I don't understand how in the solution below, $f'(x) = 0$ at points where $f(x) \neq 0$. Doesn't…
user19405892
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Evaluate a double integral.

To find: $$I =\int\int_Rx(1+y^2)^{\frac{-1}{2}}dA$$ R is the region in the first quadrant enclosed by $y=x^2$, $y=4$, and $x=0$ $$y=x^2, y=4,x=0, (x= y^\frac{1}{2})$$ $$R=((x,y), 0 \le y \le 4, x^2 \le y \le 4)$$ i.e. $$R=((x,y), 0 \le x \le…
27182818sheena
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Find $\lim_{n \to \infty} \left( \frac{3^{3n}(n!)^3}{(3n)!}\right)^{1/n}$

Find $$\lim_{n \to \infty} \left( \frac{3^{3n}(n!)^3}{(3n)!}\right)^{1/n}$$ I don't know what method to use, if we divide numerator and denominator with $3^{3n}$, I don't see that we win something. I can't find two sequences, to use than the…
Gjekaks
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How are the average rate of change and the instantaneous rate of change related for ƒ(x) = 2x + 5?

How are the average rate of change and the instantaneous rate of change related for ƒ(x) = 2x + 5 ? Should I figure out what is similar between them to solve this question? I don't understand how they correlate
Kate.K
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When antidifferentiating, are we impliclty restricting to an interval?

I was asked this question by a student I am tutoring and I was left a little puzzled because his textbook only defines antiderivatives on intervals (which leads me to believe its author would answer the question in the title in the affirmative). To…
user18063
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Prove that $a-b=b-a\Rightarrow a=b$ without using properties of multiplication.

Yesterday my Honors Calculus professor introduced four basic postulates regarding (real) numbers and the operation $+$: (P1) $(a+b)+c=a+(b+c), \forall a,b,c.$ (P2) $\exists 0:a+0=0+a=a, \forall a.$ (P3) $\forall a,\exists (-a):…
Deathkamp Drone
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For what values of $n$ does every tangent line to the graph $y=x^n$ intersect the graph exactly once?

Let $n$ be a positive integer. For what values of $n$ does every tangent line to the graph $y=x^n$ intersect the graph exactly once? I said we have $\frac{dy}{dx} = nx^{n-1}$ and at the point $(x_0,y_0)$, we have $y = nx_0^{n-1}x+(y_0-nx_0^n)$.…
Puzzled417
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Find the minimum value of this integral.

Find the minimum value of this integral: For what value of $k > 1$ is $$ \int_k^{k^2} \frac 1x \log\frac{x-1}{32}\, \mathrm dx $$ minimal? After applying Newton-Leibniz, I got $k = 3$ and then did 2nd derivative test, it gave me positive result,…
Hyperbola
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Find the minimum and maximum of $h(x) = \dfrac{1}{1+|x|}+\dfrac{1}{1+|x-a_1|}$

Let $a_1 \in \mathbb{R}$. Find the minimum and maximum of $h(x) = \dfrac{1}{1+|x|}+\dfrac{1}{1+|x-a_1|}$. This question seems hard to solve since we don't know what $a_1$ is. We want both $|x|,|x-a_1|$ to be as small as possible or as large as…
user19405892
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Why is being onto necessary for a function to have inverse?

I know that a function needs to be one-to-one so that it can have an inverse but could someone please explain why a function (in addition to being one-to-one) needs to be onto so that it can have inverse? We define the function $f:A\rightarrow B$…
user99885
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If $f$ is continuous with $f(x) = f(2x),f(1) = 3$, then what is $ \int_{-1}^{1}f(f(x))\,dx$?

If $f(x)$ is a continuous function such that $f(x) = f(2x)$ and $f(1) = 3\;,$ Then $\displaystyle \int_{-1}^{1}f(f(x))\,dx$ $\bf{My\; Try::}$ Here $-\infty
juantheron
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