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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 is the practical difference between a differential and a derivative?

I ask because, as a first-year calculus student, I am running into the fact that I didn't quite get this down when understanding the derivative: So, a derivative is the rate of change of a function with respect to changes in its variable, this much…
Faqa
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Don't see the point of the Fundamental Theorem of Calculus.

$$\frac{d}{dx}\int_a^xf(t)\,dt$$ I would love to to understand what exactly is the point of FTC. I'm not interested in mechanically churning out solutions to problems. It doesn't state anything that isn't already known. Prior to reading about…
JackOfAll
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Is there a function with the property$ f(n)=f^{(n)}(a)$

Is there a not identically zero, real-analytic function $f\colon\mathbb R\to\mathbb R$, which satisfies $$f(n)=f^{(n)}(a),n\in\mathbb N \text{ or }\mathbb N^+?$$ and $a\in \mathbb R$ I saw a special case when $a=0$ I try to solve it by…
mnsh
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Derivative of max function

I was just wondering what the derivative of $f(x) = \max(0,1-x)^{2}$ would be. What technique do you use to determine this derivative?
phil12
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Calculus conjecture

When I was a senior in high school in 2001, as I took calculus, I made the following conjecture that proves resistive to attack. It goes like this: For every positive integer $n$, there are exactly $n$ positive real zeroes of $$\frac…
Roman Chokler
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Is $f(x)=1/x$ continuous on $(0,\infty)$?

I've never actually done a delta-epsilon proof, so I thought I'd try my hand at one. I decided to try it out for $f(x)=1/x$. If I understand correctly from the wikipedia article, I want to show for any $\varepsilon>0$, there exists a $\delta>0$ such…
yunone
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Function which is continuous everywhere in its domain, but differentiable only at one point

I am new in this forum. My question: Suppose a real valued function $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous everywhere. Is it possible to construct $f$ that is differentiable at only one point? If possible, please give an…
netsurfer
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Why does L'Hôpital's rule work?

Can anyone tell me why the L'Hôpital's rule works for evaluating limits of the form $\frac{0}{0}$ and $\frac{\infty}{\infty}$ ? What I understand about limits is that when you divide a really small thing (that is $\rightarrow0$) by another really…
Green Noob
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Is differentiating on both sides of an equation allowed?

Let's say we have $x^2=25$ So we have two real roots ie $+5$ and $-5$. But if we were to differentiate on both sides with respect to $x$ we'll have the equation $2x=0$ which gives us the only root as $x=0$. So does differentiating on both sides of…
Dèsjardins
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Taylor's Theorem with Peano's Form of Remainder

The following form of Taylor's Theorem with minimal hypotheses is not widely popular and goes by the name of Taylor's Theorem with Peano's Form of Remainder: Taylor's Theorem with Peano's Form of Remainder: If $f$ is a function such that its…
Paramanand Singh
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When to do u-substitution and when to integrate by parts

I'm in my first semester of calculus, so the problems I'm facing are about as hard as those on KhanAcademy calculus playlist. I'm currently doing integration, a somewhat difficult part of the course. Doing derivatives is mechanics; finding the…
jacob
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Is there a function $f: \mathbb R \to \mathbb R$ that has only one point differentiable?

While studying complex variables, I could learn that $f(z)=|z|^{2}$ has only one point which is $z=0$ that $f$ being differentiable and $f$ being not differentiable at any other points. Then, I was wondering if there is a function $f: \mathbb R \to…
Emily
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All derivatives zero at a point $\implies$ constant function?

Suppose $f: \mathbb{R} \rightarrow \mathbb{R} $ is a continuous function, and there exists some $a \in \mathbb{R}$ where all derivatives of $f$ exist and are identically $0$, i.e. $f'(a) = 0, f''(a) = 0, \ldots$ Must $f$ be a constant function? or…
donburi
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Find an expression for the $n$-th derivative of $f(x)=e^{x^2}$

I need to find an expression for $n$th derivative of $f(x) = e^{x^2}$. Really need help.
Mykolas
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What comes after tetration ? And after ? And after ? etc.

The power is to the multiplication what the multiplication is to the addition. We can put those this way: Addition Multiplication Exponentiation Tetration What comes after tetration? What comes whatever comes after tetration? Is this generalized…
Olivier Grégoire
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