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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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Differential bounded function on real number field

Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ be a bounded differentiable function. Then for any $\varepsilon > 0$, there exists $x \in \mathbb{R}$, s.t. $|f^{\prime}(x)| < \varepsilon$. Is this statement above true? I think it's true, because if…
user1133961
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prove $(\sin x)^{-2}-x^{-2}\leq 1-\frac{4}{{\pi}^{2}},x\in(0,\pi/2]$

$(\sin x)^{-2}-x^{-2}\leq 1-\frac{4}{{\pi}^{2}},x\in(0,\pi/2]$ How to deal with this problem? Observing that when $x=\pi/2$, the above inequality becomes equality. Firstly, denote $f(x)=(\sin x)^{-2}-x^{-2}$ and then take derivative of $f(x)$. We…
monotone operator
  • 687
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$f\in C^{\infty}$ and $|f^{(j)}(x)|\leq M$- show that if $f(1/k)=0 $ for $k\in \mathbb{N}$ so $f=0$

I'd love your help with the following question: let $f\in C^{\infty}$ and there's $M$ such that $|f^{(j)}(x)|\leq M$ for every $j\in \mathbb{Z}_{+}$ and for all $ x\in [-1,1]$. I need to prove tht if $f(1/k)=0 $ for every $k\in \mathbb{N}$, so…
user6163
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Solve $\;6^x+10^x =3^x +13^x$

Here's what I've tried so far. Obvious solutions for equation $(1)$ are $x_1=0 $ and $x_2=1$ $$(1) \Leftrightarrow 6^x-3^x=13^x-10^x \Leftrightarrow \frac{6^x-3^x}{3}=\frac{13^x-10^x}{3} $$ Let $\,f(t)=t^x \hspace{0.1em},\;x\in \mathbb{R}\;$,$\;f$…
ericforman
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Calculus limit question

If $f:(0,+\infty)\rightarrow\mathbb{R}$ continuous $$f(x)=\lim_{n\to\infty}\frac{3^n(x^3+ax^2+3x+1)+x^n(2x^3+6x^2+6x+2)}{2x^n+3^n} \hspace{0.5em} \forall x\in(0,3)\cup(3,+\infty) ,a\in \mathbb{R}$$Prove $f(x)=(x+1)^3 \hspace{0.2em},x>0 $ Ok so…
ericforman
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Proof of The Mean Value Theorem

I find the proof of the Mean Value Theorem not intuitive because it uses Rolle's theroem on an auxiliary function.I am sure that there must be another proof which is longer and intuitive but I can't find it in any calculus or analysis book.I wonder…
Narek Margaryan
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If $f$ is differentiable in $(1,\infty )$ and $\lim_{x\to\infty }f'(x)=L<\infty $ then $\lim_{x\to\infty }f(x)=l\le\infty$?

I need to prove or disprove this: If $f$ is differentiable on $(1,\infty)$ and $\lim\limits_{x\to\infty}f'(x)=L\lt \infty$, then $\lim\limits_{x\to\infty}f(x)=\ell\leq\infty$. After I didn't find any function to disprove with, I started to think…
user6163
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Find the minimum value of the expression $\sqrt{\frac{1}{3}x^4+1}+\sqrt{\frac{1}{3}y^4+1}+\sqrt{\frac{1}{3}z^4+1}$

Let $x, y, z$ be positive real numbers such that $x + y + z = xyz$. Find the minimum value of the expression $$\sqrt{\frac{1}{3}x^4+1}+\sqrt{\frac{1}{3}y^4+1}+\sqrt{\frac{1}{3}z^4+1}$$ My attempt: By using Cauchy-Schwarz inequality and AM-GM…
Mathxx
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What is the difference in usage for $dx$ before or after the integrand?

In almost any calculus book I can think of (for example, "The Calculus with analytic geometry" by Louis Leithold), and even the books of analysis (for example, the Rudin's "Principles of analysis" and "Real and complex analysis"), one can find that…
Arturo Ortiz Tapia
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Calculus Made Easy -Exercises IX, Question 10 (rates of change)

I am working with this exercise, which should be pretty easy, as a similar one is exemplified in the book. A spherical balloon is increasing in volume. If, when its radius is r feet, its volume is increasing at the rate of 4 cubic feet per second,…
Manuel Del Río Rodríguez
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In calculus, can the $\delta$ always be expressed in terms of $\epsilon$?

This question is based on this question. I was wondering why the author did not end the proof by expressing the $\delta$ in terms of $\epsilon$ like he did in prior examples. Is this because it is not always possible to express the $\delta$ in terms…
mauna
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Question about limits involving exponents and bases

While working through a problem set on using derivatives to sketch curves, I noticed something when playing around with an online function grapher: Let $f(x)=x^n -x$, where $n \in \{3, 5, 7, 9, 11...\}$. Let $M$ denote the local maximum of the…
Ryan
  • 245
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Why a closed interval is not necessarily finite?

In the book I am reading now, Calculus, a Complete Course b Adams and Essex, in the section on continuous functions there is the term "a closed, finite interval". The book defines "a closed interval" as to be consisting of all real numbers $x$…
Kaveh Rad
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Existence of nonzero real number $r$ such that $\pi^r$ is rational

So far I couldn't find any related post of the title which is Is there a nonzero real number $r$ such that $\pi^r$ is rational? Is this a know problem? Or a corollary of some theorem?
one potato two potato
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Can anyone explain this process of solving? (Differentiation)

I'm at differentiation of algebraic functions. There's an example in the module that I couldn't quite get how it led to that. $y=\frac {(x+1)^3}{x^2}$ It's solved by using a combination of quotient and power rules. I'll enumerate how it's…
Hal
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