Mathematics Stack Exchange
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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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At what point of mathematical education can you start inventing new math?

I am a 2nd year student doing an honors program in math and statistics. Everything that I have been learning has been formulas, theorems, and mathematical concepts that other people have discovered/invented/created. Some very simplistic formulas…
Mark
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Show $ \varlimsup_{n\rightarrow\infty}{\sqrt[n]{a_{1}+a_{2}+\cdots+a_{n}}}=1 $ if $\displaystyle \varlimsup_{n\rightarrow\infty}{\sqrt[n]{a_{n}}}=1 $

Let $\{a_{n}\}$ be a positive sequence with $\displaystyle \varlimsup_{n\rightarrow\infty}{\sqrt[n]{a_{n}}}=1 $. How can we show that $$ \varlimsup_{n\rightarrow\infty}{\sqrt[n]{a_{1}+a_{2}+\cdots+a_{n}}}=1 $$ I am not sure the problem is true. If…
pxchg1200
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Calculus Proof Problem

Let $f$ be a continuous function whose domain includes $[0,1]$, such that $0 \le f(x) \le 1$ for all $x \in [0,1]$, and such that $f(f(x)) = 1$ for all $x \in [0,1]$. Prove that $\int_0^1 f(x)\,dx > \frac34$. I've been stuck on this question for a…
Mimi
  • 201
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Is the maximum function of a continuous function continuous?

Suppose $f(x)$ is continuous on the closed interval $[a,b]$. Define $m(x)=\max_{a\leq s\leq x}\, f(s)$, $a\leq x\leq b$. Is $m(x)$ continuous necessarily? Thank you.
Knightgu
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Is $\sqrt x$ continuous at $0$? Because it is not defined to the left of $0$

If a function has a limit from the right but not from the left, is it still continuous?
Rodrigo Stv
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Suppose $f$ is continuous on $[0,2]$ and $f(0) = f(2)$. For which $a\in(0,2)$ must there exist $x,y\in[0,2]$ so that $|y − x| = a$ and $f(x) = f(y)$?

Suppose $f$ is continuous on $[0,2]$ and $f(0) = f(2)$. For which $a\in(0,2)$ must there exist $x,y\in[0,2]$ so that $\lvert y − x\rvert = a$ and $f(x) = f(y)$ I'm really unsure how to approach this problem ...we did a similar problem where…
ky370211
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Indefinite Integral of Absolute Value of x? Is there a closed form solution?

Been searching the net for awhile and everything just comes back about doing the definite integral. So just thought to ask here. Title says it all. Is there a closed form solution for the indefinite integral $\int |x| dx$ ?
Wikkyd
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How is the shape of the curve $f(x)$, near $x=a$, affected by $f'''(a)$

In introductory calculus classes we learn the utility of calculating $f'(x)$ and $f''(x)$ for sketching the curve $f(x)$. My question is given $f'''(a)$, how does this value affect the shape a curve $f(x)$ for $x$ near $a$? Consider an example where…
Applied Squared Mathematician
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Can the fundamental theorem of calculus be proved without an appeal to mean value or Rolle's theorem or its immediate consequences?

I think the answer is in the negative. Here are two of the ways I know. Both of them use the Mean Value Theorem. The first one use in an indirect way, and the second uses it more forthrightly. The first proof goes something like this. Prove that…
abel
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Show $\frac{d}{dx} \tan^3{x}-3 \tan{x}+3x = 3 \tan^4{x}$

How do I go about doing this? Show $\frac{d}{dx} \tan^3{x}-3 \tan{x}+3x = 3 \tan^4{x}$. My work. $$ \begin{align*} \frac{d}{dx} \tan^3 x -3 \tan x+3x &= 3 \tan^2 x \sec^2 x - 3 \sec^2 x + 3 \\ &= 3 \frac{\sin^2 x}{\cos^2 x} \frac{1}{\cos^2 x}…
Jiew Meng
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Should $f(x) \equiv 0$ if $0\le f'(x)\le f(x)$ and $f(0)=0$?

Assume $f(x)$ is a real-function defined on $[0,+\infty)$ and satisfies the followings: $f'(x) \geq 0$ $f(0)=0$ $f'(x) \leq f(x)$ Should we always have $f(x) \equiv 0$ ? Thanks for any solution.
Xucheng Zhang
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Calculate $f^{(25)}(0)$ for $f(x)=x^2 \sin(x)$

Calculate $f^{(25)}(0)$ for $f(x)=x^2 \sin(x)$. The answer is too short for me to understand, and the answer is $- 25 \cdot 24 \cdot 8^{23}$
jacob
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How do you take the derivative with respect to a function?

I'm trying to figure out how to take a derivative that looks like $\displaystyle \frac{d}{d(\ln(a))}$, of a function $F(a)$, where $a = a(t)$. In the paper I'm reading (where this appears), they give the following result in the case that $F(a) =…
tentaclenorm
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prove $f(x)$ has at least $2n$ roots

Let $f(x)$ be continuous function in $(0,\pi)$,for any integer $k$ where $1\leq k\leq n$,we have $$ \int_{0}^{\pi}f(x)\cos kxdx=\int_{0}^{\pi}f(x)\sin kxdx=0 $$ Prove that:$f(x)$ has at least $2n$ roots on $(0,\pi)$. I can only solve the cases when…
pxchg1200
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