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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Signs of coefficients of polynomial

Below, there is a plot of function: $a(x+d)^2 +b|x+d| +c$. What can you learn about coefficients $a,b,c,d$? I know the answer: $a>0, b<0, c>0, d<0$ but I want to understand why is it so. $a>0$ because (1) it is even-order polynomial and (2) its…
Igrek
  • 157
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Can one legally take the derivative of a derivative?

Let $$ f(n,x) = \frac{d^n}{dx^n} \cos(x) = \cos(\frac{\pi n}{2}) \: \cos(x) - \sin(\frac{\pi n}{2}) \: \sin(x)$$ Can one take the derivative with respect to $n$ so that one has $$ \frac{\partial}{\partial n} f(n,x) = -\frac{\pi}{2}…
DecarbonatedOdes
  • 968
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if $1=\int _{ 1 }^{ \infty }{ \frac { ax+b }{ x(2x+b) } dx } $ then $a+b=?$

I have the following question: if $1=\int _{ 1 }^{ \infty }{ \frac { ax+b }{ x(2x+b) } dx } $ then $a+b=?$ a) $0$ b) $e$ c) $2e-2$ d) $1$ I tried finding it's anti-derivative but that doesn't seem to help. I also tried to get it to $\int _{ 1 }^{…
Paz
  • 789
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Given $f$ is continuous at $[0, \infty] $and $\lim_{x \to \infty} \frac {f(x)}{x} = 1$ , prove that $f$ have global minimum at $[0, \infty]$

I tried to show that $f$ $\;$ is not bounded, else the limit must be $0$. Then i tried to prove that $\lim_{x \to \infty} f(x) = \infty$ $\;$, to find a way to impose Weierstrass's second theorem but failed to do so.
john_gayl
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How to find the range of a function?

I need to find the range of $\frac{x^2+6}{2x+1}$ I know that $x$ cannot be $-1/2$. I graphed the function on Desmos and I can see that there is a vertical asymptote at $x=-1/2$ However, I'm having trouble finding the range of this function? I can…
user8290579
  • 886
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Determine all functions satisfying $f(x)=\frac{n}{2}\int^{x+\frac{1}{n}}_{x-\frac{1}{n}} f(t)\,dt$

Question: Determine all functions $f:\mathbb{R}\to \mathbb{R}$ that satisfy the following two properties. The Reimann integral $\displaystyle \int^b_a f(t)\,dt$ exists for all real numbers. For every real number $x$ and every integer $n\geq 1$, we…
73rd
  • 140
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How do I solve for $x$ s.t. $\sum_i^n \frac{1}{x-a_i}=0$?

I am trying to solve for $x$ s.t. $$\sum_i^n \frac{1}{x-a_i}=0$$ or $$\sum_i^n \frac{1}{x-a_i}\rightarrow0$$ $a_i$ is an arbitrary real number. A trivial solution is $x \rightarrow \infty$, and sometimes there aren't other solutions. But when there…
AetbeUT
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Proving that there are only two real roots

Prove that $$ f(x)=2x^4-8x+2$$ has exactly two real roots I am aware of how to prove the existence of the two roots 1.4933.. and 0.2509.. by using Intermediate-Value theorem. However, I am unsure what theorem to use in showing that there is only…
Sharmin Aeheli
  • 91
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Split area defined by curve into equal parts

I have a curve $y=x^2$ defined for the region: $0\le x\le2$. What's the best way to work out the values $b$ and $c$ so that the three areas defined by the shaded regions are equal? I believe this is the regions defined between: $y=x^2$ and…
rwmiller
  • 41
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How to solve $2(k-1)-2z-z\frac{U^\prime}{U}=0$?

Consider the function $$f_X(x)= \frac{(x^2)^{\frac{k-1}{2}}e^{-\frac{x^2}{2}}}{\sqrt{\pi}\Gamma(\frac{k}{2}) 2^{\frac{k}{2}}} U(\frac{1}{2} , \frac{k+1}{2},\frac{x^2}{2}), \quad k>0, \quad x\in R,$$ where $U(a,b,z)$ is the Trichome's…
Masoud
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Why if $x,y\in\mathbb{R}^n-\{0\} $ and $\|x\| + \|y\| =\|x+y \|$, then exist $\lambda > 0$ s.t. $x=\lambda y$

Why if $x,y\in\mathbb{R}^n-\{0\} $ and $\|x\| + \|y\| =\|x+y \|$, then there exists $\lambda > 0$ such that $x=\lambda y$? I don't know how to proceed.
Daniel Tinoco
  • 65
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If $a_n>0$ for all $n$, and $\lim_{n\to\infty}a_na_{n+1}=A$, $\lim_{n\to\infty}a_na_{n+2}=B$, $\lim_{n \to \infty}a_na_{n+3}=C$, dis/prove $A=B=C$

I have hard time proving the following: Let $a_n$ be a sequence such that $a_n>0$ for all $n$ and: $$\lim_{n \to \infty}a_na_{n+1}=A$$ $$\lim_{n \to \infty}a_na_{n+2}=B$$ $$\lim_{n \to \infty}a_na_{n+3}=C$$ I try to prove or disprove that…
Rowar
  • 63
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4 answers

Find the second real root for cubic $x^3+1-x/b=0$

A cubic of the form $$x^3+1-x/b=0$$ has has three real roots Using the Lagrange inversion theorem one of the roots is given by $$x = \sum_{k=0}^\infty \binom{3k}{k} \frac{b^{3k+1} }{(2)k+1} $$ How do you find the second one? I cannot find any info…
CarP24
  • 300
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2 answers

Integrate $\int\frac{x^4}{\sqrt{a^2-x^2}} \, dx$ using trigonometric substitution

Integrate $\int\frac{x^4}{\sqrt{a^2-x^2}} \, dx$ using trigonometric substitution Ok, so it's been a really long time since I've done a problem like this but after doing a little bit o studying, this is how far I've…
cele
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Derivative of absolute value function.

If you differentiate the absolute value function the result could be the Sign Function. By definition, the Sign Function is defined at 0, but there is no derivate of absolute value function at zero. So, the question is: Is the Sign Function a valid…
Guilherme Namen
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