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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Is integral of a function always bigger than the function?

Is $|\int_a^b f(x)dx| \ge |f(b) - f(a)|$ true?
kptlronyttcna
  • 793
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What is the relationship between $x$ and $x^{1+\sin(x)}$ with respect to their growth speed?

How to calculate \begin{align} \lim_{x\to +\infty} \frac{x^{\sin(x)+1}}{x} \end{align} By looking at $x^{1+\sin(x)}$ plot I can see that since it oscillates between $1$ and $x^2$ the limit does not exist. So my main question is: If the limit does…
KFkf
  • 826
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Is this another identity of exponential formula

As written in Wiki, $e^x=\lim_{n \to \infty}\left(1+\frac{x}{n}\right)^n$. However, does anyone agree that $$e^x = \lim_{n \to \infty}\left(\frac{1+0.5\frac{x}{n}}{1-0.5\frac{x}{n}}\right)^n ?$$
chuackt
  • 305
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The limit of a sum (a Riemann sum?)

$$ \lim_{n \rightarrow \infty} \sum_{k=1}^n \left( \sqrt{n^4+k}\ \sin\frac{2k\pi}{n} \right) = ? $$ I tried to transform it into a Riemann sum, to use Taylor-series of the sine function, to estimate, but nothing. Any help would be great.
Lacitek
  • 33
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Limit of Product of Two Functions

Given $\lim_{x\rightarrow a} f(x)=l$ and $\lim_{x\rightarrow a} g(x)=m$, I tried to prove that $lim_{x\rightarrow a} f(x)g(x)=lm$. Although the statement is simpler, proof is not obvious, and I faced some minor problems while writing the proof. Let…
Groups
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If $\int\frac{1-5\sin^2 x}{\cos^5 x\cdot \sin^2 x}dx = \frac{f(x)}{\cos^5 x}+\mathcal {C}$. Then value of $f(x)$.

If $\displaystyle \int\frac{1-5\sin^2 x}{\cos^5 x\cdot \sin^2 x}dx = \frac{f(x)}{\cos^5 x}+\mathcal {C}$. Then value of $f(x)$. $\bf{My\; Try::}$ Given $$\displaystyle \int\frac{1-5\sin^2 x}{\cos^5 x\cdot \sin ^2 x}dx = \underbrace{\int…
juantheron
  • 53,015
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Find x-coordinate, given the derivative and horizontal tangent.

dy/dx = (2x-y)/(x+y) Horizontal tangent at y=8. How does one determine the x-coordinate of the point? I'm not sure.
Donny
  • 31
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Find slope of tangent line using m$_{tan}=\lim\limits_{x\to a}\frac{f(x)-f(a)}{x-a}$ and the point $P = (5,\frac{2}{5})$

Perhaps I'm missing something simple here, but every time I attempt this problem I get the same answer that does not make sense. The question says, use the definition m$_{tan}=\lim\limits_{x\to a}\frac{f(x)-f(a)}{x-a}$ to find the slope of the line…
Sabien
  • 625
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$(f_n(x))_{n=1}^{\infty}$ , $g \in C[\mathbb{R}]$. Prove: $(g\circ f_n)_{n=1}^{\infty}$ uniformly converges iff $g$ is uniformly continues

Let $(f_n(x))_{n=1}^{\infty}$ a series of uniformly continuous functions, $\mathbb{R}\to\mathbb{R}$ which uniformly converges to the function $f$, and a continuous function $g$ :$\mathbb{R}\to\mathbb{R}$. I need to give an example where $(g\circ…
Jozef
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Are all endpoints discontinuous?

I learned that something is a limit if the left limit and right limit exist and are equal. But then doesn't this mean that if I have a function on $[a,b]$, that the endpoints $a$ and $b$ are discontinuous because $a$ doesn't have a left limit and…
Johannes
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Quotient rule, one sign wrong

After using the quotient rule on $$y=\frac{\cos x}{x}$$ I got $$\frac{-x\sin x -\cos x}{x^2}.$$ However the answers says it should be $$\frac{-x\sin x + \cos x}{x^2}.$$ So who's right?
Abby
  • 221
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If $ p^2=a^2 \cos^2 \theta+b^2 \sin^2 \theta $, show that $p + p'' = \frac{a^2b^2}{p^3}$

if $ p^2=a^2 \cos^2 \theta+b^2 \sin^2 \theta $, show that $p + p'' = \frac{a^2b^2}{p^3}$ My try : $2pp' = (b^2-a^2)\sin 2\theta$ $p'^2 + pp'' = (b^2-a^2)\cos 2\theta$ Thats it ! it doesn't simplify no matter what I try. Any help ?
AgentS
  • 12,195
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Inequality proof for a $\mathcal{C}^1$ function.

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be continuously differentiable. Suppose $|f_x (x,y) | \leq K$, $|f_y (x,y) | \leq K$ for all $(x,y)$. Prove that $$|f(x_1, y_1) - f(x_2, y_2)| \leq \sqrt 2 K \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$ My…
teoo
  • 31
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Check existence of limit

$$ \lim_{k\rightarrow\infty} \int_o^1 |\cos{(kx)}|\,dx $$ $$ \int_o^1 |cos(kx)|\,dx = \frac{1}{k}\int_o^k |\cos{y}|\,dy =\frac{f(k)}{k} $$ where I used $y=kx$ and $f(x)=\int_0^x |\cos{y}|\,dy$. How can I proceed or what would be the easier…
Kal S.
  • 3,781
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If $y^{\frac{1}{m}} + y^{\frac{-1}{m}}=2x$, show that $x^2y_{n+2}+(2n+1)xy_{n+1} + (n^2-m^2)y=0$

My try : I have taken $y_1,y_2$ and tried to get a recursive relation between them but couldn't find any pattern. Please help.
AgentS
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