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 there a mathematical theorem that states the equivalence of curved and straight lines

Any curve can be considered to be made of sufficiently small straight lines. What is the name of the theorem which states this fact? Thank you. I asked the same question at Hacker News. [Edit: in the Hacker News question, the OP included the…
Zeynel
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Derivative Counterexamples - Calculus

I need counterexamples for the following (I guess these claims are not correct): If $ lim_{n\to \infty} n\cdot (f(\frac{1}{n}) - f(0) ) =0$ then $f$ is differentiable at $x=0$ and $f'(0)=0$ . If f is defined in a neighberhood of $a$ including $a$…
criticism
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proof that a function is decreasing

I need to prove that $$\sqrt{3+x^{1/3}}+2\over x-1$$ is a decreasing function in $(1,\infty)$ with the definition of a decreasing function; the problem I have is that this expression is difficult so please I need your help
David Hernandez
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A $\sin^n x$ integral

By trying to derive volume of N-sphere I came the integrals like: $\int_0^{\pi} \sin^n x dx $ Wolfram Mathematica was able integrate it giving the following: $$\int_0^{\pi} \sin^n x dx = \sqrt\pi\frac{\Gamma(\frac{1+n}{2})}{\Gamma(1+\frac{n}{2})}…
Winten
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Calculation of $\int_{0}^{\pi}\frac{1}{(5+4\cos x)^2}dx$

Calculation of $\displaystyle \int_{0}^{\pi}\frac{1}{(5+4\cos x)^2}dx$ $\bf{My\; Try}::$ Using $\displaystyle \cos x = \frac{1-\tan^2 \frac{x}{2}}{1+\tan^2 \frac{x}{2}}$ Let $\displaystyle I = \int_{0}^{\pi}\frac{1}{\left(5+\frac{4-4\tan^2…
juantheron
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How do I calculate the derivative using the chain and product rules?

How can I calculate the derivative of the following function using the chain and product rules? $y=30e^{-0.2x} \cdot \cos (1.5x) + 100$ I know I will have to use: $y=vu'+uv'$ And I've found the answer using Wolfram Alpha - I just can't figure out…
Parachuting Panda
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Determine (without using a calculator) which of the following is bigger: $1+\sqrt[3]{2}$ or $\sqrt[3]{12}$

I have encountered the following question in a highschool book in the subject of powers. and, it seems I can't solve it.... Determine (without using a calculator) which of the following is bigger: $1+\sqrt[3]{2}$ or $\sqrt[3]{12}$ Any ideas? Thank…
topsi
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Prove that $\frac{d^2x}{dy^2}$ equals $-\frac{d^2y}{dx^2}\left(\frac{dy}{dx}\right)^{-3}$

The actual question is : $$\frac{d^2x}{dy^2}$$ equals : $$1. \frac{d^2y}{dx^2}^{-1}$$ $$2. …
A Googler
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If $|x_1-t|+|x_2-t|+\cdots+|x_n-t| = |y_1-t|+|y_2-t|+\cdots+|y_n-t|$ for all $t$ then $\{x_1,\ldots,x_n\} = \{y_1,\ldots,y_n\} $

Prove that if $|x_1-t|+|x_2-t|+\cdots+|x_n-t| = |y_1-t|+|y_2-t|+\cdots+|y_n-t|$ for all values of $t$ then $\{x_1,\ldots,x_n\} = \{y_1,\ldots,y_n\} $ (All the variables are real numbers). I tried to prove that minimums are equal then continue by…
hhsaffar
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How would I differentiate $\sin{x}^{\cos{x}}?$

How can I differentiate $\displaystyle \sin{x}^{\cos{x}}$? I know the power rule will not work in this case, but logarithmic differentiation would. I'm not sure how to start the problem though and I'm not too comfortable with logarithmic…
iii
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help with summation index

I try to exprex the following doble summ $$\sum _{k=0}^n \sum _{m=k+1}^n \frac{a_k \left(2^{2 m} n \binom{m+n}{2 m}\right)}{m+n}$$ it comes from acceleration series and i need to express in this way $$\sum _{j=1}^n \sum _{k=0}^{j-1} \frac{2^{2…
capea
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What's the limit of this sequence?

$\lim_{n \to \infty}\frac{1}{n}\bigg(\sqrt{\frac{1}{n}}+\sqrt{\frac{2}{n}}+\cdots + 1 \bigg)$ My attempt: $\lim_{n \to \infty}\frac{1}{n}\bigg(\sqrt{\frac{1}{n}}+\sqrt{\frac{2}{n}}+\cdots + 1 \bigg)=\lim_{n \to…
Twnk
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The limit of an inverse trig function.

$$\lim_{x\to\infty}\tan^{-1}(\dfrac{x}{4})$$ I have a question about why this limit is $\pi/2$. If the argument of $\tan^{-1}$ goes to infinity, doesn't the slope (since it is the tangent function) also go to infinity? I don't see why it would go to…
Kot
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If $f(x)g(x) = x$, then is it possible that $f$ and $g$ are differentiable and $f(0)=g(0)=0$?

True/False: If $f(x)g(x) = x$, then it is possible that $f$ and $g$ are differentiable and $f(0)=g(0)=0$. If true, explain. If false, then give a counterexample. Not even sure where to start, please help!
user100500
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Telescoping Series confusion

My professor gave us this example on her notes: $$\sum_{n = 1}^\infty \left(\frac{3}{n(n+3)}+\frac{1}{2^n}\right)$$ So I know we're supposed to find the partial fraction, which ends up…
FrostyStraw
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