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.

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Derivative of the sine function when the argument is measured in degrees

I'm trying to show that the derivative of $\sin\theta$ is equal to $\pi/180 \cos\theta$ if $\theta$ is measured in degrees. The main idea is that we need to convert $\theta$ to radians to be able to apply the identity $d/dx \sin x = \cos x $. So we…
somebody
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Can endpoints be local minimum?

My textbook defines local maximum as follows: A function $f$ has local maximum value at point $c$ within its domain $D$ if $f(x)\leq f(c)$ for all $x$ in its domain lying in some open interval containing $c$. The question asks to find any…
Phil
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Simple Maclaurin Series $e^{\tan(x)}$

In a multichoice online test that I did the other day, I was required to select the Maclaurin series for $e^{\tan(x)}$. It was necessary for me to find the first four terms in order to establish which answer was correct. In the end of year exam, I…
Charles Salmon
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Prove that $ f(x) = e^x + \ln x $ attains every real number as its value exactly once

Prove that the function $$ f(x) = e^x + \ln x $$ attains every real number as its value exactly once. First, I thought to prove that this function is a monotonic continuous function. But then I wasn't sure if that is how to prove the result, and…
D_R
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What is the derivative of $x^n$?

If $n$ is an integer I can evaluate the limit in the "difference quotient" to see that the derivative of $x^n$ is $nx^{n-1}$. If $n=p/q$ is rational then I can write x as a $q$th root of $x^p$ and since $p$ is an integer I can evaluate the limit in…
Geoffrey Critzer
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Is the catenary the trajectory of anything?

Notice that the parabola, defined by certain properties, is also the trajectory of a cannon ball. Does the same sort of thing hold for the catenary? That is, is the catenary, defined by certain properties, also the trajectory of something?
user27325
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How to calculate $\int_{-a}^{a} \sqrt{a^2-x^2}\ln(\sqrt{a^2-x^2})\mathrm{dx}$

Well,this is a homework problem. I need to calculate the differential entropy of random variable $X\sim f(x)=\sqrt{a^2-x^2},\quad -a
bigeast
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A problem with minimizing a function

I have the following cost function: $\mbox{BSP Cost}=\sum_{i=1}^{\frac{n}{G}}G^{2}\left\lceil \frac{i}{p}\right\rceil +g\left(p\right)\sum_{i=1}^{\frac{n}{G}}Gi+l\left(p\right)\frac{n}{G}$ I would like to minimize it by choosing an appropriate G…
Amir Rachum
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prove $f'(x)=f(x)$

Given a function $f$ satisfying the following two conditions for all $x$ and $y$: (a) $f(x+y)=f(x)\cdot f(y)$, (b) $f(x)=1+xg(x)$, where $\displaystyle \lim_{x\rightarrow 0}g(x)=1$. Prove that $f'(x)=f(x)$. The only thing I know is that $f'(x)=f(x)$…
steve
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Derivative of Lambert W function.

I'm trying to find the derivative of the Lambert W function which is defined such that: $$W(x)e^{W(x)}=x$$ Through implicit differentiation I get: $$W(x)e^{W(x)}W'(x)+W'(x)e^{W(x)}=1$$ $$W'(x)(W(x)e^{W(x)}+e^{W(x)})=1$$ And using $W(x)e^{W(x)}=x$ I…
Ahmed S. Attaalla
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Can we speak of derivatives of sets?

Suppose we have a monotone sequence of sets: $$A_1,\ldots,A_n$$ $$A_i \subseteq A_{i+1}$$ I think this is a function from $\mathbb N$ to a space of sets. Can we define a function from $\mathbb R$ to a space of sets? Could we then define a derivative…
blue-dino
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Is the max of two differentiable functions piecewise-differentiable?

The question here asks: Given that $f$ and $g$ are two real functions and both are differentiable, is it true to say that $h=max(f,g)$ is differentiable too? Convincing arguments have been presented there that the answer is No. So, how about a…
hBy2Py
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Is using L'Hospital's Rule to prove $\lim \limits_{x \to 0} \sin{x}/x$ circular?

If $\frac{d(\sin{x})}{dx}= \cos{x}$ is proven using the limit $\lim \limits_{x \to 0}\frac{\sin{x}}{x}=1$ (as it is in most textbooks), would it be circular to then use $\frac{d(\sin{x})}{dx}= {\cos{x}}$ and L'Hospital to prove the limit $\lim…
john themathteacher
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Why does $\int_{0}^{\infty}\frac{dx}{1+(x \sin x)^2}$ diverge?

I'd like your help with understanding and showing why $\int_{0}^{\infty}\frac{dx}{1+(x \sin x)^2}$ diverges. As I see it the "problematic spots" where the function may blow are backed up by the sum with $1$. What can I do in order to show that it…
Jozef
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Prove the integral inequality on interval [-1,1]

Let $f(x)$ be continuous function on $[-1,1]$ which satisfies: 1. $f(-1)\ge f(1)$. 2. $x+f(x)$ is non-decreasing. 3. $\int_{-1}^1 f(x)dx=0$. Prove that $\int_{-1}^1 f^2(x)dx \le \frac2 3$
HLong
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