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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Roots of a functionwith condition $\int_0^\pi f(x) \sin x dx = \int_0^\pi f(x) \cos x dx =0.$

Let $f:[0,\pi] \rightarrow \Bbb R$ be a continuous function which satisfies $\int_0^\pi f(x) \sin x dx = \int_0^\pi f(x) \cos x dx =0.$ Show that $f$ has at least two roots.
Arman
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What is the derivative of $y=\ln(\cot^{-1}\ x)$?

What is the derivative of $y$? $$y=\ln(\cot^{-1}\ x)$$ If I take out the exponent $-1$, so $$y=-\ln(\cot\ x)$$ Getting the derivative would be $$dy=-\frac{1}{\cot\ x}\ (-\csc^{2}\ x)\ dx$$ $$dy=\frac{1}{\sin\ x\ \cos\ x}\ dx$$ But if we don't take…
user188811
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Prove that $h$ is of class $C^r$

Let $g$ be a $C^r$function defined in a neighborhood $U$ of $\mathbf{x_0}$ in $\mathbb{R}^n$. Show that if $\phi:\mathbb{R}^n\to \mathbb{R} $ is a $C^r$ function whose support lies in $U$, then the…
user239726
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How to obtain the series of the common elementary functions without using derivatives?

First, I'm a freshman student of physics, not of mathematics, so please excuse my ignorance of mathematics :) Well, I'm reading the book "Huygens and Barrow, Newton and Hooke" by Vladimir Arnold, and one excerpt (at the page 43) called my attention…
João Marques Wolken
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If two real polynomials $f(x)$ and $g(x)$ of degrees $m \geq 2$ and $n \geq 1$ satisfy $f(x^2+1)=f(x) g(x) ~~\forall~~x \in \mathbb R, $ then :

If two real polynomials $f(x)$ and $g(x)$ of degrees $m \geq 2$ and $n \geq 1$ respectively satisfy $$f(x^2+1)=f(x) g(x) ~~\forall~~x \in \mathbb R, $$ then : $(A)~ f$ has exactly one real root $x_0$ such that $f~'(x_0) \ne 0$ $(B)~ f$ has exactly…
MathMan
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Fourier series of $\cos^2x$

This is my first step with Fourier series and I'm stuck at the beginning. So my solution: The function $f(x)=\cos^2x$ is an even function. Thus I use formulas: $a_0 = \frac{2}{\pi} \int _0 ^\pi \cos^2x\,\text dx$ (1) $a_n = \frac{2}{\pi}\int _0^\pi…
flipback
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$\lim_{n \to \infty} \frac{1}{2n}\log\binom{2n}{n}$

Please help me to solve this problem. I can find almost no clue regarding the log part. I tried to break the $\binom{2n}{n}$ part, but in vague...will the breaking help me in anyway?
user232070
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Find $\left \lfloor 1000S \right\rfloor$ where $S=\sum_{k=2}^\infty (-1)^k\log_k e$

$$S = \log_{2}{e} - \log_{3}{e} + \log_{4}{e} - \log_{5}{e} + \log_{6}{e}\cdots $$ Find the Value of $\left \lfloor 1000S \right\rfloor$. My Attempt I changed it into summation and tried to form a Taylor series of $ \ln{(\ln{x} + 1)} $. But got…
rajdeep dhingra
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Finding the equation of the curve that passes through the point $(4,3)$ if its slope is given by $\frac{dy}{dx} = 3x-5$

I tried substituting the $4$ into the $3x-5$ equation, so my slope would be represented as $3(4)-5 = 7$. Then my equation for the line would be $y-3 = 7(x-4)$. That means the equation of the line would be $y = 7x - 25$. However, I'm trying to submit…
etree
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Steps for calculating tangent line of a function

I want to calculate the tangent line of $f(x) = Ax^3 + Bx^2 + Cx + D$. But I am not sure what are the required steps. I remember that I should first get the derivative, which is $3Ax^2 + 2Bx + C$ I believe. Then something like $Y-Y_1 = M(x-x_1)$ or…
Dumbo
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If $\int_a^b f(x)\,dx=0$, prove that $f(c)=0$ for at least one $c$ in $[a,b]$

Assume $f$ is continuous on $[a,b]$, if $\int_a^b f(x)\,dx=0$, prove that $f(c)=0$ for at least one $c$ in $[a,b]$. The problem didn't state anything about the function $f$, is it safe to assume either: $f$ is an odd function and implies that…
shinobi20
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Finding the darkest point between two lamps

I was working on some exercises when I came over a rather curious question. Two lamps have intensities 40 and 5 candle-power and are 6 m apart. If the intensity of illumination I at any point is directly proportional to the power of the source,…
E.O.
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integration of $(1-x)^mx^n$ from $0$ to $1$, $m$ and $n$ positive integers

trying to integrate $\int_0^1 (1-x)^m x^n dx$, $m$ and $n$ positive integers. I know the answer is a fraction containing gamma functions but don't know how to get there
brayton
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Find the derivative of the function $F(x) = \int_{\tan{x}}^{x^2} \frac{1}{\sqrt{2+t^4}}\,dt$.

$$\begin{align} \left(\int_{\tan{x}}^{x^2} \frac{1}{\sqrt{2+t^4}}\,dt\right)' &= \frac{1}{\sqrt{2+t^4}}2x - \frac{1}{\sqrt{2+t^4}}\sec^2{x} \\ &= \frac{2x}{\sqrt{2+t^4}} - \frac{\sec^2{x}}{\sqrt{2+t^4}} \\ &= \frac{2x-\sec^2{x}}{\sqrt{2+t^4}}…
KKendall
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