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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Old MIT exam question, how do I solve it?

I've been working through old MIT practice exam papers, and I found a question that stumped me. It goes: What value for the constant $c$ will make the function $e^{-x}\sqrt{1+cx}$ approximately constant, for values of $x$ near $0$? (Show your…
Korgan Rivera
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Integral of $\int_0^{2\pi}\cos^n(x)\,dx$.

Consider $$\int_0^{2\pi}\cos^n(x)\,dx,\qquad n\text{ a positive integer}$$ For $n$ odd, the answer is zero. Is there a slick way to find a closed form for $n$ even?
Peter
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Using differentiation to solve equations

Lets say that I have an equation that can't really be solved via elementary means, for e.g: $$ e^x = 4x$$ Logically, what is wrong with me using equating derivatives (or integrals for that matter)? For e.g: $$ \dfrac{d}{dx} (e^x) = \dfrac{d}{dx}…
John Tan
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For which values of $a$, $b$ and $c$, if $a + b = c$, then $\frac{1}{a} + \frac{1}{b} = \frac{1}{c}$?

I have a problem in my homework, which I have tried to solve, but I have just ideas, no real mathematical solutions. The problem is the following: Suppose we have three real numbers $a$, $b$, and $c$ which satisfy the equation: $$a + b = c$$ Is…
user168764
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Integral with Tanh: $\int_{0}^{b} \tanh(x)/x \mathrm{d} x$ . What would be the solution when 'b' does not tends to infinity though a large one?

two integrals that got my attention because I really don't know how to solve them. They are a solution to the CDW equation below critical temperature of a 1D strongly correlated electron-phonon system. The second one is used in the theory of…
zoran
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Why doesn't it work when I calculate the second order derivative?

Let $y=y(x)$ be determined by the equation \begin{align*}\begin{cases} x=t-\sin{t}\\ y=1-\cos{t}.\end{cases} \end{align*} I understand the solution: $$\frac{d^2y}{dx^2}=\frac{d(\frac{dy}{dx})}{dt}\frac{1}{\frac{dx}{dt}}=-\frac{1}{(1-\cos{t})^2}$$…
longtemps
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Why does the higher order derivative test work?

I'm an AP Calculus BC student, so all I know about derivatives is the increasing/decreasing/ relative max/min function relation with first derivative (first derivative test), concavity (second derivative test). I showed that you can create a third…
Mario James
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Solve the integral $\int_0^\infty x/(x^3+1) dx$

I'm new here! The problem: integrate from zero to infinity x over the quantity x cubed plus one dx. I checked on wolfram alpha and the answer is that the indefinite integral is this: $$\int \frac{x}{1+x^3} dx = \frac{1}{6}\left(\log(x^2-x+1)-2…
Chad
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Find the limit of $ \lim_{n\to\infty}\frac{n}{\ln{n}}\left(\frac{1}{p+1}-na_{n}^{p+1}\right) $

Problem:Let postive real sequence$\{a_{n}\}$ satisfy $\displaystyle\lim_{n\to\infty}a_{n}\left(\sum_{i=1}^{n}a_{i}^{p}\right)=1$,where $p>-1$,Find the limit. $$ \lim_{n\to\infty}\frac{n}{\ln{n}}\left(\frac{1}{p+1}-na_{n}^{p+1}\right) $$ Here is my…
pxchg1200
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The Geodesics of a Sphere.

I need to find the geodesics of a sphere. Then in polar coordinates $$x=a \sin\theta \cos\phi \\y=a \sin\theta \sin\phi\\ z=a\cos\theta$$ Then $ds^2=dx^2+dy^2+dz^2$. Can someone please tell me how is $ds^2=a^2(d\theta^2+\sin^2\theta \, d\phi…
sam_rox
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Is it possible to be both a relative max/min and an inflection point?

Can anyone find a function $f$ with an $x$ such that $f'(x)=0$, $f(x)$ is either a relative max/min and $(x,f(x))$ is an inflection point? In other words, suppose you're using the second derivative test, see that $f''(x)=0$ and then note that it's…
hoyland
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Two functions discontinuous, but sum continuous

I have some exercise that asks me this: Find $f$ and $g$ discontinuos such that $f+g$ is continuous. This is what I tought: $$f = \mbox{sign}(x)$$ $$g = -\mbox{sign}(x)$$ Where $\mbox{sign}(x)$ is the function that maps to $1$ if $x\ge0$ and $-1$ if…
Marter Js
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Example where $\lim_{x\to 0}f(x^2)$ exists but $\lim_{x\to 0}f(x)$ does not.

Can somebody give me an example where $$\lim_{x\to 0}f(x^2)$$ exists but $$\lim_{x\to 0}f(x)$$ does not?
aaaa
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Convergence of composition of functions sequences

Let $X$ be a metric space, $f_n: X \to X$, $g_n: X \to X$, $f_n(x) \to f(x)$, $g_n(x) \to g(x)$ ($n \to \infty$). Is $f_n(g_n(x)) \to f(g(x))$ ? Here: 1) pointwise convergence; 2) uniform convergence. So, there are 2 cases in my question. In the…
Dina
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Derivative of product of three functions: product rule

I am trying to find the derivative of $f(x)= xe^x \csc x$, and I am not too sure how to even start. Is it two terms or three? $xe^x$ and $\csc x$ or is it $x$, $e^x$ and $\csc x$? I can't get a proper answer either way. With two terms I get…
user138246
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