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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How many faces does the $n$-dimensional cube $I^n$ have?

I am taking an online Coursera Calculus course, and this question popped up as one of the challenge problems, reproduced below. I had a difficult time understanding the answer, which was $I^n$ has $2n$ faces. This problem concerns the boundary…
sir_thursday
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proof of a function times the nth derivative of a function formula

Prove by induction or otherwise that when u and v are functions of x $$vD^nu=D^n(uv)-n{D}^{n-1}(uDv)+\frac{n(n-1)D^{n-2}(uD^2v)}{2}-...$$an question from differential calculus by Ferrar so far I have checked so base case n=1 and…
Cow
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How To Find the Integral of e When the Exponent Has Multiple Variables

My calculus class just started going over "Separation of Variables" where you have something like $\frac{dy}{dx} = x$ and have to solve for $y$. I understand that to do this, one must get dx on one side and dy on the other, and then take the…
Jack
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What point on $y= -x^2$ is closest to what point on $y = 5 -2x$?

I'm studying calculus on my own with the MIT OCW book. Question 61 on page 113 asks: "What point on $y= -x^2$ is closest to what point on $y = 5 -2x$?" I have been looking all over the internet for an example but can only find examples of this when…
maybedave
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Why would $2xdx + 2ydy=0$ and $2x + 2y(\frac{dy}{dx})= 0$ be the same thing?

The reason I asked this question is that I am trying to differentiate the relation $x^2 + y^2 = 25$. In an attempt to understand what's going on I gave the expression $x^2 + y^2x$ a name. "S" is essentially a function of two variables; it takes…
YUKITERU_AMANO
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Find the limit $\displaystyle\lim_{n\rightarrow\infty}{(1+1/n)^{n^2}e^{-n}}$?

Find the limit $\displaystyle\lim_{n\rightarrow\infty}{(1+1/n)^{n^2}e^{-n}}$? I found the limit as $e^{-1/2}$ using l'Hospital rule. I guess I made a mistake. Because the limit seems to be 1. Also, can we find the limit without L'Hospital rule?
mtm
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Second partial derivatives vs total second derivative

Suppose that second partial derivatives exist at $(a,b)$ and $\dfrac{\partial^2f}{\partial x\,\partial y}(a,b)\neq\dfrac{\partial^2f}{\partial y\,\partial x}(a,b)$. (For example $f(x,y)=\begin{cases}xy\frac{x^2-y^2}{x^2+y^2}\quad\textrm{for }…
Barbara
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Curiosity about the area between the arclength and the connected line

For example, $f(x)=x^2$ I'm curious if it's possible to find the area, for example, between the function $f(x)$ and the connected line between $f(1)$ and $f(2)$?
user82602
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Can the product of two functions with no inflection points have an inflection point?

If $f(x)$ and $g(x)$ are functions with no inflection points, can $h(x)=f(x)\cdot g(x)$ have an inflection point? Edit: I experimented a bit with a few functions (like $x^2\cdot x^2$) in a graphing calculator but I couldn't find a good example. I…
Milo
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Integral substitutions.

For integrals of the form: $$\intop_a^bg(t)dt,$$ we can apply the $tanh$-substitution to transform the integral into a doubly infinite integral, i.e: $$\intop_a^bg(t)dt = \frac{b-a}{2}\intop_{-\infty}^\infty g \left( \frac{b+a}{2}…
David Simmons
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How can I integrate $\int\frac{\ln(1+xy)}{1+x^2} dx$?

$$\int\frac{\ln(1+xy)}{1+x^2} dx$$ Give me some idea how to solve to solve this. Is it possible to use some complex analysis to solve this. Or some nice substitution will work?
user80658
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Convergence test for series

Let $\space \displaystyle\sum_{n=1}^{\infty} \space \left | \frac{\cos(n\pi)}{n+1}\right |$. Does this series converge or not? The serie is valid for the natural numbers, so it can be writen as $\space \displaystyle\sum_{n=1}^{\infty} \space …
user24047
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Differentiability Using Little Oh Notation

I'm trying to get a deeper understanding of the derivative of a function. I have been reading from the following page: I have been thinking about why this is an equivalent statement to the original, but I've been having trouble with it. What I've…
cuppajoeman
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Derivate as a division of differentials

It's always said that $\frac{df(x)}{dx}$ is the derivate of a function with respect to x and it musn't be understood as a division of $df$ and $dx$. I've read that it is true that $\frac{df(x)}{dx}$ is the quotient of $df$ and $dx$. But this cannot…
jinawee
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Prove inequality with powers

$4^{79}<2^{100}+3^{100}<4^{80}$ The only thing I can think of is the following: $\ln(3) = 1.098612288$ and $\ln(4) = 1.38629436112$ so $\frac {\ln(3)}{\ln(4)} = 0.79248125036$ so $\frac {79}{100} < \frac {\ln(3)}{\ln(4)} < \frac {80}{100} $ I don't…
Sal.Cognato
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