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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Question about singularities and path integrals

I'm given a vector field that has an obvious singularity at a point $(a,b)$. In order to learn more about the singularity I place a circle around it with the singularity at it's center. The line integral for the field across the circle gives me…
JDD
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How to solve $1.1^x +1.5^x = 3.46$

I'm drawing a total blank as to how to solve this for x ... $$ 1.1^x + 1.5^x = 3.46 $$ I thought I could differentiate and substitute the first derivative back into the equation, but I must be rustier than I thought because I'm it's not working out…
Jack CL
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Motivation for a particular integration substitution

In an old Italian calculus problem book, there is an example presented: $$\int\frac{dx}{x\sqrt{2x-1}}$$ The solution given uses the strange substitution $$x=\frac{1}{1-u}$$ Some preliminary work in trying to determine the motivation as to why one…
NoClue
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Examples where the product of reciprocal derivatives isn't one?

I'm trying to understand why it isn't a good idea to treat derivatives like fractions. Could someone give me an example of a function $y$ such that $$\frac{dy}{dx} \cdot \frac{dx}{dy} \not = 1$$ This inspired my question, so I would appreciate it if…
user179487
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Find the second derivative of some implicit function?

I have a function given implicitly, you know. X and Y on both sides. Then it says, assume y = y(x). That's fine. I should be able to find y'(0), but what about y''(0)? How do you treat the dy/dx parts when taking the second derivative? Edit: I…
Algific
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Oblique asymptotes of $f(x)=\frac{x}{\arctan x}$

Find the oblique asymptotes of the function $\displaystyle f(x)=\frac{x}{\arctan x}$. I tried finding the slope and $y$-intercept of the line going to $+\infty$ first, because the function is even (symmetrical in $x=0$). $\displaystyle…
rae306
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The Shortest Distance Between 2 Points On The Earth

Assuming that the earth is a perfect sphere with radius 6378 kilometers, what is the expected straight line distance through the earth (in km) between 2 points that are chosen uniformly on the surface of the earth?
SNEHIL SINHA
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Rate Of Change Of Shadow

A spotlight on the ground shines on a building $12m$ away. If a man $2m$ tall walks from the spotlight towards the building at a speed of $1.6m/s$, how fast is the length of his shadow on the building decreasing when he is $4m$ from the…
Kermit the Hermit
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Alternate ways to find the limit of a given sequence

I need to find the $\lim_{n\to\infty}\{x_n\}$ where $\{x_n\}$ is defined as $$\{x_n\}_{n\ge1}=n^{\frac{1}{n}}\;,\;n\in \mathbb{N}$$ Now if I had a function $f:\mathbb{R}-\{0\}\to\mathbb{R},\quad f(x)=x^{\frac{1}{x}}$, then I could easily find it's…
goku996
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Prove that if $f(x)$ is continuous in $\mathbb R$ then exists $x$ such that $f(x)f(x+1) \ge 0$

I have a homework question to prove that if $f(x)$ is continuous in $\mathbb R$ then exists $x$ such that $f(x)f(x+1) \ge 0$. I am failing to see why this is true and how I can prove this. Can someone please help me out? Thanks :)
Jason
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Surface area of sphere $x^2 + y^2 + z^2 = a^2$ cut by cylinder $x^2 + y^2 = ay$, $a>0$

The cylinder is given by the equation $x^2 + (y-\frac{a}{2})^2 = (\frac{a}{2})^2$. The region of the cylinder is given by the limits $0 \le \theta \le \pi$, $0 \le r \le a\sin \theta$ in polar coordinates. We need to only calculate the surface from…
Haresh
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Integral equals to the intermediate value

Let $f$ be twice continuously differentiable. Prove that there exists $\xi\in (-1,1)$ such that $$\int_{-1}^1 xf(x)dx=\frac{2}{3}f'(\xi)+\frac{1}{3}\xi f''(\xi).$$ What I have tried is as follows. $$\frac{2}{3}f'(\xi)+\frac{1}{3}\xi…
xldd
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Find $\lim_{n\to\infty}\left(1+\dfrac{1}{n}\right)\left(1+\dfrac{1}{2n}\right)\ldots\left(1+\dfrac{1}{2^{n-1}n}\right)$

Find $\displaystyle\lim_{n\to\infty}\left(1+\dfrac{1}{n}\right)\left(1+\dfrac{1}{2n}\right)\left(1+\dfrac{1}{4n}\right)\ldots\left(1+\dfrac{1}{2^{n-1}n}\right)$. (This is not my homework. One of my friends gave this to me.)
Grobber
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What is $f '(x)$ for $f(x)=(x-3)^3$?

What is $f '(x)$ for $f(x)=(x-3)^3$? I'm thinking it is $3x^2 - 18x + 27$ but my textbook says it is $3x^2 - 18x - 27$
user3184807
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Is $f(x) = \sum_{n=-\infty}^\infty \frac{2^n \log x}{1+(2^n \log x)^2}$ for $x \geq 2$ constant?

I want to find a non-constant, continuous function $f: [2, \infty) \rightarrow \mathbb{R}$ which satisfies $f(x) = f(x^2)$ for all $x \in [2, \infty)$. A friend of mine suggested to try something of the form $$f(x) = \sum_{n=-\infty}^\infty…
Huy
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