For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

learn more… | top users | synonyms

1
vote
0answers
7 views

Helicity is Conserved

In fluid mechanics, the helicity is defined as $$\int_{R^3} u(x,t)\cdot \omega(x,t),$$ where $u(x,t)$ is a smooth solution of the Euler equations $$\partial_tu + (u \cdot \nabla) u = -\nabla p$$ ...
0
votes
1answer
23 views

Finding the volume of a solid bounded by a sphere and a paraboloid

I am working on a problem that requires me to find the volume of the solid bounded by the sphere $x^2 + y^2 + z^2 = 2$ and the paraboloid $x^2 + y^2 = z$. I know that to do this, I must use triple ...
0
votes
1answer
22 views

Supremum vs Integral

Let $h$ be a positive function defined on $(0,\infty)$. Is the following inequality always true ? $$ \sup_{r<t<\infty}h(t)\leq\int_{r}^{\infty}h(t)\frac{dt}{t} $$
4
votes
1answer
41 views

Closed-form of $\int_0^\infty \frac{1}{\left(a+\cosh x\right)^{1/n}} \, dx$ for $a=0,1$

While I was working on this question by @Vladimir Reshetnikov, I've conjectured the following closed-forms. $$ I_0(n)=\int_0^\infty \frac{1}{\left(\cosh x\right)^{1/n}} \, dx \stackrel{?}{=} ...
1
vote
4answers
63 views

calculate-binomio-newton

i am help Calculate: $$(C^{16}_0)-(C^{16}_2)+(C^{16}_4)-(C^{16}_6)+(C^{16}_8)-(C^{16}_{10})+(C^{16}_{12})-(C^{16}_{14})+(C^{16}_{16})$$ PD : use $(1+x)^{16}$ and binomio newton
1
vote
0answers
15 views

Calculus book for computer science students

I'm going to teach calculus I and II to undergraduate computer science students and I would like to know if someone here knows some book or site with easy calculus applications in computer science. ...
0
votes
1answer
20 views

Finding a tangent line with implicit differentiatio

Find the tangent line on point $P$ for this curve $(x + 2)^2 + (y - 3)^2 = 37$ on $P(4,4)$ I tried implicit differentiating $2(x + 2) + 2(y - 3)y' = 0$ I'm not sure if solving for $y'$ is the ...
2
votes
0answers
16 views

Does $\mathfrak T^r(\Bbb R^m)$ count as an vector space?

Here $\mathfrak T^r (\Bbb R^m)$ denotes all the $r$-th tensors (multi-linear functions) acting upon the elements $(u_1,\cdots,u_r)$ from the product space $\displaystyle \prod^r \Bbb R^m$. And the ...
17
votes
3answers
112 views

Evaluate $\displaystyle\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^\frac{1}{n} \ \mathrm{d}x$

Evaluate $$\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^\frac{1}{n} \ \mathrm{d}x$$ I simplified the limit to $$\dfrac{1}{2}\lim_{n \to \infty} \int_{0}^{\frac{1}{2}} ...
-1
votes
0answers
31 views

Quick Integration by parts question [on hold]

If while doing integration by parts I get a sum of +infinity and -infinity, can I obtain that the Integral diverges?
3
votes
4answers
75 views

Simple differential equation( introduction but need some basic explanation)

I have a couple of questions before I dig deeper into my calculus book. First: I have learned that $\frac{d}{dx}\frac{x}{y}$=$\frac{y x'-x y'}{y^2}$ never really gotten a proper explanation for ...
5
votes
2answers
72 views

The value of the integral $\int_0^2\left(\sqrt{1+x^3}+\sqrt[3]{x^2+2x}\:\right)dx$

The value of definite integral $$\int\limits_{0}^{2}\left(\sqrt{1+x^3}+\sqrt[3]{x^2+2x}\:\right)dx$$ is $$(A)\,4 \quad(B)\,5 \quad (C)\,6 \quad(D)\,7$$ My attempt: I tried using ...
0
votes
3answers
44 views

Application of Fubini's Theorem to a simple function

I'm trying to solve the integral: $$\int_0^2\int_0^{x/2}xy^2dydx$$ Using both sides of Fubini's Theorem - that is, doing $dydx$ and then obtaining the right intervals of integration and calculating ...
0
votes
1answer
24 views

Compute the definite integral of f(x) as a limit of Riemann sums.

$[-1, 0] = [a,b]$ and $f(x) = 4x-1$. When I attempt to solve this problem, the limit I'm taking keeps blowing up to infinity. How should this problem be set up?
1
vote
1answer
28 views

Number of points of discontinuity

Find the number of points where $$f(\theta)=\int_{-1}^{1}\frac{\sin\theta dx}{1-2x\cos\theta +x^2}$$ is discontinuous where $\theta \in [0,2\pi]$ I am not able to find $f(\theta)$ in terms of ...
-1
votes
1answer
21 views

Let $ \alpha \neq 0 $ isn't $ n \times n $ identity matrix and $ P $ is $ n \times m $ matrix. Let $ P = \alpha P $ . When $ P \neq 0 $? [on hold]

Let $ \alpha \neq 0 $ isn't $ n \times n $ identity matrix and $ P $ is $ n \times m $ matrix. Let $ P = \alpha P $ . When $ P \neq 0 $ ?
-6
votes
1answer
61 views

Solve math then decoding, I'll admit, I'm stumped. [on hold]

Raise 10666369614354225120 to 16479139101483902597. Take the result and divide by 18074899840695706733. Take the rest of that division. Encode it to hex F__-_C__-D__3-58__
3
votes
4answers
71 views

How to prove that the line perpendicular to the radius is the tangent in the calculus sense?

Let $P=(p_1,p_2)$ be a point on an semicircle and $r$ be the line perpendicular to the radius $\overline{OP}$, like the picture below. Euclid showed (Book III, Proposition 16) that $r$ does not ...
1
vote
0answers
41 views

“Maximum point lies on a curve” implies tangential derivative is zero there.

Given a differentiable function $f:\mathbb{R}^2\to\mathbb{R}$, suppose that it has a local maximum at the point $(x_0,y_0)$. Let $\gamma$ be a smooth curve passing through $(x_0,y_0)$. Does it follow ...
2
votes
1answer
128 views

How to calculate the integral?

How to calculate the following integral? $$\int_0^1\frac{\ln x}{x^2-x-1}\mathrm{d}x=\frac{\pi^2}{5\sqrt{5}}$$
0
votes
1answer
125 views

An Infinite series I

By decompising fractions one can show that \begin{align} \sum_{n=1}^{\infty} \frac{1}{n \, (n+1)^{2} \, (n+3)} = \frac{65}{72} - \frac{\zeta(2)}{2}. \end{align} The fraction can also be seen in the ...
-5
votes
1answer
56 views

what is the limit of $(-2)^{1/(2n+1)}$ as $n\rightarrow\infty$? [on hold]

what is the limit of $(-2)^{1/(2n+1)}$ as $n\in\mathbb{Z}, n\rightarrow\infty$? and what is the limit of $(-2)^{2/(2n+1)}$ as $n\in\mathbb{Z}, n\rightarrow\infty$?
0
votes
3answers
33 views

Modelling interest with differential equations (Interpretation)

I am having trouble interpreting the meaning of this differential equation model for interest on an account. The problem is as follows: Assume you have a bank account that grows at an annual ...
3
votes
1answer
75 views

Where did I make a mistake?

This is an excerpt from a dynamical systems paper: They provide a proof of this Lemma, and numerical simulations also show it should be true. It's clear the equilibrium point on each axis is ...
1
vote
2answers
31 views

finding the volume of the solid via disk or washer method

the question is: $y = 1/4x^2$, $x = 2$, $y = 0$; about the $y$-axis I tried to draw it out, but I can't figure this stuff out. The graphing is the hardest part for me because I don't know what to do ...
-2
votes
1answer
22 views

Does the gradient function at a point give the direction of greatest increase and also perpendicular at the same time?

So say if I have a cone and took the gradient and then evaluated it at a point would this vector that points in the direction of greatest increase also be perpendicular and is this true for all ...
10
votes
1answer
270 views

About the integral $\int_{-1}^1 \frac{1}{\pi^2+(2 \operatorname{arctanh}(x))^2} \, dx=\frac{1}{6} $

Here is a question that naturally arose in the study of some specific integrals. I'm curious if for such integrals are known nice real analysis tools for calculating them (including here all possible ...
-1
votes
0answers
15 views

Show Uniform convergence of a Serise

Im trying to show that the serise $ \sum_{1}^{infinite}\frac{x^2}{(1+x^{2})^{n}} $ is not Uniform converge in the domain $(-\infty$ to $ \infty )$ . I managed to show that the sum of the serise when $ ...
0
votes
0answers
22 views

General formula for sinusoidal taylor series centered at any a?

I understand that to find a taylor series centred at a particular a value you need to find a formula for the nth derivative, but this is tricky for cos(x) and sin(x). Is it possible to have a formula ...
2
votes
1answer
46 views

Evaluating the indefinite integral $\int\sqrt{\cos2x}\sin^32x\,dx$

I have tried to integrate the following indefinite integral but I'm not sure if I get the right answer. Please tell me if I'm wrong and if so, please indicate what went wrong. $$ ...
1
vote
1answer
38 views

Quadric equation-physics

$$\frac{[(\omega_0^2-\omega^2)-2i\omega\gamma]^2}{[(\omega_0^2-\omega^2)^2+4\gamma^2\omega^2]^2}=\frac{1}{[(\omega_0^2-\omega^2)^2+4\gamma^2\omega^2]}$$ I don't understand how can I get to that ...
2
votes
1answer
37 views

How to prove that $f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt},$ for all invertible functions.

A while ago, I found that: $$f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt}.$$ I managed to prove it for a few functions, and I believe that it may be the case for all ...
1
vote
0answers
22 views

Quadric equation for a physics problem

[($\omega_0^2$-$\omega^2$)-2i $\omega$ $\gamma$]$^2$=1. I don't understand how can I get to that solution '1'. Any hint will be very thankful.
1
vote
3answers
84 views

limit of an integral question

Let $f : [0, \infty) \to \Bbb R$ be bounded and continuous. Prove that $\lim \limits _{h \to \infty} h \int \limits _0 ^\infty e ^{-hx} f(x) \, d x = f(0)$. Our intuition was to use l'Hospital's rule ...
1
vote
2answers
48 views

If the area bounded by $y=x^2+2x-3$ and the line $y=kx+1$ is the least, find $k$ and the least area. [on hold]

What concept to use in the Application of Integral question? Please help me
1
vote
1answer
55 views

A question about differential function

If $f(x)=f'(x^{2})+2x$, then $f(1)=?$ and $f''(1)=?$ Sorry. I am going to check the original problem, and then i will update.
1
vote
1answer
27 views

Help with a problem regarding sequence divergence.

There are two forms of definition of sequence divergence. By negation of the sequence convergence we have A sequence $x_k$ diverges iff $∀x∈\Bbb{R}∃ϵ>0∀N∈\Bbb{N}∃k>N$ st. $|x_k-x|>ϵ$. ...
-2
votes
0answers
53 views

Integral calculation question [duplicate]

Calculate the following integral: $\int \limits _0 ^\frac \pi 2 \ln (\sin x) \Bbb d x$. We used the substitution $x=2t$ and then used the identity $\sin 2t = 2 \sin t \cos t$ but now we're stuck. ...
0
votes
1answer
36 views

What does it mean to say $f(x) \sim g(x)$, i.e. $f(x)$ behaves like $g(x)$ when $x \to \infty$?

If $\lim_{x\to\infty}\frac{f(x)}{g(x)}=\infty$, then $f$ grows faster than $g$. Same if $\lim_{x\to\infty} \frac{g(x)}{f(x)} = 0$. Would $f$ behave like $g$ if $\lim_{x\to\infty}\frac{f(x)}{g(x)} = ...
0
votes
1answer
47 views

Function with increasing property.

Prove that $\frac{1}{2}(x+2)^{-3/2}-(\frac{1}{2}x+3)(x+3)^{-3/2}$ is increasing function for $x\ge4$. I tried it by taking its first derivative but by first derivative for me its difficult to say it ...
24
votes
2answers
2k views

Why does $ \int_0^1 \lceil { x\sin({1 \over x})} \rceil = 1 - \frac{\log(4)}{2\pi} $?

One time I was bored and played around a bit with integrals and wolfram alpha and tested the following integral: http://www.wolframalpha.com/input/?i=integral_0%5E1+ceil%28x*sin%281%2Fx%29%29 Note: ...
3
votes
3answers
79 views

Evaluate $\iint_{R}(x^2+y^2)dxdy$

$$\iint_{R}(x^2+y^2)dxdy$$ $$0\leq r\leq 2 \,\, ,\frac{\pi}{4}\leq \theta\leq\frac{3\pi}{4}$$ My attempt : Jacobian=r $$=\iint_{R}(x^2+y^2)dxdy$$ $$x:=r\cos \theta \,\,\,,y:=r\cos \theta$$ ...
1
vote
2answers
26 views

Improper integral convergence question

Prove that the following integral converges: We divided the integral to 2 integrals (one from 0 to 1/2 and the other from 1/2 to 1). We managed to prove that the integral from 1/2 to 1 converges ...
1
vote
2answers
31 views

Connection between Fréchet derivative and the directional derivative in finite euclidean space

In the lecture notes I am reading, the following statement is made: Let $U$ be an open subset of $R^n$, and define the function $e:U \to R$. $e$ is said to be differentiable if for every $u \in U$ ...
-1
votes
2answers
77 views

Improper rational/trig integral comes out to $\pi/e$ [on hold]

During my studying to integration I find this integration. So I tried to prove but I got stuk. So I need help in this integration. $$\displaystyle\int_{-\infty}^{\infty} \frac{x \sin (x)}{1+x^2} ...
2
votes
4answers
152 views

How should I go about solving this definite integral?

The integral is: $$\int_{-1}^1\sqrt{4-x^2}dx$$ I'm having difficulty figuring out how to go about this. I attempted to use u-substitution, both by substituting $u$ for $\sqrt{4-x^2}$ entirely, and ...
2
votes
2answers
22 views

Convegence of $\sum_{i\in J}a_i$ implies that index set is countable

Let $J$ be a uncountable set and $\{a_i\}_{i\in J}$ be a set of non-negative real numbers. Prove that $\sum_{i\in J}a_i<\infty$ implies that there is a countable set $H\subset J$ such that $a_i=0$ ...
0
votes
1answer
39 views

How do I convert a sum to an algebraic expression?

Something something Riemann sum to integral is the most that I remember. I just don't remember how we did it or whether or not that would be the best method for doing it. Let $ \theta(n) = ...
4
votes
3answers
79 views

Challenging $\lim_{x \rightarrow 10} \frac{1}{\lfloor x \rfloor} = \frac{1}{10}$ for $\epsilon=\frac{1}{2}$.

Consider the (incorrect) claim that $$\lim_{x \rightarrow 10} \frac{1}{\lfloor x \rfloor} = \frac{1}{10}.$$ How might I find the largest $\delta$ such that I can challenge $\epsilon = 1/2$? Clearly ...
-1
votes
2answers
84 views

How to integrate this formula 3 [on hold]

$$\int_0^1 \left( \frac{2e^{2x}}{x}+\frac{1}{xe^x}-\frac{e^{2x}}{x^2}+\frac{1}{x^2e^x} \right) \, dx$$ I tried many times but still could not get it. Any help?