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

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0
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1answer
33 views

Finding an integral.

Evaluate $$\!\int x^5\sqrt{x} + x\sqrt[4]{x}\ \mathrm{d}x$$ My attempt: I tried to factor out a $\sqrt{x}$ and I got $$\sqrt{x}\int\! x^5+x\sqrt[3]{x} \ \mathrm{d}x$$ But here I cannot factor a ...
-3
votes
3answers
28 views

write an expression [on hold]

A word processor determines the width of the body of text on a page. The page is 11 inches wide and has two equal size margins of x inches on each side of the text. Write a formula that gives the ...
1
vote
1answer
24 views

Prove the convergence of sequences

Let $x_{n} = 0 $ if $n < 100 $ and $x_{n} = 1$ if $n \geq 100$ prove $x_n$ converges and find its limit. I started by letting $\epsilon > 0$, as per normal, and choosing $n \geq 100$ as well as ...
0
votes
0answers
13 views

Interpreting a 3 dimensional Graph?

The points P(a, b, c) and Q(2, 3, 5) are symmetric in the sense given. Find a, b, c. About the xy-plane. How does one find the answer to this without graphing. I just do not understand how one would ...
7
votes
2answers
105 views

A reason for the value of $\int_{0}^{1}\log{(x)}\log{(1-x)}\,\mathrm{d}x$

In this .pdf document, which is just a list of Putnam-style undergraduate-level problems from various sources, the third question is as I have stated it below (up to a change of notation). ...
3
votes
2answers
56 views

Problem with two variable limit where $\lim\limits_{(x,y) \to (-1, 8)} xy = -8$ using only definition

So we just started with two-variable limits. The definition is quite straight forward though my head is still giving it a few spins. I thought doing a couple of examples would help me. Come the second ...
0
votes
1answer
13 views

Using Distance Formula to Graph and Find Equation

My Question : Write an equation for the plane which passes through the point P(3, 1,−2) and satisfies the given condition. Parallel to the xy-plane. How does one use the equation : $$(x − a)^2 + (y − ...
2
votes
0answers
38 views

How to evaluate the integral $\int\frac{1-e^{-2y} -\frac{2}{k}\ln{(1+ky)}}{(1+ky)e^{-2y}-1}dy$ [on hold]

Please help me in doing this integration. $\int\frac{1-e^{-2y} -\frac{2}{k}\ln{(1+ky)}}{(1+ky)e^{-2y}-1}dy$
3
votes
0answers
36 views

Subdifferential of integral

I am currently trying to extend my knowledge about subdifferentials. Now I am stuck at a particular property of the subdifferential. In this "paper" ...
0
votes
2answers
38 views

Physical interpretation for the curl of a field

I was supposed to compute the curl of a field for a fairly simple assignment and got the following : $$\nabla \times F = (0,0,y-e^{x+y}) \text{ ; } (x,y) \in [0,1]\times [0,1]$$ However, I'm unable ...
1
vote
2answers
56 views

Wolfram Alpha “x = derivative x”

Asking Wolfram Alpha $x = \text{derivative } x$, I was expecting $e^x$, being that the derivative of $e^x$ is $e^x$, Wolfram Alpha however yields $x = 1$. Is this stating that the derivative of a ...
1
vote
1answer
33 views

limit of sum $\dfrac{(-1)^{n-1}}{2^{2n-1}}$

What is: $$\sum^{\infty}_{n=1}\dfrac{(-1)^{n-1}}{2^{2n-1}}$$ I have done a Leibniz convergence test and proved that this series converges, but I do not know how to find the limit. Any suggestions?
1
vote
1answer
31 views

Analytically Understanding The Definite Integral As A Limit Of Sums

With naive intuition one can obviously see that the definite integral as infinite subdivisions of an area under a curve, within the finite interval "a to b", from which the function of integration ...
0
votes
2answers
29 views

Find all points such that function has all partial derivatives in that point.

Find all points $(x,y) \in \mathbb{R}^2$ such that function has all partial derivatives in that point.$$ f(x,y) = \begin{cases} \frac{\sin(xy^2)}{y} &\mbox{if } y>0 \\ xy^2 & \mbox{if } y ...
1
vote
2answers
459 views

Very basic calculus question - do you think there's a typo?

I'm helping someone with their homework before they go back to A-Levels, and I came across the following question which I think is miswritten: Find the gradient at $x=1$ of the equation ...
3
votes
2answers
60 views

Are all operations functions?

I have looked at Wikipedia(I know it's not completely reliable) but on it an operation is formally defined as: "A function ω is a function of the form $ω : V → Y$, where $V ⊂ X_1 × … × X_k$." and I ...
3
votes
3answers
87 views

How to find $\lim_{n\to\infty}\left(\frac{\pi^2}{6}-\sum_{k=1}^n\frac{1}{k^2}\right)n$?

How to find $\lim_{n\to\infty}\left(\frac{\pi^2}{6}-\sum_{k=1}^n\frac{1}{k^2}\right)n$? It is well-known that $\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{k^2}=\frac{\pi^2}{6}$, so ...
0
votes
0answers
11 views

Solutions for the dependency problem

Currently I read about the dependency problem of interval arithmetic. Mainly it's the problem that in the equation $X-X$ for $X$ being an interval the following is calculated: $$X-X=\{x-y:x\in X, y\in ...
3
votes
4answers
75 views

Show $\frac{\sin(x)}{x}>\cos(x)$ for $0<x<\pi$ using the Mean Value Theorem

I'm trying to show the inequality $$\frac{\sin(x)}{x}>\cos(x)$$ by for $0<x<\pi$ using the Mean Value Theorem, but I don't know how to start. I can show that $\sin(x)<x$, but I can't see ...
1
vote
2answers
40 views

Calculate double integral $\iint_A \sin (x+y) dxdy$

Calculate double integral $$\iint_A \sin (x+y) dxdy$$ where: $$A=\{ \left(x,y \right)\in \mathbb{R}^2: 0 \le x \le \pi, 0 \le y \le \pi\}$$ How to calculate that? $x+y$ in sin is confusing as i do not ...
3
votes
2answers
36 views

Double integral $\int\int_A y dx dy$

Calculate Double integral $$\iint_A y dxdy$$ where: $$A=\{(x,y)\in\mathbb{R}^2 : x^2+y^2\le4, y \ge 0 \}$$ I do not know what would be the limit of integration if i change this to polar coordinates. ...
0
votes
1answer
73 views

Why is it incorrect to integrate by $d(2x)$?

I tried to prove the volume of a cone. If you let the radius be $r$ and let the height be equal to the radius, then all you need to do is integrate the area of a circle with radius $r$ by $dr$. ...
0
votes
0answers
24 views

Bungy Jump Model

Let's say there is a rope that has been designed so that it's modulus of elasticity is known. I have been given the information that the rope is stretched to twice it's natural length when there is a ...
1
vote
1answer
97 views

Solve this integral:$\int_0^\infty\dfrac{\arctan x}{x(x^2+1)}\mathrm dx$

I occasionally found that $\displaystyle\int_0^{\frac{\pi}{2}}\dfrac{x}{\tan x}=\dfrac{\pi}{2}\ln 2$. I tried that $$\int_0^{\frac{\pi}{2}}\dfrac{x}{\tan x}=\int_0^{\frac{\pi}{2}}x \ \mathrm ...
-1
votes
2answers
26 views

A Crucial Observation On Li's Criterion for the Riemann Hypothesis?

In 1997, Xian Jin Li formulated an interesting criterion whose validity is completely equivalent to the Riemann Hypothesis, namely: Define the real number $(n-1)!\lambda_n$ to be the $n-th$ ...
0
votes
1answer
16 views

Limit of a Monotonic Increasing and Non-Bounded Function

I have made a solution for the following question and I'm wondering if it's correct. I think that something is missing here. Can you help me complete the solution? Let $f$ be a function. The ...
0
votes
0answers
27 views

How does one integrate a function where the numerator is a polynomial of a degree n, and the denominator is a polynomial under root of degree m<n?

How does one integrate a function where the numerator is a polynomial of degree $n$, and the denominator is a polynomial under root of degree $m$ $(m<n)$? A random example being ...
1
vote
0answers
23 views

ramanujan type sum about functional equation

Could you prove the following series numericaly i could not verify the computer take a lot of time $$\sum _{k=1}^{\infty } -\frac{16 x^2 \left(\pi \coth \left(\frac{\pi ^2 (2 k-1)}{2 x}\right)-\pi ...
0
votes
1answer
21 views

Problem: conservative and not conservative $F=\left( y+\frac{y}{x^2+y^2}, x-\frac{x}{x^2+y^2}\right)$

I don't know how I can solve this problem: Consider $$F=\left( y+\frac{y}{x^2+y^2}, x-\frac{x}{x^2+y^2}\right).$$ Proving that $F$ is not conservative in $\mathbb{R}^2-(0,0)$ but is conservative ...
0
votes
3answers
63 views

How to prove that this function is integrable on $[0,1]$

Here I tried to find two step functions, one of them is less than $f$ on $[0,1]$ whereas one of them is greater than $f$ on the same closed interval, to prove this function is Riemann-integrable on ...
0
votes
1answer
37 views

Number of solutions of the differential equation ${dy}\over {dx}$=$y^{1/3}$ $y(0)=0$

The given differential equation is ${dy}\over {dx}$=$y^{1/3}$, $y(0)=0$ I got the solution $$y^{2/3}={{2}\over {3}}x$$ $$i.e. y^{2}={{8}\over {27}} x^{3}$$ $$i.e. y= \pm \sqrt{{{8}\over ...
2
votes
1answer
65 views

Is there a way of solving integrals where the numerator is an integral of the denominator?

Is there a way of solving integrals where the numerator is an integral of the denominator? I was evaluating the integral $$\int \frac{x-\sin x}{1-\cos x}\mathrm{d}x$$. I separated the numerator into ...
-2
votes
0answers
9 views

sturm liouville Problem finding function eigenvalues given [on hold]

find a problem whose eigenvalues are 1, cosx, cos2x effort done: calculated ao, a1, a2.... using Fourier series formula
-3
votes
0answers
27 views

Verification of an indefinite integral with trigonometric functions [on hold]

I was making this integral $\int \frac{dx}{\sin(x) + \cos(2x)}$ and i end up with this result: $\frac {2}{\sqrt3}\ln({\frac{\tan(x/2) + 2 -\sqrt3}{\tan(x/2) + 2 +\sqrt3}})\ - \frac ...
0
votes
3answers
26 views

Solving ODE y'(x)=2 x y(x), using power expansion

I have this equation: $$y'(x)=2 x y(x)$$, I want to solve this ODE with differential equation with power expansion. I get a problem cause I do not get how to equate the coefficients. $$2 x ...
2
votes
1answer
28 views

Circle Packing, Estimate only of number of smaller circles in a circle.

Given x number of circles of radius r what is a good approximate size Radius for a bigger circle which they fit in. To explain in actual problem terms. I want to move units in a video games which ...
0
votes
2answers
39 views

Solving for a Limit Given a Limit

$$ \text{Given}\; \lim_{x \to 1} \frac{f(x)-4}{x-1} = 10, \;\text{evaluate}\; \lim_{x \to 1} f(x) $$ I'm wondering if anyone can give me some tips on how to approach this problem. I ...
11
votes
3answers
146 views

Show that $(1+\frac{1}{n})^n+\frac{1}{n}$ is eventually increasing

I would like to find a way to show that the sequence $a_n=\big(1+\frac{1}{n}\big)^n+\frac{1}{n}$ is eventually increasing. $\hspace{.3 in}$(Numerical evidence suggests that $a_n<a_{n+1}$ for ...
1
vote
2answers
35 views

Find Taylor series for $f(x)=e^x$ at $c=3$. Then simplify the series and show how it could have been obtained directly from the series $f$ at $c=0$.

Find the Taylor series for $f(x)=e^x$ about the point $c=3$. Then simplify the series and show how it could have been obtained directly from the series for $f$ about $c=0$. Taylor's Theorem: ...
1
vote
0answers
30 views

does the equivalence class of an element in a set is the set itself?

what does equivalence class mean? I am trying to understand I think that I am a little confused so let's take this example: Suppose that we have the relation $2x+3y$ is a number less than or equal ...
0
votes
1answer
18 views

Let $l(x)$ be the linear approximation of $f(x) = x^{2/5}$ at $a = 32$. Approximation?

I'm still a bit confused on how to figure out linear approximations. What are the basic steps to solving a problem like this? Thanks so much! Let $l(x)$ be the linear approximation of $f(x) = ...
0
votes
0answers
25 views

Name for kind of big O notation with leading coefficient

Context: As known the big O notation $O(f(n))$ describes a function $g(n)$ such that there is a constant $C \ge 0$ with $\limsup_{n\to\infty} \left|\frac{g(n)}{f(n)}\right| \le C$ (I assume that ...
2
votes
1answer
49 views

How many terms required in $e =\sum^∞_{k=0}{1\over k!}$ to give $e$ with an error of at most ${6\over 10}$ unit in the $20$th decimal place?

How many terms are required in the series $e =\sum^∞_{k=0}{1\over k!}$ to give $e$ with an error of at most ${6\over 10}$ unit in the $20$th decimal place? Here is what I have: $$e\approx ...
-4
votes
2answers
104 views

proof of elementary inequality $\frac{1}{\sqrt[n+1]{3}-1}-\frac{1}{\sqrt[n]{3}-1}< 1 $

I would like to prove $$\frac{1}{\sqrt[n+1]{3}-1}-\frac{1}{\sqrt[n]{3}-1}< 1 $$ for all $n$, but can't find a way. Thank you for any help Enjoy!
1
vote
1answer
40 views

Why is t used instead of delta t?

Consider a tank that holds $V$ liters of water. Let $x_0$ kg of salt be dissolved in the water at time $t_0$. Suppose that $V_o$ amount of the mixture is leaving the tank in every time interval, ...
2
votes
2answers
37 views

Calculus - Derivative help [on hold]

I'm sure this problem is much simpler then I think, but how do I derivative this function: Thank you, Yaniv
2
votes
2answers
72 views

Result of $\int \limits_{-\infty}^{+\infty}x^2\times\exp\left(\dfrac{-x^2}{2}\right)\mathrm{d}x$ [duplicate]

I would like to read a very thorough and explained calculation process for a couple of integrals. For the life of me I just can't figure out the result on my own, and no resource on the web were able ...
4
votes
5answers
79 views

Does the limit $\lim\limits_{x\to0}\left(\frac{1}{x\tan^{-1}x}-\frac{1}{x^2}\right)$ exist?

Does the limit: $$\lim\limits_{x\to0}\frac{1}{x\tan^{-1}x}-\frac{1}{x^2}$$ exist?
1
vote
1answer
25 views

Limit Help: $\lim_{x\to\infty} xe^{-a\frac{x}{\ln x}}$

I feel dumb for asking this, but I couldn't quite show that this limit is 0 (which I think is correct) whenever $a>0$: $$\lim_{x\to\infty} xe^{-a\frac{x}{\ln x}}.$$ I tried using L'Hospital's ...
4
votes
0answers
77 views

Another integral related to Fresnel integrals

How would we prove this result by real methods ? $$\int_0^{\infty } \frac{\sin \left(\pi x^2\right)}{x+2} \, dx=\frac{1}{4} \left(\pi-2 \pi C\left(2 \sqrt{2}\right)-2 \pi S\left(2 ...