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

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15 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
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0answers
15 views

If $F=(\frac{-2xy}{(1+x^2)^2+y^2},\frac{1+x^2}{(1+x^2)^2+y^2} )$…

If $F=(\frac{-2xy}{(1+x^2)^2+y^2},\frac{1+x^2}{(1+x^2)^2+y^2} )$ and $a=(0,0),\ b=(x,0),\ c=(x,y)$ compute: $$f(x,y)=\int_{[a,b]}F+\int_{[b,c]}F$$ Someone could explain to me how I could solve that ...
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1answer
50 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 $dr$. ...
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0answers
7 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
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1answer
12 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 $n-th$ ...
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1answer
15 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 ...
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0answers
23 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 ...
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0answers
18 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 ...
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1answer
16 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 ...
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3answers
53 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 ...
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1answer
34 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
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1answer
61 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 ...
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0answers
7 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
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0answers
24 views

Verification of an indefinite integral with trigonometric functions

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 ...
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3answers
24 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
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1answer
27 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
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2answers
36 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 ...
10
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2answers
114 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 ...
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2answers
32 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: ...
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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 ...
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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) = ...
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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
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1answer
47 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 ...
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2answers
102 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!
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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
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2answers
36 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
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2answers
69 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
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5answers
76 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?
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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
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0answers
72 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 ...
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0answers
24 views

Parametrize the given curve. [on hold]

$x^2 + y^2 = 121\;$ satisfying the condition $\;\displaystyle c(0) = \left(\frac{11}2, \frac{11\sqrt{3}}2\right)$ Thank you! 
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0answers
25 views

Find $C^1$ class function such that

Given: $$g: \mathbb{R}^3 \rightarrow \mathbb{R}, g(x,y,z)=z^3-3xyz-x-8$$ Decide whether in the neighbourhood of the point $(x,y)=(0,0)$ there exist $C^1$ class function $z=z(x,y)$, such that ...
1
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0answers
25 views

Lagrange multiplier vs KKT

Suppose task 1: maximize $f(x, y)$ subject to $g(x, y) = 0$ and $h(x,y) = 0$ Suppose task 2: maximize $f(x, y)$ subject to $g(x, y) \geqslant 0$ and $h(x,y) = 0$ According to wiki for the first ...
3
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2answers
44 views

Use Taylor's Theorem with $n=2$ to prove that the inequality $1+x<e^x$ is valid for all $x\in \mathbb{R}$ except $x=0$.

Use Taylor's Theorem with $n=2$ to prove that the inequality $1+x<e^x$ is valid for all $x\in \mathbb{R}$ except $x=0$. Taylor's Theorem: $$ f(x)=\sum_{k=0}^n{1\over ...
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1answer
34 views

Computing limits example: Swaping limit to $0$ into infinity.

I have found the following example: $$ \lim_{x\to 0^{+}} \frac{e^{\frac{-1}{h}}}{h} = \lim_{z\to\infty} ze^{-z} = 0 $$ Could you explain to a kid nice and slowly why does the limit of $x$ to ...
1
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1answer
96 views

$\int_0^1(1+\log(x))\sin(x)dx$ How to solve this Integral?

$$\int\limits_0^1(1+\log(x))\sin(x)dx$$ Someone has challenged me to solve this, I solved it without bounds, I have no idea how to do it with those limits.. Is $u=1+\log(x)$ right substituion? or ...
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0answers
21 views

Question concerning continuity of composite functions

Consider two functions, $a(r)$ and $b(r)$. If a is continuous at $c$,and $b$ is continuous at $a(c)$ , then $b(a(c))$ is continuous at $c$ .(This is a theorem stated in the text Thomas' Calculus) Now ...
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1answer
34 views

Equation Of Common Normal

How to find equation of common normal between two random non-intersecting conic sections (say a parabola and an ellipse) ? What should be the general approach ?
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1answer
27 views

Boolean Algebra, stuck

I'm having trouble simplifying this Boolean Algebra equation. Can anyone help? XY'Z + X'Y'Z + XYZ + XY'Z
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1answer
21 views

Double integral proving that a function is a probability density

If $$g(x,y)=f(x+y)/(x+y)$$ for $x,y>0$ and $$\int_0^{\infty} f(z) \, dz = 1$$ How do you show that $$\int_0^{\infty} \int_0^{\infty} \frac{f(x+y)}{x+y} dx \, dy = 1$$ as well?
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3answers
57 views

Give the equations that are a tangent to the parabola $y = x^2 + 5x + 6$ and pass through $(1,1)$

I have been given the question: Give the equations that are a tangent to the parabola: $y = x^2 + 5x + 6$ and pass through the point $(1,1)$ I have tried two different methods for solving this. ...
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0answers
32 views

A simple Laplace transformation problem

Could someone please help me with my misunderstanding here? Here's the question: "A mass M is attached to a spring of stiffness $\omega^2M$ and is set in motion at t=0 by an impulsive force P. The ...
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0answers
34 views

convergence of general harmonic series

My question is about determining the convergence of a general harmonic series using the integral test. According to the following resource: pg.32 , we can see that for a general harmonic series ...
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1answer
42 views

Integrate $\int_{0}^{2}{\int_{y^2}^{4}{ycos(x^2)dxdy}}$

I'm asked to evaluate $\int_{0}^{2}{\int_{y^2}^{4}{ycos(x^2)dxdy}}$. Letting $f(x)=cos(x^2)$ We have have that $\int_{y^2}^{4}{cos(x^2)dx} = F(4)-F(y^2)$ by the FTC. This gives us \begin{align*} ...
1
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2answers
49 views

I need help with this trigonometric integral [on hold]

I dont know how to do this integral $\int \dfrac{dx}{\sin(x) + \cos(2x)}$ i have tried the fundamental trigonometryc identity $(\sin x)^2 + (\cos x)^2 = 1$ but that does not work out the way i ...
1
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1answer
29 views

Mean value formula integrals

Let $f: B(0,R) \rightarrow \mathbb{R}$ be a continuous function. Then I was wondering whether $$\frac{1}{\text{area}(\partial B(0,r))} \int_{\partial B(0,r)} (f(x)-f(0)) dS(x) \rightarrow_{r ...
0
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1answer
26 views

Arc length of polar curve

I was trying to determine the arc length of the polar curve $r = f(\theta) = a(1 - \cos \theta)$, and it was going well until I got to the definite integral. I know that $f'(\theta) = a \sin \theta$, ...
1
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1answer
21 views

Rules for manipulating differential/ Leibniz notation?

What are the rules on manipulating Leibniz Notation? dy/dt = -(y-3)/2 Can I treat it like a fraction and do this? dy = -(y-3)/2 dt can I "convert" the dy into a derivative? (dy/dt)(1/(y-3)) = ...
1
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3answers
44 views

Show that the function is continuous

To show that the function $f: \mathbb{R}^2 \rightarrow\mathbb{R}$ with $f=\left\{\begin{matrix} \frac{x^3-y^3}{x^2+y^2} & , (x,y) \neq (0,0)\\ 0 & , (x,y)=(0,0) \end{matrix}\right.$ is ...
2
votes
1answer
42 views

Taylor series question

I've been struggling with this problem: Find the Taylor series representation for $xe^{2x}$ I was able to find the Taylor series for $e^{2x}$ (centered at a=k) in a previous exercise which I ...