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

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0
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0answers
36 views

Power series expansion of $z\mapsto \frac{1}{1+z^2}$ around arbitrary point $x\in \mathbb R$

Determine the power series expansion of $z\mapsto \frac{1}{1+z^2}$ around $x\in\mathbb R$ with the respective radius of convergence. At first I tried working with Cauchy's integral formula to ...
0
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0answers
8 views

Vector Calculcus identities involving second order tensors

I was reading a paper on Vortex Dynamics and came across the expressions $$\nabla(\vec{A}\cdot\vec{B})=(\nabla\vec{A})^T\cdot\vec{B}+(\nabla\vec{B})^T\cdot\vec{A}$$ and ...
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0answers
30 views

How to derivative the function which have constraint like $x^2+y^2+z^2 = 1$

For example , we have function like $f(x,y,z) = 1 - 2\times x^2 + y + z$ with constraint $x^2+y^2+z^2 = 1$ when I need to compute the derivative of $\frac{∂f}{∂x}$ , should it be ...
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2answers
39 views

Find the integration for $3x e^{-3x}$ from $0$ to Infinity؟

Find $$\int_{x=0}^\infty3xe^{-3x}\,\mathrm{d}x$$ Could you please help me find the integral of the product of two functions?
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1answer
50 views

All Continuous function can be drawn? [duplicate]

I googled and came to know that there are many continuous functions which cannot be drawn by hand, like Cantor, Weierstrass functions etc. Now this question was asked in a college admission ...
0
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1answer
52 views

Can a non-continuous function be differentiable?

I learnt that a function is called "differentiable " if the derivative (if it exists) of the function is continuous. Suppose a function $f: \mathbb{R}\to\mathbb{R}$ is differentiable, i.e., $f'(x)$ is ...
1
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1answer
64 views

Calculate $\lim \limits_{x \to \infty} x\int_{0}^{x}e^{t^2-x^2}dt$

I'm trying to find this limit $$\lim \limits_{x \to \infty} x\int_{0}^{x}e^{t^2-x^2}dt$$ From the graph I can see that it equals $1/2.$ I've looked into making substitution in order to modify the ...
2
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3answers
69 views

Limit of ($\sqrt{x^2+8x}-\sqrt{x^2+7x}$) as $x$ approaches infinity

I've been stuck on this one problem for 3 days now, I don't know how to proceed. Any help would be appreciated. The problem is asking for the $$\lim_{x\to\infty} (\sqrt{x^2+8x}-\sqrt{x^2+7x}) $$ ...
2
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2answers
55 views

Curve sketching without a computer program

How to sketch the curve x^6 + y^6 = (x^4)*y without using a computer program ? Could someone give me the step by step ?
3
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0answers
32 views

Are these Infinite Series Representations of Special Functions?

I am not sure how to google the answer for this question. Anyway, in trying to compute the velocity of a charged particle in an electromagnetic field, I came across these two infinite series ...
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1answer
72 views

Integrate $\int x \frac{f'(x)}{f(x)} dx$

I need your advice in integrating $\int ln(f(x)) dx = \int x \frac{f'(x)}{f(x)} dx$, where $f(x)$ is a probability density function. So it is the same as $\int x \frac{F(x)}{f(x)} dx$. How can I ...
1
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0answers
14 views

How can my model for a compound angular arm be adjusted to account for a special case?

I'm creating a robot. More specifically, I'm creating a robotic arm that can pick up objects and place them accurately in egg-carton slots. This is for the Science Olympiad(I'm a high school ...
2
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4answers
45 views

$\sum (-1)^{n+1}\log\left(1+\frac{1}{n}\right)$ convergent but not absolutely convergent

I need to prove that: $$\sum_{n=1}^{\infty} (-1)^{n+1}\log\left(1+\frac{1}{n}\right)$$ is convergent, but not absolutely convergent. I tried the ratio test: $$\frac{a_{n+1}}{a_n} = ...
1
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1answer
15 views

Using differentials to find the maximum error that a diameter can have so that the area error is within $1\%$

I am given the following problem: The area of a circle was computed using the measurement of its diameter. Use differentials to find the maximum error that a diameter can have so that the area ...
2
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4answers
293 views

How did the answer key get $h=40-2r$?

A cone has radius of $20\ \rm cm$ and a height of $40\ \rm cm$. A cylinder fits inside the cone, as shown below. What must the radius of the cylinder be to give the cylinder the ...
0
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1answer
39 views

Show that $\lim{\sup{A}}$ is the supremum of all limit points of $A$

Definition. Let $A$ be a nonempty subset of $\mathbb{R}$. $x \in \mathbb{R}$ is called an almost upper bound of $A$ if there are only finitely many $y \in A$ for which $y \geq x$. Similarly we define ...
1
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1answer
60 views

For what value of $v_0$ is the solution periodic?

A solution of the second-order differential equation $$ x''+x-x^3=0 $$ satisfies the initial condition $x(0)=0$ and $x'(0)=v_0$. For what value of $v_0$ is the solution periodic? I have tried ...
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2answers
66 views

If $\sum_{n=0}^{\infty}a_{n}x^n $ converges for $|x| < R$ , then $\sum_{n=0}^{\infty}na_{n}x^n $ converges for $|x| < R$

I have the following statement: If $\sum_{n=0}^{\infty}a_{n}x^n $ converges for $|x| < R$ , then $\sum_{n=0}^{\infty}na_{n}x^n $ converges for $|x| < R$ as well. I couldn't find a ...
1
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1answer
14 views

Using differentials find the approximate decrease on the area of a circle when the radius of it decreases from $r = 1 cm$ to $r = 0.8 cm$

I am asked to solve the following problem: Using differentials find the approximate decrease on the area of a circle when the radius of it decreases from $r = 1 cm$ to $r = 0.8 cm$ What I have ...
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1answer
18 views

Using differentials to find the percentual error on calculating the volume of a cube

I am given the following problem: The size of a cube measures $20 \, \rm{cm}$ with a percentage error of $\pm 2 \%$. Use differentials to estimate the error on calculating its volume. What I ...
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0answers
20 views

Related rates of $y^2-3xy+x^2=25$ given $x$ and $y$ are dependent of $t$

I am given the following problem: Two variables $x$ and $y$ are dependent of $t$ and are related according to $$y^2-3xy+x^2=25$$ If $x$ varies $1$ unit when $x = 0$ then find ...
0
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2answers
28 views

Prove given $f(x)$ integrable on $[a,b]$ then $g(x)=f(x-c)$ is integrable on $[a+c,b+c]$

I've tried to prove that if $f(x)$ is an integrable function on $[a,b]$ then $g(x)=f(x-c)$ is integrable on $[a+c,b+c]$. I'm not sure about my way of proof so I would appreciate a lot your feedback. ...
1
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2answers
76 views

limit $\lim_{x\to 0}\frac{\tan x-x}{x^2\tan x}$ without Hospital

Is it possible to find $$\lim_{x\to 0}\frac{\tan x-x}{x^2\tan x}$$ without l'Hospital's rule? I have $\lim_{x\to 0}\frac{\tan x}{x}=1$ proved without H. but it doesn't help me with this complicated ...
0
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2answers
37 views

Let $f(x)=x^3+ax^2+bx+c$ Prove that for $a^2\lt 3b$ there exists only one $x_0$ such that $f(x_0)=0$

Let $f(x)=x^3+ax^2+bx+c$ Prove that for $a^2\lt 3b$ there exists only one $x_0$ such that $f(x_0)=0$ Now we know this $x_0$exists because of the IVT. Also graphing different values of a and b, I've ...
1
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0answers
21 views

Point which do not lie in domain of $f (\frac{2}{x-2})$

If $$f(x)=\frac{1}{x^2-17x+66}$$ then the points which are not in the domain of $f (\frac{2}{x-2})$ are: $(A) \frac{7}{3}$ $(B) \frac{24}{11}$ $(C) \frac{8}{3}$ $(D) 2$ I have already found that ...
0
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2answers
41 views

If $a_0, a_1,\ldots,a_n \in R$ and $\frac{a_0}{1}+\frac{a_1}{2}+\cdots+\frac{a_n}{n+1}=0$, then exists $x\in(0,1)$ s.t. $a_0+a_1x+\cdots+a_nx^n=0$

Prove or disprove: If $a_0, a_1,\ldots,a_n \in R$ satisfy $\frac{a_0}{1}+\frac{a_1}{2}+\cdots+\frac{a_n}{n+1}=0$, then exists $x\in(0,1)$ such that $a_0+a_1x+\cdots+a_nx^n=0$ Let's define f(x)= ...
1
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3answers
43 views

Expand a function to power series

I have the following function and i try to expand it to a power series - $$F(x) = \frac{2x}{(x^2+1)^2}$$ around $X = 0$ I tried to substitute $t = -x^2$ and got stuck. I would like to get some help ...
2
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2answers
38 views

Double integral of log squared

Reading up on integral equations, the text states as matter of fact that $$\int_0^1 \int_0^1\ln^2|x-y| \, dx \, dy < \infty$$ Wolfram|Alpha confirms that the double integral in fact evaluates to ...
0
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2answers
32 views

Prove that $x-a \sin(x)=b$ has one real solution, where $0\lt a \lt 1 $

$a,b \in \mathbb{R}$. Prove that $x-a\sin(x)=b$ has one real solution, where $0\lt a \lt 1 $. I need some sort of starting hint as to how to prove this. I can define $g(x)= x-a\sin(x)-b$ but more ...
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3answers
63 views

How is this series rearranged?

I'm stuck at this. How is RHS rearranged? Is it a change of index? $$ \sum_{n=1}^{2N} \frac{1}{n} - \sum_{n=1}^{N} \frac{1}{n} = \sum_{n=N+1}^{2N} \frac{1}{n} $$ I'm stuck here: $$ \sum_{n=1}^{2N} ...
0
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0answers
26 views

Find the absolute max and min values of a multivariable function bounded by a circular boundary

Find the absolute minimum and maximum values of $f (x, y) = xy e^{−2x^2 −2y^2}$ on the set $\Delta = $ {$(x,y)\in\mathbb{R^2} | x^2+y^2\le1$} i know i should take the partial derivatives and set ...
-2
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0answers
32 views

I want to draw one expression of fraction which have sin and cos. [on hold]

I want to know what shape of next expression. $a$ is a real number and $0<a<1$. $t$ moves $0<t<\frac{2\pi}{a}$. $x=\frac{\cos(a+1)t}{\cos(at)}$ $y=\frac{\cos(a+1)t}{\sin(at)}$
0
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1answer
25 views

The behavior of the trajectory of the phase portrait

For the plane autonomous system $$ x' = ax+by $$ $$ y' = cx+dy $$ If the solution to this system is, says, $ \binom{x}{y}= c_{1}\binom{1}{1}e^{-5t} + c_{2}\binom{1}{2}e^{-t} $, then it is ...
3
votes
2answers
71 views

How should I understand $\displaystyle\sum_k f(k) \approx \int dk \ f(k)$?

I am trying to understand the statement: $$\sum_{k\geq 0} f(k) \approx \int dk\ f(k).$$ When we do an integral, $dk$ is an infinitesimal and so we are roughly speaking, summing over all values of ...
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2answers
25 views

Prove that $M_1^2\leq 2M_0M_2$, if $2M_1t≤2M_0+M_2t^2$

Let $0\leq M_1,M_2,M_3\in\mathbb{R}$ and $\forall \ t\in\mathbb{R}:\ 2M_1t≤2M_0+M_2t^2$. Prove that $M_1^2\leq 2M_0M_2$. I tried assigning different values to $t$, but this didn't help.
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0answers
41 views

Show that $(a,b) +\alpha = (a+\alpha,b+\alpha)$

Assume $a<b$ are real numbers. Show that $(a,b) +\alpha = (a+\alpha,b+\alpha)$. How do we add a number to an interval? Is this considered to be a translation? If so, isn't obvious this is true ...
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1answer
30 views

The sum of series in an interval

I have the following series - $$ \sum_{n=1}^\infty nx^{2n-1} $$ I found that its convergence interval is $[-1,1]$ but how can i calculate the sum in this interval ? i would like to get some hint ...
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0answers
26 views

Are harmonic functions ($\Delta u=0$) with compact support unique?

Does anyone how to solve the problem $$ \Delta u=0$$ with $u \in R^n$ and $u(x) \to 0$ as $|x| \to \infty$? Is $u=0$ the only solution? Many thanks.
1
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1answer
49 views

How to integrate $\frac{dx}{(x^2+k^2)^m}$, with $m$ positive integer.

How can I integrate: $$\int \frac{1}{(t^2+k^2)^m}\, dt$$ without trygonometric substituition? where $t= (x+(p/2))$ and $k= (1-(p^2/4))$ coming from an equation with complex roots: $x^2 + px + q.$ ...
0
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1answer
65 views

what does $\frac{\text{d}x}{x}$ mean?

I saw in a lecture recently the Gamma-function written like $$\Gamma (k) = \int_0^\infty e^{-x} x^k \frac{\text{d}x}{x}$$ and the professor said, that the integral was with respect to the measure ...
4
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1answer
35 views

A question about a route of a point that travels in a particular way through the plane

I don't know exactly how to classify this question. It is not from any homeworks, just something I've been wondering about. Let's say that in the beginning of an experiment ( the beginning is $t=0$ ...
1
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3answers
42 views

The function $A$ is given by the formula $f(t)=4-t$ and integral from $A(x) = \int_0^x f(t) dt$

Sketch a graph of $y=A(x)$ for $0≤x≤4$. Calculate the values of $A(0)$, $A(1)$, $A(2)$, and $A(3)$. Determine the values of $A'(0)$, $A'(1)$, $A'(2)$, and $A'(3)$. So I know the indefinite ...
7
votes
1answer
168 views

mathematics was recreated on a foundation of number concepts rather than geometrical ones

In Richard Courant and Fritz John's book Introduction to Calculus and Analysis Volume I, says In modern times mathematics was recreated and vastly expanded on a foundation of number ...
3
votes
2answers
61 views

Prove that: $\frac{2\pi i}{(1 - e^{2i\pi/n})\prod_{k=0, k \neq 1}^{n-1} (e^{i\pi/n} - e^{i(2k-1)\pi/n})} = \frac{\pi/n}{\sin(\pi/n)}$

I am trying to find $\int_0^{\infty} \frac{dx}{1 + x^n}$ using contour integration. I did the computation by taking the contour $[0,R] \cup \gamma_R \cup [R e^{2i\pi/n}, 0]$, with $\gamma_R$ the arc ...
3
votes
2answers
83 views

Volume enclosed by $(x^2+y^2+z^2)^2=x$

I need to calculate the volume of solid enclosed by the surface $(x^2+y^2+z^2)^2=x$, using only spherical coordinates. My attempt: by changing coordinates to spherical: ...
0
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0answers
17 views

Can $0 \preceq A\preceq B$ and $span(A)\subseteq span(B)$ lead to $\sigma_i(A)\leq \sigma_i(B)$?

Can $0 \preceq A\preceq B$ and $span(A)\subseteq span(B)$ lead to $\sigma_i(A)\leq \sigma_i(B)$? In the question, $A\preceq B$ means that $B-A$ is a positive semi-definite matrix, $span(A)\subseteq ...
2
votes
8answers
60 views

Is there a simple, intuitive way to see that $f(x)=x-\sqrt{x^2-1}<1$ if $x>1$

Is there a simple intuitive way to show that $f(x)=x-\sqrt{x^2-1}<1$ if $x>1$? I sense it could be done more simple than this: 1 - take the derivative $f'(x)=1-\frac{x}{\sqrt{x^2-1}}<0$ if ...
3
votes
1answer
60 views

Determining convergence of a series $\sum_n (-1)^n \sin a_n $

I need to determine if the following series is convergent: $$\sum_{n=2}^\infty (-1)^n\sin\left(\frac{\sin(3n)}{\sqrt{n}} + \frac{1}{n\ln^2(n)}\right).$$ I've tried to use alternating series test but ...
1
vote
0answers
34 views

Looking for a closed form sum of a series

I am wondering if there is a closed form solution of the following series $$ 2 \sum_{n =1}^{\infty} \frac{\sin (\pi a n) }{\pi n} (1-\cos (\pi n)) \exp \left(-(\pi a n)^2 t^{2 b}\right) $$ where ...
0
votes
0answers
21 views

Antiderivative of the greatest integer function

One of my homework problem wants me to prove that the greatest integer function $f(x)=[x]$ does not have an antiderivative. While thinking, I got to this expression, $$\int_0 ^x [t] \, dt = ...