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

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1
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1answer
36 views

Double integral with coordinate transformation

I want to integrate $\int_B \cos\frac{x-y}{x+y}dB$ where the area B is characterized by the inequalities $\frac{1}{2} - x \leq \frac{1}{2}$ $1 \leq 1 + y$. $\frac{1}{2} \leq x + y \leq 1$ using ...
-1
votes
2answers
189 views

can anyone help with this integral $\int\frac{1}{x\sqrt{x^{2}+x-1}}dx$ [closed]

Can anyone help with this integral $$\int\frac{1}{x\sqrt{x^{2}+x-1}}dx$$ Thanks in advance
1
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2answers
184 views

Bisector of two lines in the euclidean space $\mathbb{E}_3$

Let $$r: \begin{cases} x + z = 0 \\ y + z + 1 = 0\end{cases}$$ and $$s: \begin{cases} x - y - 1 = 0 \\ 2x - z -1 = 0\end{cases}$$ be two lines in the euclidean space $\mathbb{E}_3$. It is easily ...
2
votes
0answers
118 views

0's reciprocal (Theoretical) [closed]

Background: I am in 8th grade and I like to study around advanced mathematical subjects. However, I do not know enough to be sure in my conjectures. Therefore, I would like your help. I have a ...
0
votes
2answers
165 views

The third derivative of the first principles definition of of a derivative

So the $\lim\limits_{h\to 0}\frac{f(x+h)-f(x)}{h}$ this is what I learned to find the first derivative and by taking this concept and trying to find the second derivative using this method I came up ...
2
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1answer
145 views

Why do derivatives pop up in abstract algebra?

Why is it that derivatives, which are used in calculus, are also useful in abstract algebra concerning abstract polynomial rings? What is the connection between calculus and algebra here?
4
votes
1answer
157 views

Limit of $n\left(e-\left(1+\frac{1}{n}\right)^n\right)$

I want to find the value of $$\lim\limits_{n \to \infty}n\left(e-\left(1+\frac{1}{n}\right)^n\right)$$ I have already tried using L'Hôpital's rule, only to find a seemingly more daunting limit.
2
votes
1answer
62 views

How to find the lines through the point $(2, 2, 0)$?

In the real space $\mathbb{R}^3$ is a cylinder with an axis $x = 0$, $y = z$ and a radius of $2$. Find the lines through the point $(2, 2, 0)$, which are parallel to the plane $x + y + z = 2$, and ...
0
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3answers
86 views

Uniform convergence of $\sin(x^n)/x^n$

Let $f_n(x)=\sin(x^n)/x^n$. Show that $f_n$ converges uniformly to $1$ in the interval $(-\delta,\delta)- \{0\}$, with $0<\delta<1$. So far what I've tried is to take the infinite norm of the ...
0
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2answers
67 views

A family of parabola are given: $f(x)=mx^2+(m+2)x+3, m \neq 0.$ Let's observe the set of points in plane $xOy$ through which no parabolas pass

Which of the following are correct? a.) $S$ is a line segment b.)$S$ is a line without two points c.)$S$ contain at least 4 rays d.)$S$ is made of one parabola and two rays e.)$S = \emptyset$ The ...
1
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1answer
49 views

Integration by Parts Problem: Help in understanding why a part of it equals 0

$$4I= \int_0^{\infty} \frac{4x^3 +\sin(3x)-3\sin x}{x^5} \ \mathrm{d}x $$ $$=\frac{-1}{4} \underbrace{\left[\frac{4x^3+\sin(3x)- 3 \sin x}{x^4} \right]_0^{\infty}}_{=0} +\frac{1}{4} \int_0^{\infty} \...
0
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1answer
86 views

Is infinity even and odd at the same time?

For a real number $x < -1$, what is the result of $\lim\limits_{n \to \infty}x^n$? At first I thought of "$\infty$" as the solution, but that's only the case if $n$ is an even number. For an odd ...
0
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6answers
85 views

Why does this set a compact?

Consider the following set in $\mathbb{R}^n$: $$S = \{ x \ :\ \|x\| = 1 \}$$ Why is this set a compact? I don't know about closed, but it doesn't look bounded to me.
-2
votes
2answers
55 views

Calculate the arc length of the curve $\int _Lydl\:$,where $L$ - the arc $y=\sqrt{1-x^2}$ [closed]

Calculate the arc length of the curve $\int _Lydl\:$,where $L$ - the arc $y=\sqrt{1-x^2}$ Use the formula $$ L=\int _a^b\sqrt{1+(y')^2}dx$$ We shall find the derivative. What's next? How to find the ...
0
votes
1answer
59 views

some vector calculus frequently occurs in the study of PDE

As I work through PDE by L.C. Evans, as well as reading papers in the literature of linear/nonlinear PDE. The terms "inward pointing normal" and "outward pointing normal"frequently occur. For instance,...
2
votes
1answer
58 views

Prove the dominated convergence for $f_n(x)=\frac{x^n}{x^2+3x+2}$

Suppose that we have $f_n:[0,1]\rightarrow\mathbb{R},f_n(x)=\frac{x^n}{x^2+3x+2}$. $f_n$ don't have uniform convergence because the pointwise isn't continuous: $$\lim_{n\to\infty}f_n(x)\neq\lim_{n\...
1
vote
1answer
61 views

Find the constant $C$ such that the following is a p.d.f.

Find the constant $C>0$ such that $C\dfrac{\sqrt{xy}}{x^2+y^2}$ is a pdf on the region $0<x,y;xy\leq\dfrac{1}{2}$ I have been stuck with this problem for quite some days. I tried routine ...
3
votes
1answer
69 views

Changing integration order in $\int_0^4\int_0^1\int_{2y}^2 \frac{4\cos (x^2)}{2\sqrt{z}}\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z$

I have this integral $$\int_0^4\int_0^1\int_{2y}^2 \frac{4\cos (x^2)}{2\sqrt{z}}\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z$$ I tried to change the order of integration in the following way $$\int_0^1\...
1
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2answers
117 views

Applying a constant force on pendulum and integrating for velocity

We apply a constant horizontal force $ F $ to a weight that's connected to a solid rod. I'd like to find out the velocity of the weight when it's at its horizontal position. See image below. My ...
0
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3answers
44 views

Help with definite integral using U-subsitution

I have the following integral I need to use U-subsitution for: $\int_{0}^{1} \frac{x^{3}}{(x^{2} + 1)^{\frac{3}{2}}}dx$ I'm struggling with mostly the first step. I know I have to rewrite the ...
0
votes
2answers
35 views

Simple Integration by substitution problem

I was given the following function: $$\int e^x \tan(e^x+3)^4dx$$ and was requested to replace $e^x=u$. Isn't the answer $$\int u\cdot \tan(u+3)^4du$$ Am I missing something?
2
votes
1answer
70 views

Interchanging Inverse Laplace Transform

I have a function $f(|\boldsymbol{k}|,s,\theta)$ for which I am interested in its inverse Laplace transform. I am also interested in the function's mean value for constant $|\boldsymbol{k}|$, but ...
3
votes
3answers
92 views

Cauchy's Integral Question Complex Number

I have a question and I'm kind of stuck, I was wondering if you were able to help me move forward. The question is, Use Cauchy's integral formula to evaluate, $$ \int_{|z| = 1}\frac{e^{2z}}{z^2}dz $...
8
votes
2answers
255 views

Evaluate $\lim_{n\to\infty}nI_n$ with $I_n=\int_0^1\frac{x^n}{x^2+3x+2}dx$

We have to evaluate: $$\lim_{n\to\infty}nI_n$$ with $$I_n=\int_0^1\frac{x^n}{x^2+3x+2}\:dx.$$ There is an elegant way to solve this problem? Here is all my steps: My first ideea was to find ...
4
votes
1answer
98 views

Proving uniform continuity of trigonometric function arctan

In my assignment I have to prove that the following function is uniformly continuous: $$f(x) = \arctan x \cdot \sin\left(\frac{1} {x}\right) $$ in the open interval $(0,\infty) $. I thought I'd ...
0
votes
1answer
233 views

How to solve a one sided limit?

Let $f(x)$ be some function (defined over the reals). How do I find the limit as $x$ approaches $a$ from either the positive or the negative side? (i.e how do I find either $\lim\limits_{x \rightarrow ...
5
votes
3answers
427 views

Curves With Known Arc Length [closed]

I would appreciate if you could list as many (planar) curves with known closed-form analytical expressions for the arc length as possible. Please include formulas for both the curve and the arc ...
1
vote
6answers
112 views

find the minimum value of $\sqrt{x^2+4} + \sqrt{y^2+9}$

Question: Given $x + y = 12$, find the minimum value of $\sqrt{x^2+4} + \sqrt{y^2+9}$? Key: I use $y = 12 - x$ and substitute into the equation, and derivative it. which I got this $$\frac{x}{\...
1
vote
2answers
71 views

Angle between two quarters of ellipses

I must find the angle between two quarters of ellipses at their common point by the parametric equations: $R_1(t) = 3\cos (t)i + \sin (t)j$ for $0 \leq t \leq \pi/2$ and $R2(s) = \cos (s)i + 3\sin (...
0
votes
1answer
32 views

How to check if a function is partially differentiable

Sorry for this basic request. $$$$Could you please tell me how to check if a function is partially differentiable (I don't know if this is the right term), both over an interval, as well as at a point?...
4
votes
3answers
127 views

Problem with Differentiation under the Integral Sign

Problem: Evaluate: $$\displaystyle\int_{0}^{\infty} \dfrac{1}{x} \left(\tan^{-1}(\pi x) - \tan^{-1}x\right)dx.$$ My incorrect attempt: $$\displaystyle\int_{0}^{\infty} \dfrac{1}{x} \left(\tan^{-...
1
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2answers
51 views

Find the curve, given that $r'(t) = Cr(t)$

I need to find the curve, given that $r'(t) = Cr(t)$ (where $C$ is a constant), for all real $t$ and $r(0)=i+2j+3k$. To start, I know that the equation I will need is the $$K=\frac{||r'(t) \times r''...
0
votes
1answer
85 views

Sketch the level curves (contour lines) of the function?

Consider the function $f\colon\mathbb{R}^2\to\mathbb{R}$ given by $$ f(x,y)=e^{x^2+y^2}-1. $$ My attempt: Let $$e^{(x^2+ y^2)} - 1 = C$$ $$\Leftrightarrow e^{(x^2+ y^2)} = C+1$$ $$\Leftrightarrow x^...
0
votes
1answer
50 views

Condition under which a set is compact

I'm studying at university real analysis and in class the teacher said that a set is compact if and only if is closed and bounded. But I don't really understand the concept, more widely: what really ...
1
vote
1answer
34 views

An estimate for dyadic numbers

I would like to prove that for some positive $\delta<1/2$ we have the following inequalities $$ |\frac{1}{2^{n+1}} - \frac{1}{2^{m+1}}| \leqslant \delta\left( |\frac{1}{2^{n}} - \frac{1}{2^{m+1}}| ...
1
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2answers
43 views

Is the limit of the function is $0$?

According to wolfram alpha,$$\lim _{x \to \ 0} \frac{\cos x}{\sin x} \cdot \sin\left(\frac 1 x\right)=0$$ In my question, the function is defined only in $(0,\infty)$ I want to prove it through the ...
0
votes
6answers
96 views

How can I expand $f(x)$ in powers of x?

$f(x)=\frac{1-x}{1+x}$. The closest thing I know to this would be $\sum_{k=0}^\infty x^k=\frac{1}{1-x}$ but I don't know how to use it to write $f(x)$
1
vote
4answers
106 views

Calculate the sum $\sum_{n=3}^{\infty}\frac{4n-3}{n^3-4n}$

$$\sum_{n=3}^{\infty}\frac{4n-3}{n^3-4n}$$ I think it is related to power series, because it is the topic, but I have no idea how to get there. Could you give a hint?
0
votes
2answers
69 views

Let $\{a_n\}$, $\{b_n\}$ be sequences bounded above. prove $\limsup (a_n + b_n)\le \limsup(a_n) +\limsup(b_n)$ [duplicate]

I'm having trouble starting this proof from the definition I know that there exists a limit point $x$ such that $M \ge x > M- \frac \epsilon2$ but not quite sure how to go about it.
5
votes
2answers
154 views

Can a “difference formula” be a “sum formula” too?

Consider the following "product-to-difference" formula: $$\sin p\sin q=\frac{1}{2}[\cos(p-q)-\cos(p+q)].$$ I wonder whether the right-hand side can be expressed as a weighted sum of cosine functions ...
4
votes
3answers
160 views

Is something wrong in my solution?

We have $f:(-1,\infty)\rightarrow\mathbb{R},f(x)=\frac{x}{x+1}$ and need to show that $a_n=\sum_{k=1}^n f(k)-\int_0^n f(t)dt$ is bounded. Here is all my steps: $f'>0\Rightarrow f$ is strictly ...
1
vote
1answer
51 views

Find the length of the curve parametrized by $x(\theta) = r(\theta - \sin\theta)$

I am really having difficulties in Calculus II and i am trying to do as many examples as i can. I bumped into this question and I dont have any slightest idea where i should start. Help appreciated. ...
2
votes
2answers
41 views

Finding limit with two variable

$\lim_{(x,y)\to(0,0)} (x^2+y^2)\ln(\sqrt{x^2+y^2}) = ? $ I have tried this; $\lim_{x\to0}(\lim_{y\to0}(x^2+y^2)\ln(\sqrt{x^2+y^2})) = 0 $ $\lim_{y\to0}(\lim_{x\to0}(x^2+y^2)\ln(\sqrt{x^2+y^2})) = 0 ...
3
votes
1answer
58 views

Proving a statement about a continuous function for which $\forall x\in\mathbb{R},\exists y>x : f(y)>f(x)$

Suppose $f$ is a function which is continuous on $\mathbb{R}$. Also, for all $x\in \mathbb{R}$, there exists $y>x$ such that $f(y)>f(x)$. I must prove that if $\lim_{x\to\infty} f(x)=L$ then $f(...
2
votes
1answer
125 views

How to evaluate $\lim\limits_{n\to +\infty}\frac{1}{n^2}\sum\limits_{i=1}^n \log \binom{n}{i} $ [closed]

I have to compute: $$ \lim_{n\to +\infty}\frac{1}{n^2}\sum_{i=1}^{n}\log\binom{n}{i}.$$ I have tried this problem and hit the wall.
3
votes
2answers
67 views

Help with Differentiating through the Integral Sign

Problem: Evaluate $$\int_{0}^{\infty} \dfrac{\sin^3{x}}{x \cdot e^x} dx=\dfrac{A\pi}{B}-\dfrac{\tan^{-1} (C)}{D},$$ My attempt through Differentiation under the Integral Sign: Using $\sin^3x=\frac{3\...
0
votes
1answer
31 views

expresion for the sum of the pentagonal numbers

How do I find the expresion for the sum of pentagonal numbers? I am doing a research paper on this and I have tried using the sn expresion for aritmetic series and replacing the variables but I am ...
1
vote
3answers
79 views

Find the derivative of $ f(x) = x^9 - x^7$ using limit definition

I can find this without using the limit definition (I think the formula is $\frac{f(x+h) - f(x)}{h}$ My first steps to solving are $f'(x) \lim\limits_{h\rightarrow 0} \frac{(x+h)^9 - (x+h)^7 - x^9 ...
3
votes
1answer
57 views

Give an example of a continuous function $f: (-1, 1) \rightarrow \mathbb{R}$ which attains a maximum at 0, but is not differentiable at 0

I need a little help with this exercise: Give an example of a continuous function $f: (-1, 1) \rightarrow \mathbb{R}$ which attains a maximum at 0, but is not differentiable at 0 I thought of the ...
0
votes
2answers
210 views

Find a particular solution that satisfies the initial condition $yy' - 2e^x = 0$

This is actually the second part of a larger question: The given information is: $$y'= x/\left(x^2 + 1\right)$$ I have calculated this from the first part of the question:$$y = \ln\left(x^2 + 1\...