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

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1
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2answers
38 views

Express g's Fourier coefficients using f's ones, if $g(x)=f(x+c)$.

The Fourier coefficients are defined (in our course) as: $$\hat{f(n)}={1\over 2\pi}\int_{0}^{2\pi}{f(t)e^{-int}dt}$$ I am asked to express g's coefficients as a combination of f's ones, given ...
2
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3answers
44 views
0
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0answers
29 views

finding the supremum

Let $A=\{x:\frac{[b\cdot n]}{n}\}$ when $n\in \mathbb{N}$ find the supremum we know that $b\cdot n-1<[b\cdot n]<b \cdot n$ therefore $b- \frac{1}{n}<\frac{[b\cdot n]}{n}<b$ so b is a ...
9
votes
3answers
157 views

difficult problem in riemman integrals

Could anyone help me with the following problem? Because i have stuck. problem Let $f:[a,b]\rightarrow [0,\infty)$ be continuous and not the zero function. Prove that $$\lim_{n\to \infty} ...
1
vote
3answers
623 views

Cartesian Product of large circle and small circle

I am bit confused with cartesian product. Suppose I have a set in R2 space and another set in R2 space so would it be right to say that cartesian product of those is in R4 space?? Following to this ...
0
votes
0answers
19 views

simple performance calculation dependent on different factors

I have to decide performance of some exam, Performance is dependant on few factors: factor1 : x1 out of y1 where x1<=y1, e.g. I checked 6 options out of 10, nearer the value of x1 to y1, better ...
6
votes
4answers
106 views

$\int \limits_0^{\infty} x^2 \exp(-2x^2) dx$

How to evaluate this integral? $$\int \limits_0^{\infty} x^2 \exp(-2x^2) dx$$ I found similar problem, but don't know how to apply them here. What do I have to substitute?
1
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4answers
158 views

To check the convergence of an integral (2)

I tried to check if this integral is convergent: $\int _{-\infty }^{\infty }\left(\frac{\sin\left(x\right)\ln\left|x\right|}{1+x^2}\right)dx\:$ so, there are 3 points to check: $\pm\infty$ and $0$. ...
1
vote
1answer
136 views

Evans PDE, Problem 8 Chapter 2 clarification on $|x-y|$

Hi I am attempting problem 8 (Chapt2 Evans PDE). Again I found the solution on the internet. enter link description here I understood much of everything of the proof except for one line. " Since ...
0
votes
1answer
83 views

Why does a line integral not depend on the parametrization you use?

I have a question about my calculus course: Why is it true that a line integral over a certain functiondoes not depend on the parametrization you use?. For example, take a function $f(x,y,z)$ of 3 ...
3
votes
2answers
222 views

How to prove Raabe's Formula [duplicate]

For quite some time, I've been trying to prove Raabe's Formula, or in other words: $$\int_a^{a+1} \ln\bigg(\Gamma(t)\bigg)dt=\dfrac{1}{2}\ln(2\pi)+a\ln(a)-a$$ This is how I tried: ...
5
votes
1answer
194 views

Mean value theorem for the second derivative, when the first derivative is zero at endpoints [duplicate]

Suppose $f:[a,b]\to \mathbb R$ has derivative up to order $2$, and $f'(a)=f'(b)=0$. Prove there is a point $c$ at $(a,b)$ such that $$ |f''(c)|\geq 4\frac{|f(b)-f(a)|}{(b-a)^2}. $$ If it was ...
1
vote
5answers
62 views

Show that this limit is related to Euler number

I am calculating the limit $\lim_{n \rightarrow \infty} \left( \frac{n!^{\frac{1}{n}}}{n} \right)= \frac{1}{e}.$ I got this limit from wolframalpha, but don't know how to show this.wolframalpha
0
votes
0answers
18 views

How can I apply the Magnus expansion here?

Suppose I have two operators $A(t),V(t)$. There is also a parameter $t \in [0, \infty]$. Moreover I have a continuous function $f(t)$ which satisfies $A(s)f(t)=0$ for all $s \in [0,\infty]$. How can I ...
0
votes
1answer
34 views

finding infimum

find the infimum and supremum of $E=\{x \in \mathbb{R}:x=\frac{2}{n}+(-1)^n, n\in \mathbb{N}\} $ $Max(E)=2$ therefore it is also the $Sup(E)$ Let assume that there is $-1<m: m\in E$ so $-1$ ...
0
votes
1answer
27 views

Multivariable calculus: what principle is this step based on?

The background is that I was asked to solve the following problem using Green's formula $L$ is a Jordan curve (smooth and closed) which encloses the origin point in $xOy$ plane. Caculate this ...
0
votes
0answers
20 views

Exponential operator on function; can it be simplified?

Suppose I have two operators $A(t),V(t)$. There is also a parameter $t \in [0, \infty]$. Moreover I have a continuous function $f(t)$ which satisfies $A(s)f(t)=0$ for all $s \in [0,\infty]$. How can I ...
4
votes
3answers
664 views

To prove a sequence is Cauchy [duplicate]

I have a sequence: $ a_{n}=\sqrt{3+ \sqrt{3 + ... \sqrt { 3} } } $ , it repeats $n$-times. and i have to prove that it is a Cauchy's sequence. So i did this: As one theorem says that every ...
1
vote
0answers
79 views

Determine null, extreme and inflection points of function $f(x)=\frac{x+e^x}{x-e^x}$

This function has a null point, but I can't compute it from equation $f(x)=0$ which gives $$\frac{x+e^x}{x-e^x}=0$$ $$x+e^x=0$$ How to compute this equation? Extreme points can be computed from ...
2
votes
2answers
132 views

Pre-calculus - Deriving position from acceleration

Suppose an object is dropped from the tenth floor of a building whose roof is 50 feet above the point of release. Derive the formula for the position of the object t seconds after its release is ...
0
votes
2answers
54 views

Finding the derivative of an integral with variable limits: ${\mathrm{d} \over \mathrm{d}x}\int_{x}^{x^2}{1 \over -2y}e^{-5xy^{2}}\mathrm{d}y$?

How do you compute the derivative $${\mathrm{d} \over \mathrm{d}x}\int_{x}^{x^2}{1 \over -2y}e^{-5xy^{2}}\mathrm{d}y$$ where the integral has variable limits?
1
vote
1answer
42 views

help with wrong result for $v(x) = \frac{\sqrt[3]{x-1}}{(x+2)^2}$

I need to differentiate this: $$v(x) = \frac{\sqrt[3]{x-1}}{(x+2)^2}$$ I used this formula: $$ \frac{f'(x)g(x) - f(x)g'(x)}{g^2(x)}$$ Where: $$ f'(x) = \frac{1}{3\sqrt[3]{(x-1)^2}}$$ and $$ g'(x) ...
0
votes
1answer
51 views

There are two periodic functions $f(x)$ and $g(x)$, provide an example when $f(x)*g(x)$ is unbounded, and $f(x)+g(x)=0$ has infinitely many solutions

There are two periodic functions $f(x)$ and $g(x)$ which are defined on $\mathbb{R}$, provide an example when $f(x)\cdot g(x)$ is unbounded, and $f(x)+g(x)=0$ has infinitely many solutions ?
1
vote
4answers
107 views

Limit $(e^x+x)^{1/x}$, when $x\to 0$

Can I expand e^x in the limit $$\large{\lim _{x\to 0}(e^x+x)^{1/x}},$$ just as $1+x$ according to the Taylor expansion? I mean is it normal to think about limits of a form $(1+x+o(x))^{1/x}$ just as ...
1
vote
3answers
55 views

what is the value of $\int \sin(x)\cos(x)dx$? $\frac{\sin^2(x)}{2}$ or $\frac{-\cos^2(x)}{2}$ or $\frac{-\cos(2x)}{4}$

$\int \sin(x)\cos(x)dx = \frac{\sin^2(x)}{2}$ because $$\frac{d}{dx}\frac{\sin^2(x)}{2}=\sin(x)\frac{\sin(x)}{dx}=\sin(x)\cos(x)$$ but also ...
2
votes
2answers
62 views

To check the convergence of an integral

I tried to find out if this integral is convergent or divergent, $$\int _0^{\frac{\pi}{2}}\left(\frac{\ln\left(\sin x\right)}{\sqrt{x}\:}\right)\:dx$$ I know that the problematic point is near ...
9
votes
4answers
223 views

Why doesn't using the approximation $\sin x\approx x$ near $0$ work for computing this limit?

The limit is $$\lim_{x\to0}\left(\frac{1}{\sin^2x}-\frac{1}{x^2}\right)$$ which I'm aware can be rearranged to obtain the indeterminate $\dfrac{0}{0}$, but in an attempt to avoid L'Hopital's rule ...
10
votes
3answers
7k views

The shortest distance between any two distinct points is the line segment joining them.How can I see why this is true?

On a euclidean plane, the shortest distance between any two distinct points is the line segment joining them. How can I see why this is true?
1
vote
2answers
65 views

Problem with understanding a Differential in Multivariable Calculus

I have just started with Partial Differentiation and the book from where I'm learning (Mathematical Methods in the Physical Sciences) had the following problem on approximations using differentials ...
4
votes
1answer
29 views

Differential Equation with Cross Products [without separating into system of equations]

I need to solve the following equation: $$ \frac{d m}{d t}=-m\wedge b-\alpha m\wedge (m\wedge b), $$ where $b$ is constant However, I was instructed specifically not to separate the calculation into ...
0
votes
1answer
33 views

Finding the volume of a cone with and oblique base.

The base of $S$ is an elliptical region with boundary curve $9x^2+4y^2=36$. Cross-sections perpendicular to the $x$-axis are isosceles right triangles with hypotenuse in the base. The base of $S$ is ...
0
votes
1answer
25 views

How proceeded author from $(3)$ to get $a_n<f(n)-f(0)<1$

We have $f:(-1,\infty)\rightarrow\mathbb{R}, f(x)=\frac{x}{x+1}$ and $$a_n=\sum_{k=1}^n f(k)-\int_0^1 f(x)dx$$ We need to prove that $a_n$ is bounded. I don't need another method, I want to help ...
2
votes
4answers
52 views

$f$ is integrable & continuous over $[a,b]$ , $\int_{a}^{b}f(x)dx \geq 0$ for any subinterval $(\alpha,\beta)$ of $(a,b)$, then $f \geq 0$ in $[a,b]$

Some known things about this problem are: if $f(c) < 0$, $a < c < b$, then $f(x) < f(c)/2$ in some neighborhood of $c$, but I am not exactly sure how to use this to get to my goal of ...
1
vote
1answer
74 views

If $k$ is a non-zero constant, determine by inspection the indefinite integral of $\int e^{kx} dx$.

I have to solve this exercise: If $k$ is a non-zero constant, determine by inspection the indefinite integral of $\int e^{kx} dx$. By inspection, I guess it means that it should be solved by ...
1
vote
3answers
69 views

prove continuity

Let $ f:\Bbb R \to \Bbb R $ satisfy the property $ f(x+y)=f(x)+f(y)$ for all $x,y$ in $ \Bbb R $ I have to show that 1)$f(0)=0 , f(-x)=-f(x),$ for all $x$ in $\Bbb R$, and $f(x-y)=f(x)-f(y)$ $y$ in ...
0
votes
1answer
41 views

Derive Inverse Laplace transform by differentiating F(s) and Integrating f(t) (5.5-21)

Request: Please check my work. I cannot duplicate the answer in text although it is very close. I believe the problem lies in how I take the derivative. Is there a better way to calculate the inverse ...
1
vote
2answers
121 views

Calculus. Why are these statements equal?

I'm taking calculus and I've been stumped on this for a while now, Google isn't helping because idk what to search for... OK my question is about the change in a quotient. $$ \delta\left(\frac ...
4
votes
2answers
100 views

Real Analysis question about proving limits using boundedness and monotony

I am trying to get a hang of solving problems that ask to "prove the limit". One of the questions on our previous homework was: The solution starts with . How do we know that sn is bounded below ...
0
votes
1answer
65 views

Can't put expression in terms of $y:\ x = \frac{2y-1}{y+1}.$

Disclaimer: This is not a student posting his homework assignment. I am an adult learning Calculus. I think this is a great forum, Ok, you know the formula for the derivative of an inverse: ...
10
votes
4answers
645 views

Integral becomes improper after a substitution

I'm suprised about the following phenomenon which I would like to discuss with you. Consider the proper integral $$\int_{\pi/4}^{\pi/2}\frac{1}{\sin(x)}dx.$$ Since $\sin(x)$ is a diffeomorphism on ...
2
votes
1answer
193 views

Second Mean Value Theorem for Integrals Meaning

The Second Mean Value Theorem for Integrals says that for $f (x)$ and $g(x)$ continuous on $[a, b]$ and $g(x)\ge 0$ $$\int_a^bf(x)g(x)\,dx=f(a)\int_a^cg(x)\,dx+f(b)\int_c^bg(x)\,dx$$ I have a ...
-3
votes
1answer
57 views

A problem about prism with triangular bases

Consider a prism with triangular base . The total area of the three faces containing a particular given vertex is $k$ . Then is the maximum possible volume of the prism $\sqrt {\dfrac {k^3} {54} } $ ? ...
1
vote
2answers
43 views

If $f$ is integrable on $[a,\ b]$ and $\int_a^b f(x) \mathrm dx >1$, then there exists a point $c$ in $(a,\ b)$ such that $f(c) > \frac{1}{b-a}$

So far for this problem, to my understanding, for something to be integrable means that $U(p,\ f) - L(p,\ f) < \epsilon$ but not sure how exactly to move beyond there to show that there exists a ...
2
votes
1answer
59 views

Evaluate $\lim_{\alpha \to \infty} e^{-t\sqrt{\alpha}}(1-\frac{t}{\sqrt{\alpha}})^{-\alpha}$

How does one show $$\lim_{\alpha \to \infty} e^{-t\sqrt{\alpha}}\left(1-\frac{t}{\sqrt{\alpha}}\right)^{-\alpha} = e^{t^2 / 2}?$$ Not homework, this is from this proof that the gamma distribution ...
-1
votes
2answers
81 views

Infinite Series for Arctan [closed]

$$ \sum_{k=1}^{\infty}\arctan\left(\frac{1}{k^2}\right) $$ Does anyone know how to determine if this infinite series diverges or converges and if it converges, what its value is?
3
votes
1answer
32 views

Generalized linear combination of probability density functions

I am working with linear non-unity combinations of independent variables in the equation form of: $$Y_i=\sum_{j=1}^N a_{ij} X_j ~~~~\forall~ a_{ij} \in \mathbb{R}, a_{ij}\neq 1$$ I am aware of the ...
0
votes
1answer
46 views

Bound of Mann iterative sequence

There is theorem in the book of Charles Chidume "Geometric Properties of Banach Spaces and Nonlinear Iterations" My question is: why if the underlined conditions are satisfied {Xn} is bounded (proof ...
5
votes
1answer
69 views

double integral problem $\iint e^{\frac{x}{x+y}}dxdy$

I'm trying to integrate $$\iint e^{\frac{x}{x+y}}dxdy$$ where $y \leq (1-x)$ and $0 \leq x,y \leq 1$. I tried to define new variables as $u=x$ and $v=x+y$, but I can't solve this either. I have ...
1
vote
1answer
79 views

When are monic polynomials of fourth degree divisible?

Note that this might be an X/Y problem, therefore I'm posting the original question too. I am asked to prove that given a monic polynomial of fourth degree which has a non-zero root, must have at ...
10
votes
2answers
2k views

find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges

Find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges. To make it more challenging, produce examples where $a_n$ and $b_n$ are ...