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

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1
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3answers
53 views

$\lim a_n = L \implies \lim a_n^2 = L^2$

I have to prove the following: $$\lim a_n = L \implies \lim a_n^2 = L^2$$ I know that $\lim a_n = L \implies \forall\epsilon>0 \ \exists n_0$ such that $n>n_o\implies |a_n-L|<\epsilon$ I ...
0
votes
1answer
29 views

max and min sine function and all intervals

I have a calculus question: The voltage signal from a standard North American wall socket can be described by the equation V(t) = 170sin(120πt), where t is time, in seconds, and V(t) is the voltage, ...
0
votes
1answer
28 views

Solve for bound of $\sigma(n)$ from harmonic series.

I am given the harmonic series, $H_n$. How can I show that for $n\geq 1$, $$\sigma(n)\leq H_n+e^{H_n}\ln{H_n}$$ By the way, if you're not familiar with it, $\sigma(n)$ is the sum of all positive ...
0
votes
1answer
35 views

Is this function increasing/decreasing and convex/concave?

The function is: $$y= 3x + \ln\left(\frac {3x - 4}{x - 1}\right)$$ After differentiating I got: $$y' = 3 + \frac 1{3x^2 - 7x + 4}\;\;\;\;\; \;\;\; y''= - \frac {6x - 7}{(3x^2 - 7x + 4)^2}$$ ...
0
votes
2answers
46 views

If the integral of $c/x$ is $c.log(x)+C$ what is the base?

This question is a follow up to an answer I gave here: How to integrate $1/x$? After the algebra I said that 'This step of course gives the argument of $ln()$ the value $e$ and note that so far we ...
2
votes
1answer
32 views

Evaluate the following trignometric sum

I am interested in the following sum $$\sum_{\text{even } n=-\infty}^{\infty}\left(-\cos^2x\delta_{n,0}+\cos x\left(\frac{1-\cos x}{\sin x}\right)^{|n|}\right).$$ Wolfram alpha returns answer ...
4
votes
2answers
91 views

Evaluation of $ \int_0^\infty\frac{x^{1/3}\log x}{x^2+1}\ dx $

The following is an exercisein complex analysis: Use contour integrals with $-\pi/2<\operatorname{arg} z<3\pi/2$ to compute $$ I:=\int_0^\infty\frac{x^{1/3}\log x}{x^2+1}\ dx. $$ I don't ...
-1
votes
1answer
11 views

Finding the Maximum of a function wrst two variables [on hold]

I need to find a maximum of a function. However, this function has two variables to take into account. My function is $$f=5x\sqrt{\frac{y}{8}}+60$$ Also, I am given the ranges for both $x,y$ as ...
1
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1answer
35 views

Suppose that $\int_{a}^b f(x) g(x) dx = 0$ and show $f = 0$

Suppose $f$ is a continuous function over $[a,b]$. Further suppose that $\int_{a}^b f(x) g(x) dx = 0$ for all continuous functions $g$. Show that $f = 0$. Let $g(x) = \epsilon$. Then since ...
-1
votes
1answer
25 views

Is there a function differentiable $f:\mathbb{R}\to\mathbb{R}$ and a sequence $(x_n)\in \mathbb{R}$ with $f'(x_n)=0$ s

Is there a function differentiable $f:\mathbb{R}\to\mathbb{R}$ and a sequence $(x_n)\in \mathbb{R}$ with $f'(x_n)=0$ such that $x_n\to 0$, but $f'(0)>0$.
0
votes
1answer
38 views

The volume of a Torus

A torus $\mathscr{T}$ with the equation $$z^2 + \left( \sqrt{x^2 + y^2} - 2 \right)^2 = 1.$$ (a) Give an equation with a close line in the plane $Oxz$ where $\mathscr{T}$ is a surface of ...
1
vote
2answers
37 views

A human way to simplify $ \frac{((\sqrt{a^2 - 1} - a)^2 - 1)^2}{(\sqrt{a^2 - 1} - a)^22 \sqrt{a^2 - 1}} - 2 a $

I end up with simplifying the following fraction when I tried to calculate an integral(*) with the residue theory in complex analysis: $$ \frac{((\sqrt{a^2 - 1} - a)^2 - 1)^2}{(\sqrt{a^2 - 1} - a)^22 ...
0
votes
0answers
31 views

Show that the graph of the function z(x,y) = xtan(y) is a minimal surface

I'm really lost on how to do this question. I know we have to use the euler equation to show this, but other than that I'm a bit lost because it's two different variables without a constraint. I was ...
13
votes
7answers
1k views

Linear Algebra with functions

Basically my question is - How to check for linear independence between functions ?! Let the group $\mathcal{F}(\mathbb{R},\mathbb{R})$ Be a group of real valued fnctions. i.e ...
1
vote
1answer
30 views

Find convergence domain of the integral

Find convergence domain of $$\int_0^\infty \! \frac{\cos^2{x}}{x^p} \, \mathrm{d}x$$ I've tried to use $\frac{\cos^2{x}}{x^p} < \frac{1}{x^p}$, but $\int_0^\infty \! \frac{1}{x^p} \, \mathrm{d}x$ ...
-1
votes
1answer
28 views

Let $I=[0,1]$ and $f:[0,1]\to[0,1]$ be a continuous function. Then exist $x\in I$ such that $f(x)=x$. [on hold]

Let $I=[0,1]$ and $f:I\to I$ be a continuous function. Then exist $x\in I$ such that $f(x)=x$. Thanks
0
votes
1answer
23 views

Proof of the Strenghtened Limit Comparison Test

I'm studying on my own using Bonar and Khoury's Real Infinite Series. I understand the proof of the "regular" Limit Comparison Test( a link to google books, page 23 ) but the book doesn't provide a ...
1
vote
1answer
21 views

Find convergence domain of integral

I need to find convergence domain of $$\int_1^2 \! \frac{\ln(x-1)}{(4-x^2)^p} \, \mathrm{d}x$$ I've tried to use estimates like $\frac{\ln(x-1)}{(4-x^2)^p} < \frac{1}{(4-x^2)^p}$ and change of ...
1
vote
1answer
30 views

Prove that if $\lim_{x\to a} f(x) =0$ and $g(x)$ is bounded, then $\lim_{x\to a} f(x)\cdot g(x)=0$ [duplicate]

Question: Prove that if $\lim_{x\to a} f(x) =0$ and $g(x)$ is bounded, then $\lim_{x\to a} f(x)\cdot g(x)=0$ Attempt: I don't really understand the meaning g(x) is bounded. I did this problem in ...
0
votes
1answer
16 views

Zero functions on open interval

Are there non-constant differentiable functions that are zero on an open interval of real line? I've tried using the product integral: $$ f(x) = \exp(\int_0^1 \log(x-u) \mathrm{d}u ) = \frac{x^x ...
0
votes
0answers
9 views

Gradient Vector of Homogeneous Functions

I've been given the definition $f:\mathbb{R}^n \to \mathbb{R}^m$ is homogeneous of degree k if $f(\lambda x)=\lambda^kf(x)$ $\forall x\in\mathbb{R}^n, \lambda>0$ and asked to show $<\nabla ...
1
vote
0answers
9 views

Problem about find the extreme of a function (Multipliers of Lagrange)

Good morning, i have a problem with this: Find the maximum and minimum distances from the origin to the curve $g\left(x,y\right)=5x^{2}+6xy+5y^{2}$ I make this: Function to optimize: ...
0
votes
1answer
28 views

Value of $f'(3^{1/5})$ from the given differential equation

A function $y=f(x)$ satisfies $$xf'(x)-2f(x)=x^4 f(x)^2$$ and given that $f(1)=-6$ and $x$ belongs to all positive real numbers then prove that $f'(3^{1/5}) =8$ I have tried in this way...... Given ...
-1
votes
1answer
39 views

Evaluate Double Integral [on hold]

Evaluate the following double integral: $$\int _0^{2\pi }\int _0^1 \left(x - 2x^2 \sin\left(y\right) \cos\left(x^2+1\right)\right) \text dx\,\text dy$$ Please note that the answer is ${\pi}$
1
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0answers
13 views

I have an problem with the function to optimize with lagrange multipliers

I need help with the restriction of the problem, because i cannot find the function to optimize. The problem: Find the maximum and minimum distances from the origin to the curve ...
1
vote
1answer
30 views

What is the best way to solve $\lim_{n\to \infty}{(e^{i \theta})^n}$?

What is the best way to solve the limit: $\lim_{n\to \infty}{(e^{i \theta})^n}$ $\theta$ is fixed, but you must have a care for cases $\ \theta > 0 , \ \theta = 0 , \ \theta < 0.$ There ...
-1
votes
1answer
52 views

Integral of $\int \frac{xe^x}{\sqrt{1+e^x}} dx$ [on hold]

I need help to solve this: $$\int \frac{xe^x}{\sqrt{1+e^x}} dx$$
2
votes
2answers
97 views

How can $\int_a^b f(x)dx $ exist if either $f(a)$ or $f(b)$ does not exist?

In class, I came across the integral: $$\int_0^1 \frac{dx }{\sqrt{1-x^2}}=\frac{\pi}{2}$$ This is easy enough to prove using a substitution or by recalling the derivative of $\arcsin x$. However, ...
0
votes
0answers
6 views

Solve inequality of composite function

There is a Calculus problem where I got stuck and need a hint to proceed: Let $f(x)=e^x+ \ln(x)-3$ , $x>0$. Solve the inequality $f(\ln(x)-1) <3 $. We can see that $f$ is increasing and that ...
0
votes
2answers
28 views

$lim_{j->\infty} (j^j)/((j+1)^{j})$ [duplicate]

Can someone please explain this limit: $$lim_{j\rightarrow\infty} \frac{j^{j}}{(j+1)^{j}}=\frac{1}{e}?$$ I got it from this series: $$\sum_1^{\infty}\frac{j!}{j^j}.$$
0
votes
1answer
14 views

How can I find the limits of this iterated polar integration?

How can compute the area of the triangle whose corners are at the origin, (1,0) and (1,1). I solved this with r integral first but I could not find the correct limits for theta integral first order. ...
1
vote
2answers
44 views

Why do I get different results for the same integral?

The variables $ a, b, s, c $ are constants, so: $$ \int \left ( a \cos(s + cx) - b \sin(s + cx) \right ) dx = \frac{a\sin(s + cx) + b\cos(s + cx)}{c} +C $$ But if $c=0$ then: $$ \int \left ( a ...
0
votes
1answer
12 views

Cost Functions and differentiation

A firm uses capital $K$,to generate output, $Q$, according to the production function $Q = K^a$.The input price is $r$ and fixed costs are $C_0 > O$. a) Derive the firms cost function in terms of ...
3
votes
4answers
60 views

$\lim_{n \to \infty} (\frac{(n+1)(n+2)\dots(3n)}{n^{2n}})^{\frac{1}{n}}$ is equal to :

$\lim_{n \to \infty} (\frac{(n+1)(n+2)\dots(3n)}{n^{2n}})^{\frac{1}{n}}$ is equal to : $\frac{9}{e^2}$ $3 \log3−2$ $\frac{18}{e^4}$ $\frac{27}{e^2}$ My attempt : $\lim_{n ...
0
votes
1answer
42 views

What does the symbol “$|_{\epsilon=0}$” mean with a derivative? [duplicate]

What does the notation "$|_{\epsilon=0}$" at the bottom of the derivative mean?
0
votes
1answer
45 views

How do I differentiate an improper integral?

I would like to differentiate a function of the type $\int_x^\infty f(x, t) dt$ with respect to $x$ ($f$ real or complex valued). Does differentiation under the integral sign apply? What are better ...
0
votes
1answer
27 views

How to integrate an equation with multiple non-independent variables

I'm a little lost with this particular equation, I have three variables which need to be integrated and can't quite wrap my mind to get the correct result. I have this: $$ \frac{dH}{dt}=8\pi ...
-2
votes
1answer
27 views

Calculate limit of function 4 [duplicate]

Why \begin{equation} \lim_{x\to-\infty}\sqrt{x^2-x-1}-x=+\infty, \end{equation} Thanks
3
votes
1answer
86 views

Is there a closed form of this integral $ \int_0^\infty \sin(xe^{-x})dx\, $?

I have tried by subsititution method and it got more complicate than before. Can anyone help me to evaluate this integral. $$ \int_0^\infty \sin(xe^{-x})dx\,. $$
1
vote
0answers
47 views

How to derive this identity: $\lim _{x\to \infty} (1 + f(x))^{\frac{1}{g(x)}} = \mathrm e^{\lim_{x \to \infty} \frac{f(x)}{g(x)}}$ [on hold]

How to prove this identity: $\lim _{x\to \infty} (1 + f(x))^{\frac{1}{g(x)}} = e^{\lim_{x \to \infty} \frac{f(x)}{g(x)}}$ I've found references to this identity but no derivation This identity is ...
2
votes
1answer
21 views

Having trouble deriving the symbols used in a quadratric approximation problem.

I'm refreshing my calculus by studying MIT OCW's Single Variable Calculus course online. The problem is 2A-11, part of Unit 2 "Applications of Derivatives". It's a problem dealing with quadratic ...
-1
votes
1answer
64 views

How can I find $dy/dx$? [on hold]

What does $dy/dx$ represent? $y = x^5$ $y = x+5$ $y = b$, $b$ is a constant Am I supposed to divide the $y$ by $x$? So, $\frac{y}{x^5}$ and $\frac{y-5}{x}$? If so, what do I do with the third one? ...
0
votes
1answer
27 views

How can I find left and right limits as $x$ approaches nonremovable discontinuity?

The book says to "simplify, find crucial numbers, determine sign in intervals then determine limits." $f(x) = \frac{(x-2)(x-1)}{(x-3)(x-2)}$ I don't understand what the directions mean. I know that ...
1
vote
0answers
25 views

Question regarding terminlogy and wording of the derivative

When doing calculus, we typically say that we "take the derivative of a function f(x)." However, rigorously, f(x) is not a function but rather the value of the function f evaluated at x. Thus, in ...
1
vote
1answer
60 views

Solving $\int_0^2\int_{y/2}^1 ye^{-x^3}\,dx\,dy$ [on hold]

So I have to solve the integral above but I wasn't really sure how to start? I know it is integrable since it is an integral over a nice boundary and that I can solve it using iterated integrals but ...
0
votes
0answers
25 views

A knotch in a tree : do you prefer geometry or integral calculus of volumes to solve? [on hold]

A notch is cut in a cylindrical vertical tree trunk. The notch penetrates to the axis of the cylinder and is bounded by two half-planes. Each half-plane is bounded by a horizontal line passing through ...
2
votes
1answer
29 views

Order of remainder term in Taylor series approximation

I'm having trouble verifying a bound on the remainder term of a Taylor series approximation. I have a $C^\infty$ function $f$ of compact support. Using the two-term Taylor series for $f$ centered at ...
1
vote
1answer
18 views

Equation of a Parabola given starting coordinates, starting angle, and vertex height

I would like to find the equation of a downward-facing parabola given a starting point (x,y) and angle L, and the height of the vertex (k). I started with three equations: $y = ax^2 + bx + c$ ...
-3
votes
1answer
52 views

How to find a general solution using substitution for $\frac{dy}{dx}$ + $e^{2x−3y}$ = −$e^x$ [on hold]

Using the substitution $y=\frac13 \log f$ to find general solution for $\frac{dy}{dx}$ + $e^{2x−3y}$ = −$e^x$
1
vote
1answer
39 views

Dude with taylor polynomial

Good night, i'm working with an problem of polynomial taylor, but i have a problem with the residue. Get a quadratic approximation $f\left(x,y\right)=\sin\left(x\right)\sin\left(y\right)$ near the ...