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

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1
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1answer
14 views

Proving $\int_{0}^{1}\sum_{n=1}^{\infty }\frac{4}{n }(\sin(\frac{2\pi nx}{3}))^3dx=\pi $

Proving $$\int_{0}^{1}\sum_{n=1}^{\infty }\frac{4}{n }\left (\sin\left (\frac{2\pi nx}{3}\right )\right )^3dx=\pi $$
0
votes
1answer
41 views

Most efficient way to integrate $\int_0^\pi \sqrt{4\sin^2 x - 4\sin x + 1}\,dx$?

$$\int_0^\pi \sqrt{4\sin^2 x - 4\sin x + 1}\,dx$$ Please help with this. I cannot do this problem in a definite way.
-6
votes
0answers
34 views

Problem concerning force and work done related to string [on hold]

Let a string has length l inches her in normal condition. To stretch it further 3 inches it requies 5lb force. How much work W will be done to stretch the spring 8 inches more from its normal ...
1
vote
0answers
12 views

Question on two basic limit equations with trig functions

Could someone please explain the reasoning behind the following two results? They appear in exercises in a book that considers them self evident, but I am missing why that is so. Thanks. ...
0
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2answers
25 views

$\sum\limits_{k=1}^{n^2}E(\sqrt{k})\quad n\in\mathbb{N}^{*}$

I found in my archives solution of this exercise Calculate $$\sum\limits_{k=1}^{n^2}E(\sqrt{k})\quad n\in\mathbb{N}^{*}$$ E represent the floor function Solution: they made Let ...
-1
votes
1answer
12 views

Triple integration in cylindrical coordinates (obtaining the formula to integrate)

Can someone help me out with this? For some odd reason, I am able to derive the boundaries but cannot figure out the formula to use. I thought it was r^2 but apparently it's wrong.
0
votes
2answers
29 views

Is this a quadratic factor or a repeated linear factor?

If I need to integrate something like $\frac{1}{y^2(1-y)}$, and I use the method of partial fractions, how do I know whether the $y^2$ in the denominator is a quadratic factor or just a repeated ...
2
votes
1answer
18 views

Directional derivative of a function

Feel like I may have gone wrong somewhere with this question: Find the directional derivative of the function $f(x,y) = \displaystyle\dfrac{2x}{x-y}$ at the point $P(1, 0)$ in the direction of the ...
1
vote
1answer
33 views

Let $y^2 = x^3 + Ax + B$ be a curve and $y = m(x - x_1) + y_1$ tangent at $x_1$. Why is $x_1$ then a double root?

Suppose we have a function $y^2 = x^3 + Ax + B$ which we differentiate implicit to find $$\frac {dy} {dx} = \frac {3x^2 + A} {2y}$$ Now suppose we know a point $(x_1,y_1)$ on the curve. Define $$y = ...
0
votes
2answers
36 views

how to find the global minimum value of the function?

Let $$f(x)=x^4+4x^3+100$$ Find the global minimum value of the function $f$, that is, give the minimum value of $f$ and the value of $x$ for which this occurs. This is not a homework problem. I have ...
2
votes
1answer
34 views

indefinite integrals equal imply integrands equal?

if there is an indefinite integral equality does it mean that there is an integrand equality? $$\int f(x)\,dx = \int h(x)\,dx \quad \overset{?}{\Longrightarrow} \quad f(x) = h(x)$$ I know that $$\int ...
0
votes
2answers
20 views

Help with limits of 2 variables

How do I find the following limits? $\displaystyle\lim_{(x,y)\to (0,0)}\dfrac{\sqrt{xy+1}-1}{xy}$ and $\displaystyle\lim_{(x,y)\to (2,0)}\dfrac{sin(xy)}{y}$
-1
votes
1answer
54 views

Compute$\int\limits_{0}^{2} \sqrt{x^2-2x+2}\ln(2+x)dx$. [on hold]

Compute: $\displaystyle \int\limits_{0}^{2} \sqrt{x^2-2x+2}\ln(2+x)dx$.
0
votes
0answers
57 views

A integration problem [duplicate]

A integration problem: $\quad \displaystyle \int_0^1 \frac {\ln (1+x)}{1 + x^2} \, \mathrm dx$ I have no idea how to deal with it, I hope your answers.
0
votes
1answer
26 views

Geometric Interpretation of Liouville's Theorem?

The only bounded entire functions in $\mathbb{C}$ are constants. Could someone please give me a geometric interpretation of the theorem above?
1
vote
1answer
41 views

Is it legit to assume this?

I have the following sequence: $$\lim\limits_{n \to \infty}\frac{(-3)^{5n}-2(-3)^n+2}{(-3)^{3n}+(-3)^n+2}$$ I had to find the limit. So I said that we can just look at the highest power of both the ...
-2
votes
0answers
14 views

Probabilitty calculation [on hold]

offee Breaks Are you a coffee drinker? If so, how many coffee breaks do you take when you are at work or at school? Most coffee drinkers take a little time for their favorite beverage, and many take ...
0
votes
2answers
74 views

Devriative of $\frac {1} {\sqrt{x+1}}$ using first principle

I am stuck at the first step: $$f(x)=\frac{1}{\sqrt{x+1}}$$ $$f'(x) = \lim_{h\to0} \frac{\frac{1}{\sqrt{h+x+1}}-\frac{1}{\sqrt{x+1}}}{h}$$ I tried multiplying by the conjugate but that didn't get me ...
0
votes
1answer
42 views

$\int_0^{\frac{\pi }{4}} \sec ^5(x) \, dx$

I am wondering how to use integration by part and the reduction formula to solve this integral $$\int_0^{\frac{\pi }{4}} \sec ^5(x) \, dx$$ The answer in the book is: $\frac{\sqrt{2} ...
0
votes
0answers
13 views

Calculate the volume of a cube in spherical coordinates

I need to calculate the volume of a cube having edge length $a$ by integrating in spherical coordinates. Any help?
0
votes
2answers
25 views

Are there extremely discontinuous functions?

Are there any functions $f:\Bbb R\to \Bbb R$ with the following property: For any $x_0\in \Bbb R$, any $\delta >0$ and any $\epsilon>0$ there is an $x$ with $|x-x_0|<\delta$ such that ...
0
votes
1answer
11 views

Inequality Graph Behavior

Let there be $\frac{(x-2)(x-4)}{x(x-1)}\leq 0$ the answer is $0<x<1$ or $2\leq x\leq4$ I understand that the way to find the solution is to look for the numbers that set to zero the right ...
0
votes
3answers
66 views

How to find the limit [1/x] when x goes to -1/3 from the right?

How to find the limit $$\lim\limits_{x \to -\frac13^+}\bigg\lfloor \frac1x \bigg\rfloor$$ where the brackets denote the greatest integer less than or equal to $\frac1x$? Note: an earlier version of ...
1
vote
1answer
15 views

Convergence for x values of function

I'm trying to determine at what x values an infinite series of the function $-sin(nx)$ converges. I think I may be over thinking this relatively simple question. But I just want to verify that I'm on ...
0
votes
2answers
37 views

How to disprove a limit of series?

Let $$a_{n} = \dfrac{7n^{3} - 3n^{4} -1}{4n^{2} + 3}$$ I want to show that this series has no limit. So, I know I need to show that $\exists \epsilon >0, \forall N\exists n > N $,$ |a_{n} - L | ...
-3
votes
0answers
42 views

Why $\frac{1}{2^+} = 0$ [on hold]

I can see why Why $\frac{1}{0^+} = 0$ because denominator is very very tiny but why $\frac{1}{2^+} = 0$ ?
2
votes
0answers
13 views

Smooth and not smooth surface parametrisation

Give an example of two parametrisations of the same surface, one which is smooth, one which is not.
0
votes
2answers
6 views

corresponding system of equation of the given solution space

The following question seems to me interesting. it gives solution space and required the corresponding system of equation. The question is the following: Consider the vectors in $R^4$ defined by ...
0
votes
2answers
24 views

boundedness of the sequence $a_n=\frac{\sin (n)}{8+\sqrt{n}}$

How can i prove boundedness of the sequence $$a_n=\frac{\sin (n)}{8+\sqrt{n}}$$ without using its convergence to $0$? I know since it is convergent then it is bounded.
3
votes
2answers
65 views

Whats the derivative of $\sqrt{4+|x|}$ using first principle

Here is my attempt: $$f(x)=\sqrt{4+|x|}$$ $$f`(x) = \lim_{h\to0} \frac{\sqrt{4+|x-h|}-\sqrt{4+|x|}}{h}$$ multiplying by the conjugate: $$\lim_{h\to0} ...
0
votes
1answer
32 views

Value of indeterminate form — $a_n \to \infty \wedge b_n \to 0$, $\lim_{n\to\infty}a_n\cdot b_n = ?$

$A_n$ and $B_n$ are sequences and $B_n\to 0$ and $A_n\to\infty$. $\lim_{n\to\infty}A_nB_n$ should be equal to $0$ OR $+\infty$ OR $-\infty$? I need to answer yes/no about this problem. I know the ...
2
votes
7answers
270 views

problem with simple induction proof

i want to prove that $\forall n\geq 5$ $$2^{n}-1 > n^{2}$$ so the basis is trivial, and in the induction step (n+1), i stuck. i get : $(n+1)^{2} = n^{2} + 2n + 1 < (2^{n} -1)+ 2n+1 = 2^{n} ...
0
votes
1answer
22 views

Values of x for which series can converge

I'm given a function $sin(nx)/(n^2)$ and I'm trying to find for which values of x the infinite series for this function would converge. It's easy to see that $sin(nx)$ is always between (-1,1), so ...
0
votes
1answer
51 views

A limit of a sequence [duplicate]

I'm trying to prove the following limit $$(\frac{2^n}{n!}) \to 0$$ But it seems difficault to me. How can I prove it? Thanks.
3
votes
0answers
48 views

The following series is an irrational number [duplicate]

$(*)$ Let $\{ x_n \}$ be a sequence where $x_n$ is either $1$ or $-1$. Then $$ \sum_{n=0}^{\infty} \frac{ x_n}{n!} \; \; \; \text{is irrational}$$ This problem arise when I was trying to prove that ...
0
votes
1answer
33 views

Evaluate the integral, and then take the derivative of it.

I'm mostly curious as to if the way I've went about solving this is correct, or if there is a more simple way to get the answer. So I first evaluated the top section And when I did that I got ...
0
votes
1answer
25 views

Limits approaching from both sides go to infinity

Suppose that $\lim_{x \to a} f(x) = \infty$. Prove that we then have $\lim_{x \to a^+} f(x) = \infty$ and $\lim_{x \to a^-} f(x) = \infty$ from the definitions using epsilon-delta methods.
1
vote
2answers
54 views

Computing the limit of a summation of sequence

How to compute the limit $$\lim _{n\rightarrow \infty }\left( \dfrac {1}{\sqrt {n^2+1^2}}+\dfrac {1}{\sqrt {n^2+2^2}}+\cdots+\dfrac {1}{\sqrt{n^2+n^2}}\right)$$ The answer is $$\ln ( \sqrt{2}+1)$$ ...
4
votes
5answers
106 views

How to integrate $\int_{0}^{1}\ln\left(\, x\,\right)\,{\rm d}x$?

I encountered this integral in the quantum field theory calculation. Can I do this: $$ \left. \int_{0}^{1}\ln\left(\, x\,\right)\,{\rm d}x =x\ln\left(\, x\,\right)\right\vert_{0}^{1} ...
1
vote
3answers
56 views

Evaluation of $\int \frac{x^4}{(x-1)(x^2+1)}dx$

Evaluation of $\displaystyle \int \frac{x^4}{(x-1)(x^2+1)}dx$ $\bf{My\; Try::}$ Let $$\displaystyle I = \int\frac{x^4}{(x-1)(x^2+1)}dx = \int \frac{(x^4-1)+1}{(x-1)(x^2+1)}dx = \int\frac{(x-1)\cdot ...
0
votes
0answers
18 views

How should i go about proving an expression of this kind?

Lets say i have a complete bell polynomial that is equal to a summation such that $$ B_n(d_1,d_2,\cdots,d_n) = \sum_{k=0}^{n}[g(x)^{-k} h(k)] $$ Where $d_n = \frac{d^n}{dx^n}[f(x)\ln(g(x)]$ and ...
1
vote
1answer
28 views

Critical points of an integrated function

Let $$F(x)=-\int_{0}^{x^2}\frac{2}{3+e^t}dt$$ Find all critical points of $F(x)$ and determine whether they are minima, maxima or points of inflection. Prove that $F(300)>F(310)$. First I ...
4
votes
2answers
84 views

How to do integration of this?

$$\int_0^\infty\frac{x \sin x }{(x^2 + a^2)(x^2 + b^2)}dx\quad\quad a > b > 0$$ I have no idea how to compute this. Any help would be greatly appreciated.
0
votes
1answer
27 views

Big-Omega proof using L'Hopital's Rule?

Prove or disprove: $15n^2$ is in $\Omega(3 \times 2^n)$ So we'd have to prove or disprove this statement: $$ \exists c \in\mathbb{R}^+,\,\exists B\in\mathbb{N}, \forall n \in\mathbb{N}, n ≥ B ...
0
votes
0answers
22 views

Converting a word problem to algebra

This is a forming of an equation, which I haven't been able to get my head around. I have a worked solution to this problem. Question: For $x\in\mathbb{R}^m$ and $\epsilon>0$, show that ...
0
votes
4answers
58 views

Calculating $\lim_{x\to\infty}\sqrt{x^2 + 5x} - x$ [on hold]

How to evaluate following limit $$\lim_{x\to\infty}\sqrt{x^2 + 5x} - x$$ I'm not sure how to factor and solve.
1
vote
0answers
33 views

Examples using LIATE rule for Integration by parts

Popular textbook contain many examples of integrals which can be computed by parts using the LIATE Rule. However there is almost no example of the case LI, that is a logarithmic function times an ...
1
vote
2answers
28 views

Show a limit of a bounded function is 0 then solving the integral

$B=\{x^2+y^2\le1\}$, & for all $\delta>0$, there is $B_\delta=\{x^2+y^2\le\delta\}$. $f$ is a continuous function and $\|\nabla f\|\le1$ on $B$, and suppose $\frac{\partial^2f}{\partial ...
2
votes
1answer
36 views

Great books on all different types of integration techniques

It's coming up to Christmas so I can ask to have all the books I can't afford from begrudging relatives! I'm really interested (mainly from looking at some of the answers cleo and other fantastic ...
3
votes
4answers
53 views

Help me, a doubt $f(x)=\cot^{-1} \frac{1-x}{1+x}$

I have a doubt $$f(x)=\cot^{-1} \frac{1-x}{1+x}$$ $$f´(x)=\frac{1}{(\frac{1-x}{1+x})^2}\cdot\frac{(-1)(1+x)-(1-x)}{1+\frac{(1-x)^2}{(1+x)^2}}$$ mm this could to be really easy but I do not ...