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

learn more… | top users | synonyms

21
votes
3answers
658 views
2
votes
1answer
52 views

Computing double integral

Find $$\iint\limits_D \sqrt{(x-10)^2+y^2}\hspace{1mm}dA$$ where $\{(x, y)\in D \mid x^2+y^2\leq 10^2\}$. I am not sure how to start, every way I have tried so far, ends up into something ugly. All ...
1
vote
2answers
65 views

Check my proof: Big O notation

I was asked the following: We are given the functions $f(n)=n^{10\log(n)}$ and $g(n)=(\log (n))^n$. Which of the following statements is true: $f(n)\in\mathcal{O}(g(n))$, $f(n) \in ...
1
vote
2answers
55 views

Fundamental Theorem of Calculus and limit

I've been reading through a paper and my question has essentially came down to this: Let $f(\beta) \to M$ as $ \beta \to 0$ and $f(\alpha) \to 0$ as $\alpha \to \infty.$ Prove that ...
1
vote
3answers
162 views

Value of the integral $\int_{\mathbb{R}} \frac{x\sin {(\pi x)}}{(1+x^2)^2}$

How do we evaluate the integral $$I=\displaystyle\int_{\mathbb{R}} \dfrac{x\sin {(\pi x)}}{(1+x^2)^2}$$ I have wasted so much time on this integral, tried many substitutions $(x^2=t, \ \pi x^2=t)$. ...
3
votes
1answer
68 views

Evaluation of $ \int \frac{x^2+n(n-1)}{(x\cdot \sin x+n\cdot \cos x)^2}dx$

Evaluation of $\displaystyle \int \frac{x^2+n(n-1)}{(x\cdot \sin x+n\cdot \cos x)^2}dx$ $\bf{My\; Solution:}$ Using $\displaystyle (x\cdot \sin x+n\cdot \cos x) = ...
5
votes
1answer
158 views

Is this a valid proof of $\lim _{n\rightarrow \infty }(1+\frac{z}{n})^n=e^z$?

Define the function $g_n\left(z\right)=\left(1+\frac{z}{n}\right)^n$ for $\:n\in \mathbb{R^+}$. Then ...
1
vote
3answers
108 views

Evaluate $\int_{1}^{e}\frac{u}{u^3+2u^2-1}du.$

I'm trying to solve $$\int_{1}^{e}\frac{u}{u^3+2u^2-1}du.$$ My first approach was to factorise and then do a partial integration. However the factorisation ...
2
votes
0answers
54 views

What is a good technique for evaluating this double integral?

The integral is: $ \int_0^1 \int_0^1 \frac{x^2 - y^2}{(x^2 + y^2)^2} dxdy $. I'm having difficulty finding an appropriate technique for evaluating it. I initially thought that polar coordinates ...
6
votes
2answers
98 views

$f$ is twice differentiable, $f + 2 f^{'} + f^{''} \geq 0$ , prove the following

Let $ f : [0,1] \rightarrow R$. $f$ is twice diff. and $f(0) = f(1) = 0$ If $f + 2 f^{'} + f^{''} \ge 0$ , prove that $f\le 0$ in the domain. Don't give complete solution, only hints.
0
votes
2answers
99 views

Compound interest with a compounding interest rate

I have an investment which pays 3% interest (r) annually but it also increases the interest rate every year by 5% (g). I re-invest all interest payments at the start of each year. How many years (t) ...
6
votes
0answers
157 views

An incorrect answer for an integral

As the authors pointed out in this paper (p. 2), the following evaluation which was in Gradshteyn and Ryzhik (sixth edition) is incorrect (and has been removed). $$ ...
0
votes
2answers
38 views

Maximum value of $f(x) = \left|\sqrt{\sin^2 x+2a^2} - \sqrt{2a^2-1-\cos^2 x}\right|\;\;,$ Where $a,x\in \mathbb{R}$

Calculation of Maximum value of $\displaystyle f(x) = \left|\sqrt{\sin^2 x+2a^2} - \sqrt{2a^2-1-\cos^2 x}\right|\;\;,$ Where $a,x\in \mathbb{R}$ $\bf{My\; Try::}$ We Can Write It as $f(x) = ...
7
votes
3answers
683 views

How do you calculate this limit $\lim_{n\to\infty}\sum_{k=1}^{n} \frac{k}{n^2+k^2}$?

How to find the value of $\lim_{n\to\infty}S(n)$, where $S(n)$ is given by $$S(n)=\displaystyle\sum_{k=1}^{n} \dfrac{k}{n^2+k^2}$$ Wolfram alpha is unable to calculate it. This is a question from a ...
0
votes
2answers
35 views

Could anybody provide a more detailed explanation of a tangent equation in its general form?

In my textbook I'm currently at the topic of a tangent line to an ellipsis and hyperbola. And there I've encountered this statement: If a curve has an equation $$ y = f(x) $$ then an equation of a ...
2
votes
2answers
90 views

Evaluate $\lim_{x \to 0} (x\lfloor\frac{1}{x}\rfloor)$

Evaluate $\lim_{x \to 0} (x\lfloor\frac{1}{x}\rfloor)$ I'm trying to solve it by using the squeeze theorem but I'm stuck. I'm looking for a function $g(x)$ such that $g(x) \leq ...
3
votes
2answers
274 views

What is the value of this double integral?

Let $C$ be the subset of the plane given by $$ C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | \ 0 \leq x^2 + y^2 \leq 1 \}.$$ Then what is the value of the double integral $$ \int_{C} \int (x^2 + y^2) ...
0
votes
2answers
95 views

How to evaluate this double integral?

Let $C$ be the subset of the plane given by $$C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | -1 \leq x = y \leq 1 \}. $$ Then how to evaluate the double integral $$ \int_C \int (x^2+ y^2) dx dy? $$ My ...
2
votes
3answers
294 views

Analysis Question from Berkeley Problems in Mathematics

I'm wondering if the following is correct. The original question(1.1.21, Fa96) asks to prove that \begin{align*} f''(x) = \lim_{h\rightarrow0}\dfrac{f(x+h) - 2 f(x) +f(x-h) }{h^2} \end{align*} I ...
2
votes
2answers
270 views

Intermediate Value Theorem, an application problem [duplicate]

The question is: if $$\frac{a_0}{1}+\frac{a_1}{2}+\dots+\frac{a_n}{n+1}=0$$ then, $a_0+a_1x+\dots+a_nx^n=0$ for some $x$ in the interval $[0,1]$. My approach is to let $f(x)=a_0+a_1x+\dots+a_nx^n$ ...
0
votes
2answers
51 views

What is the area bounded by these curves?

Let $f(x) \colon = x^2$, $g(x) \colon= x+1$. Then what is the area bounded by the graphs of $f$ and $g$ between the vertical lines $x= -1$ and $x= (1+\sqrt{5})/2$? My effort: Since $$ f(x) - g(x) ...
3
votes
3answers
148 views

Limit of a recursively defined sequence

Let $\{a_n\}_{n\in\mathbb{N}}$ be a sequence of real numbers such that: $$\forall n\in\mathbb{N},\quad a_{n+1}=a_n + e^{-a_n}.$$ Prove that: $$\lim_{n\to+\infty}\frac{a_n}{\log n}=1.$$
0
votes
3answers
88 views

If $x^n = f(x)g(x)$ as complex polynomials, must $f,g$ be of the form $x^m$?

If $x^n = f(x)g(x)$ as complex polynomials, must $f,g$ be of the form $x^m$? i.e. $x^n = x^mx^l$ where $m+l = n$. This is quite trivial, but I want to make sure I didn't miss anything. Attempt: If ...
1
vote
1answer
65 views

Finding the equation of more than one tangent line

I ran into a problem I have no idea how to begin, maybe you guys can help me out. I think maybe it has something to do with parametric equations? But this is just a guess. Find equations of both the ...
1
vote
3answers
52 views

Is the following series converging or diverging. $\sum_{n=1}^{\infty}\dfrac{n+4^n}{n+6^n}$

I know one solution. That is by Doing comparison with $\dfrac{4^n+4^n}{6^n}$ Wondering if there are more ways to do it
1
vote
1answer
61 views

Infinity limits using definition

Prove using the definition $$\lim_{x \to 0-} \frac{1}{x}$$ Definition : To all M>0 exists $\delta>0$ so to all x that appiles $-\delta<x<0$ appiles $\frac{1}{x}<-M$ EDIT: ...
2
votes
2answers
44 views

Calculus III: Find the points of the curve…

I have to find the points of the curve $$r\left( t \right) =\left( t,{ t }^{ 2 },{ t }^{ 3 } \right) $$ where the osculating plane passes through the point $\left( 2,-\frac { 1 }{ 3 } ,-6 \right)$.
24
votes
2answers
559 views

Prove $_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47$

I discovered the following conjecture numerically, but have not been able to prove it yet: $$_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47.\tag1$$ The equality holds ...
5
votes
4answers
474 views

Another method for limit of $[e-(1+x)^{1/x}]/x$ as $x$ approaches zero

I have solved this limit: $\lim_{x \rightarrow 0} \frac{e-(1+x)^{\frac{1}{x}}}{x}$ using L'Hopital's rule and series expansion. Do you have other method for solving it?
0
votes
1answer
27 views

Coordinates rotation and function change

In the Cartesian coordinates $(x,y)$, I have a vector function $\bar{f}(x)=\hat{x}A\cos(yk)$, where $A$ and $k$ are constants. I make now a 45 degrees rotation (in the same plane) to the new set of ...
0
votes
1answer
128 views

Rope question - integration

A 50-lb bucket is at the bottom of a 100-ft well. A 200 lb rope (also 100 ft long) is tied securely to the bucket. We will use rope to lift this bucket out of the wall, at a rate of 1 foot every ...
-1
votes
3answers
38 views

Optimization with contraint

Given the value K with constraint x+y = K, what can be the maximum value of x*y be? How did they derive this answer? It is equivalent to finding the maximum value of x*(K-x), which will happen when x ...
5
votes
3answers
693 views

Why does this infinite series equal one?

Why does $$\sum_{k=1}^\infty \binom{2k}{k} \frac{1}{4^k(k+1)}=1$$ Is there an intuitive method by which to derive this equality?
1
vote
2answers
62 views

Proving second derivatives

I'm asked to prove a theorem (if that is the right word) about double derivatives. I'm still struggling with understanding Leibniz notation and I could use a push in the right direction. It's easy ...
2
votes
2answers
119 views

Wrong interpretation of the indefinite integral

This might sound very useless but I'd like to see what you think. Bear in mind that I'm just a novice student. if $f$ is the original function, then it could be found this way $C+\int f'(x)\, ...
2
votes
2answers
80 views

Convergence/Divergence of a the series $\sum_{k=1}^{\infty} a_k$, where $a_1=1$ and $\forall 1\leq k\in\mathbb{N},a_{k+1}=\cos(a_k)$

I got this question: Determine wether the series $\sum_{k=1}^{\infty} a_k$ absolutely converges, conditionally converges or diverges, where $a_1=1$ and for each $1\leq k \in\mathbb{N}$, ...
1
vote
2answers
62 views

Calculus - Limit calculate help

I'm having a problem to solve this limit. $$\lim_{x \to \pi/4} \frac{\tan x-1}{\sin x-\cos x}$$ $\lim_{x \to \pi/4} \frac{\tan x-1}{\sin x-\cos x}$ = $\lim_{x \to \pi/4} \frac{\frac{\sin x}{\cos ...
0
votes
3answers
249 views

Can you factor before finding derivative?

Say the function is $y=\frac{x^2-1}{x-1}$ Can you factor functions before finding the derivative or does that not work?
0
votes
1answer
43 views

Figuring the function $f(x)$ from given information

Here is the given information in my question, So, what my question inform is that there is a cubic polynomial function (i.e $f(x)$) which has local maxima at $x=-1$. While that for $f'(x)$, it's ...
11
votes
2answers
261 views

An exercise from my brother: $\int_{-1}^1\frac{\ln (2x-1)}{\sqrt[\large 6]{x(1-x)(1-2x)^4}}\,dx$

My brother asked me to calculate the following integral before we had dinner and I have been working to calculate it since then ($\pm\, 4$ hours). He said, it has a beautiful closed form but I doubt ...
1
vote
3answers
83 views

Non integer derivative of $1/p(x)$

I need to find the $k$'th derivative of $1/p(x)$, where $p(x)$ is a polynomial and $k\in\mathbb{R}$ It dosen't have to be an explicit formula, an algorithm which finds a formula for some $k$ is fine. ...
19
votes
1answer
434 views

Integral: $\int_0^{\pi} \frac{x}{x^2+\ln^2(2\sin x)}\,dx$

I am trying to solve the following by elementary methods: $$\int_0^{\pi} \frac{x}{x^2+\ln^2(2\sin x)}\,dx$$ I wrote the integral as: $$\Re\int_0^{\pi} \frac{dx}{x-i\ln(2\sin x)}$$ But I don't find ...
1
vote
3answers
66 views

Why does my derivation of $\mathcal{L(\frac{f(t)}{t})}$ lead to a wrong answer?

I'm trying to prove that $$\mathcal{L(\frac{f(t)}{t})(s)} = \int_s^{\infty}\mathcal{L(f(t))}(u)du$$ Here's my attempt: $$\mathcal{L(\frac{f(t)}{t})}(s)=\int_{0}^{\infty} \frac{f(t)}{t}e^{-st}dt$$ ...
0
votes
2answers
45 views

Problem with proof that positive infinite series are commuative

Proof from real analysis book: Let $\sum_{n=0}^{\infty}a_n$ converge, where $a_n \geq 0, n \in \mathbb{N}$. Then the series $$ \sum_{k=1}^{\infty}a'_k = a'_1 + a'_2 + \cdots + a'_k + \cdots $$ ...
3
votes
2answers
75 views

Finding $f(x)$.

If $$f(x)=1+x+x^2+\displaystyle\int_{0}^{x}e^k f(x-k) dk$$ then how do we find the function $f(x)$? Is there a way to solve it, with or without arriving at a differential equation? This a homework ...
1
vote
0answers
62 views

Maximize profit

My book (George F. Simmons - Calculus with analitic geometri) hasthe following question: An library could buy from the book publisher the book "Rituals" with a cost of $40.0$ each. The manager from ...
1
vote
1answer
37 views

Is it possible to have a inflection on a vertical asymptote?

I found the derivative of a function to be f'(x)=8/x^3 and thus its second derivative as f''(x)=0/3x^2. After setting the second derivative to zero and doing the substitution into the parent function, ...
6
votes
1answer
64 views

Radius of convergence continuous?

Let $ f: [0,1] \rightarrow \mathbb{R} $ be analytic. Let $ r_f(x) $ be the radius of convergence of $ f $ at $ x $. Is $ r_x(f) $ continuous? Alternatively, is there an $ r_{min} $ I can choose so ...
2
votes
3answers
141 views

Can an inflection exist if there's no max/min?

Very quick question: if a function doesn't have a maximum nor minimum, can it still have a point of inflection? I believe that these two go hand in hand and without one you can't have the other but ...
12
votes
4answers
303 views

A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series \begin{equation} \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right) \end{equation} I have no idea how to ...