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

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1
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3answers
48 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
45 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
35 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 $\left( 2,-\frac { 1 }{ 3 } ,-6 \right)$.
10
votes
1answer
194 views
+100

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
184 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 Hopital rule and series expansion. Do you have other method for solving it? Thanks.
0
votes
1answer
12 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
25 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
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3answers
30 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
590 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
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2answers
37 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
89 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
54 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
56 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
63 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
36 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 ...
8
votes
2answers
159 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
50 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. ...
8
votes
1answer
142 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
48 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
37 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 $$ \sum_{k=1}^{\infty}a'_k = a'_1 + a'_2 + \cdots + a'_k + \cdots $$ ...
0
votes
0answers
56 views

Evaluate $\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx$ [duplicate]

As the title shown, how to evaluate the indefinite integral $$\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx\ ?$$ Thanks.
3
votes
2answers
64 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
30 views

Maximize profit

my book (George F. Simmons - Calculus with analitic geometri) have the 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
30 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
0answers
45 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
87 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 ...
7
votes
4answers
144 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 ...
1
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1answer
43 views

Alternative function definitions

If you go to the wikipedia page on the sine function or the log function you'll find a number of different definitions of these functions. I know that what defines a function are it's values, for ...
14
votes
1answer
231 views

Prove ${\large\int}_0^\infty\left({_2F_1}\left(\frac16,\frac12;\frac13;-x\right)\right)^{12}dx\stackrel{\color{#808080}?}=\frac{80663}{153090}$

I discovered the following conjectured identity numerically (it holds with at least $1000$ digits of precision). How can I prove it? ...
1
vote
1answer
67 views

How do you calculate the higher order deriviative of $x^n$?

I need some help figuring this out: $$D^n x^n =\ ?$$ $$D^1 x^n = n x ^{n-1} $$ $$D^2 x^n = n(n-1) x^{n-2}$$ $$D^n-1 =\ ???$$ Where does the $2$ go? and why does $x^n$ become $x$? ...
0
votes
3answers
83 views

How do I find a point on the surface of a sphere

How do I find a point on a sphere knowing its radius and center point ? I have a sphere: $$x^2+(y-1)^2+(z+3)^2=16$$ Obviously its center point is $(0,1,-3)$ and its radius is $4$. I am asked to find ...
0
votes
2answers
48 views

Volume of trig function around y-axis

I have this question and it's the first kind of question I'm doing involving finding volume so I just would like some help solving this question: Find the volume created by revolving the curve $ \ ...
4
votes
3answers
132 views

Evaluate $\int_0^1\frac{x^a-x^{-a}}{x-1}dx$

I have heard that: $$\int_0^1\frac{x^a-x^{-a}}{x-1}dx=\frac1 a-\pi\cot(\pi a)$$ when $-1<a<1$. How would I prove this? That doesn't have an elementary indefinite integral, but the definite ...
10
votes
2answers
319 views

Evaluating $\int_0^\pi\arctan\left(\frac{\ln\sin x}{x}\right)\mathrm{d}x$

I found the following integral as a by product of another one. It has a nice closed form. $$ \int_{0}^{\pi} \arctan\left(\ln\left(\sin x \right) \over x\right)\,{\rm d}x $$ Mathematica and ...
1
vote
1answer
44 views

Area of solid revolution using integration.

When we calculate the volume of a solid generated by rotating a curve around $x$-axis, We divide it into disks. So ,we get $dv = \pi r^2 dx$. where $r=y$ and then we integrate. That OK, but when ...
0
votes
4answers
158 views

Find whether the following series converges or diverges $\sum_{n=1}^{\infty}\frac{\ln n }{\sqrt{n}}$ [closed]

Looking for a witty answer. I can see that the given series converges by AST. All Ideas Appreciated
1
vote
1answer
42 views

Finding the zeros of trionometric polynomails.

I have a question about something I've struggled with for a while: Finding the zeros of trigonmetric polynomials. Let me show you a problem I am solving and you guys can tell me if I got the right ...
0
votes
1answer
26 views

Showing that derivative of conjugate is conjugate of derivative, using chain rule

I'm trying to show that the derivative of the conjugate is the conjugate of the derivative, i.e. $\dfrac{d[f(x)^*]}{dx} = [\dfrac{df(x)}{dx}]^*$, using the chain rule. Calling the conjugate * ...
0
votes
0answers
16 views

Volume of Solid of Revolution

This problem is giving me some trouble: The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method. x = (y − 5)^2, x = 4; ...
1
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0answers
30 views

Question about writing a proof with continuous functions [duplicate]

How would I write a proof for this example? We know that all polynomial functions on the reals are continuous by using the sequential definition of continuity. In particular, we know that the ...
3
votes
2answers
41 views

Find two functions given properties of their limits

I've been stuck on a seemingly simple problem regarding limit properties of two functions: Find two functions, f(x) and g(x), given the following properties: $$\lim\limits_{x \to 7} f(x)=0$$ ...
2
votes
2answers
55 views

Duo Fresnel-like integrals $(??)$

I really wonder how I can prove the following integrals. $$\int_0^\infty \sin ax^2\cos 2bx\, dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}\left(\cos \frac{b^2}{a}-\sin\frac{b^2}{a}\right)$$ and ...
1
vote
2answers
61 views

Cannot find the limit because the denominator is $0$

I need to solve one limes equation but I cannot find the way. Can you help me and explain me the way, please. The equation is this: $$\lim_{x\to -2} ...
3
votes
3answers
449 views

Is this proof of the fundamental theorem of calculus correct?

A student friend of mine recently gave me a proof of the fundamental theorem of calculus which does not correspond to any I can find in the textbooks. It starts by considering an increasing continuous ...
0
votes
1answer
66 views

solving the equation

let there be a function $ f(x)= \ln x-kx^2, k>0$ determine for whihc values of $ k$ ,the equation $f(x)=0.5$ has a single solution; attemp to solve: $$0.5 = \ln x-kx^2$$ $$kx^2 +0.5 = \ln x $$ ...
0
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1answer
73 views

Seemingly hard integrals which are made easy via differentiation under the integral sign a.k.a Feynman Integration [closed]

I recently discovered Differentiation under the integral sign a.k.a Feynman Integration and I read an article which says it can be substituted for contour integration. Therefore, I am assuming this ...
0
votes
5answers
77 views

Proving integrals of $f(\sin(x))$ and $f(\cos(x))$ are equal

Prove that for every continuous function $f$, $\displaystyle\int_{0}^{\pi/2} f(\sin(x))dx = \int_{0}^{\pi/2} f(\cos(x))dx $. I am not really sure how to tackle this, but nevertheless here is my ...
0
votes
2answers
73 views

How to evaluate a limit with high school math?

So my textbook's explanation of the derivative of e is very sketchy. They used lots of approximations and plugging things into the calculator. Basically I want to know how you can work out as h ...
1
vote
2answers
62 views

Can anyone prove that this is an envelope of a parabola?

Based on my last question I learned that this is an envelope of a parabola What is this geometric pattern called? But how can I prove it ?
2
votes
4answers
96 views

Antiderivative of $\frac{1}{1+\sin {x} +\cos {x}}$

How do we arrive at the following integral $$\displaystyle\int\dfrac{dx}{1+\sin {x}+\cos {x}}=\log {\left(\sin {\frac{x}{2}}+\cos {\frac{x}{2}}\right)}-\log {\left(\cos {\frac{x}{2}}\right)}+C\ ?$$