**82.437 calculus questions.**

The following definition of the derivative of a function is taken from Analysis on Manifolds by James Munkres:
Now what I'm confused about is what definition of a limit is being used above. There are ...

Please tell us about the convergence or divergence in the series?
\begin{matrix} \sum_{n=1}^{\infty} {\frac {5^{n}}{(3^{n+1})n}}\end{matrix}
thanks

So I was going through some questions, and these two made me curious:
1) An integrable function is always bounded
2) If a function is defined on an unbounded interval, then it cannot be integrable.
...

Trying to
Find trivial solutions of $\int_a^bf(x) dx=c , f(x)=?$
2.Generalise 1, in number of possible ways and then generalise those ways as well.
Motivation : there are unaccountably many ...

In this paper, they analyse a second-price auction game with an option for the first player to bribe the second player to stay out of he game.
In proposition 2, they show that for any $b\in(0, \Bbb{E}...

I want to evaluate the following integral using the tangent half-angle substitution $t = \tan (\frac{x}{2})$: $$\int_0^{2 \pi} \frac{1- \cos x}{3 + \cos x} ~dx$$ However, making the substitution ...

Why is $${d\sum_{j=0}^\infty {j(x/c)^{j-1}}\over d(x/c)} = {d\sum_{j=0}^\infty (x/c)^j\over d(x/c)}$$ true?

Say we want to solve $\max_x F(x)$. Say we use a (partially random) method that gives simpler approximations of $F$, say $F_1, ..., F_k$. Think simulation or something like that.
We can then solve $\...

Use method of Bernoulli's equations to solve the equation:
$$ x(dy/dx) = x^4y^3 - y $$
I don't really can understand how to use Bernoulli's equations as I had it only once and I wasn't able to ...

Prove, by induction, that the sum \begin{matrix} \sum_{k=1}^{2n} (-1)^k (2k+1)\end{matrix} is proportional to n, and find the constant of proportionality
thanks

I have this integral
$$ \int_{0}^{\pi/2} \frac{e^{a \cos^2x}}{b^2\cos^2x + c^2\sin^2x}dx.$$
I know that
$$\int_{0}^{\pi/2} \frac{dx}{b^2\cos^2x + c^2\sin^2x}=\frac{\pi}{2bc},$$
and
$$\int_{0}^{\pi/...

My question is maybe elementary but I'm having troubles with it. I have a function $f$ absolutely continuous in (a,c) and in (c,b), $f$ continuous in $c$. Is $f$ absolutely continuous in (a,b)?

Consider Tommy’s integrals:
$$a) \int_1^{\infty} \frac{\operatorname{li}(x)^2 (x - 1)}{x^4}\, dx = \frac{5 }{36}\pi ^2 $$
$$ b) \int_1^{\infty} \frac{\operatorname{li}(x)^2 (x - 1)}{x^5}\, dx = \...

Let $f(x)=\frac{1}{x^3+3x^2+3x+5}$, then what is $f^{(99)}(-1)$?
By letting $g(x)=x^3+3x^2+3x+5$, it is easy to see that $g'(-1) = g''(-1) = 0$, and so $f'(-1) = f''(-1) = f'''(-1) = 0$.
However, I ...

I am having a problem understanding,
Whether to keep a constant (arbitrary or fixed) in the solution of a differential equation
Figuring which is an arbitrary constant and which a fixed constant
...

I have been examining a function $f(r,\theta,\phi)$ where $f(r,\theta ,\phi)=\frac {1}{r\sin \theta}$ for $\theta \gt \pi /4$ $\;$ and $\;$ $f(r,\theta ,\phi) = 0$ for $\theta \lt \pi /4$. I am ...

The two-variable version of the Lagrange mean-value theorem says that given a function $f(x,y)$,
$$f(\vec{p_o} + \vec h)-f(\vec {p_o})=df(\vec {p_{\theta}})$$
Where $\vec p_{\theta}=\vec p_o + \theta ...

Let $f(x) = \sqrt{1+|x|^2}$, where $x \in \mathbb{R}^n$.
I found that $\nabla f(x) = \frac{1}{2}(1+|x|^2)^{-1/2}(2x)$, but is there an easy way to compute the Hessian?

$L1 : r = u[-3, 2, 4] + v[-4, 7, 1], u, r \in \mathbb{R}$
$L2 : r = s[-1, 5, -3] + t[-1, -5, 7] , s, t \in \mathbb{R}$
(Hint: Express each direction vector in the first equation as a linear ...

Two questions:
Show whether $\sum_{n=2}^{\infty} \frac{1}{n \ln(n^3)}$ converges or diverges.
According to wolfram, the series diverges by the comparison test, so I tried the following:
for $n$ ...

I have encountered many calculus textbooks that state the following theorem:
If $x=f(t)$ and $y=g(t)$ are differentiable at $t=t_{0}$, $f'(t_{0}) \neq 0$, and $y=F(x)$ holds for some open interval ...

The following definition of the derivative of a function is taken from Analysis on Manifolds by James Munkres:
Now my question is the following: Is the "neighborhood of $a$" an open set $U⊆A$ in $\...

How can I prove that $n!<n^n$ for every $n>1$ using the induction principle?
If I put $n=2$ I get $2<4$ so I know that $n!<n^n$ is true.
Now I don't know how to prove that $(n+1)!<(n+...

Given $n$ digits $x_1,...,x_n.$ I would like to understand how many $n$-digit numbers I can form from this set of digits.
I would like to emphasize that I can pick each digit only once, so
$x_1 x_1 ....

I am given the following problem:
Find a function of the form $f(x)=a+b\cos{(cx)}$ that is tangent to the line $y=1$ at the point $(0,1)$ and tangent to the line $y=x+\dfrac{3}{2}-\dfrac{\pi}{4}$ ...

I need to test these integrals for convergence $\int_{0}^{\infty}\frac{e^{ix}}{\log(x)},
\int_{0}^{\infty}\frac{\cos(x)}{x^p},
\int_{0}^{\infty}\cos(x^2)$ with $p\in\mathbb{R}$. However I suck ...

$\int_{1}^{\infty}\left( \ln\frac{1}{x^p}- \ln \left(\sin\frac{1}{x^p}\right)\right) dx$
I need to find for which values of p the integral is convergent?
I tried with Macloren series to expand $\sin\...

There was this question asked yesterday here:
Link: Evaluating $\lim\limits_{x→∞}\left(\frac{P(x)}{5(x-1)}\right)^x$
Consider $P(x)= ax^2+bx+c$ where $a,b,c \in \mathbb R$ and $P(2)=9$.
Let $\...

What is the relation between $\frac{d^2y}{dx^2}$ and $\frac{d^2x}{dy^2}$ ?
For example $\frac{dy}{dx}$=$\frac{1}{\frac{dx}{dy}}$

Please provide proof
The answer in the back of the book is different to both my calculations and also the online calculator I crosschecked my answer with...
The Question:
"Differentiate with respect ...

Question
Let $f:\Bbb{R}\to\Bbb{R}$ differentiable and $f(2)=3$ and $1 \geq f'(x)$ for every $x$.Find the best estimation of
$$ \int_{2}^{5} f(x)dx $$
Solution
$$\frac{f(5)-f(2)}{3}\leq 1\tag{...

consider the function
$$f(x) = 2x + 4y^2$$
and the curve
$$\gamma(t) = (\sin t, \cos t), \quad t\in (0, 2\pi).$$
Then, what is the derivate of the composite function $f(\gamma(t))$ in the point $t_0 =...

We have the series $\sum_{n=0}^\infty \frac{n}{n+1}x^n$.
Give a function $f(x)$such that this function is equal to this series for each $x\in \Bbb R$ for which the series converges.
I've calculated ...

I need help with this integral
$$\int_0^{\infty}e^{-x} \sin x \log x ~ dx$$
WolframAlpha gives the answer to be $\displaystyle \frac{1}{8} (-4 \gamma + π - 2\log(2))$, a curious expression with ...

How can we prove that the series $\displaystyle \sum^{\infty}_{n=1}\frac{n}{1+n^2}$ is convergent or divergent?
Solution I try:
$$\lim_{m\rightarrow \infty}\sum^{m}_{n=1}\frac{n}{1+n^2}<\lim_{m\...

Can somebody help me get started with the following problem? I want to solve:
$$\theta^* := \arg\min_{\theta>0} \quad \big\| x - \sum_{n=1}^N c_{n} \bf{k}_{n}(\theta) \big\|_{2}^{2}$$
where $x$ ...

Find the minimum of function $$f(x)=\frac{4x^2}{\sqrt{x^2-16}}$$ without using the derivative.
In math class we haven't learnt how to solve this kind of problems (optimization) yet. I already know ...

if $ p^2=a^2 \cos^2 \theta+b^2 \sin^2 \theta $, show that $p + p'' = \frac{a^2b^2}{p^3}$
My try :
$2pp' = (b^2-a^2)\sin 2\theta$
$p'^2 + pp'' = (b^2-a^2)\cos 2\theta$
Thats it ! it doesn't ...

Let $f(x) = \dfrac{1-\cos \{x\}}{(x^4 + ax^3 +bx^2 +cx)^2}$. If $l= \lim_{x\to 1^+}f(x), m = \lim_{x\to 2^+}f(x) $ and $n= \lim_{x\to 3^+}f(x),$ where $l,m$ and $n$ non-zero finite then:
$a+b+c=? ...

I have tried three times and keep getting an answer different to the one in the back of the book… Plus I do not know where to find an online calculator for limits as my regular one is out of ...

So I am about to start my Maths course, and the professor gave us some questions reguarding linear algebra and calculus. We haven't done any classes before, there are kind of introductory questions ...

I am trying to evaluate the following simple enough looking limit,
$$\lim_{x \to \infty} x^\alpha \cdot l^x, \ l\in(0,1),\ \alpha \in \mathbb{R}$$
It is not so bad when $\alpha \leq 0$, as the $l^x$ ...

Find all functions: $f: \mathbb{R}\rightarrow \mathbb{R}$ such that:
$$f\left ( x+ f\left ( y \right ) \right )= f\left ( x+ y^{2018} \right )+ f\left ( y^{2018}- f\left ( y \right ) \right ),\,\...

Question. Assume that $\{p_n\}$ is the sequence of prime numbers, in increasing order. Does the series
$$
\sum_{n=1}^\infty \frac{\sin p_n}{p_n}
$$
converge?
The only criterion to establish ...

Any ideas/hints on how to construct a non-decreasing function on $[0,1]$ whose set of discontinuities is not closed?

What is the Fourier cosine transform of $e^{-ax}$
I got
$$
\int_{0}^{\infty}\cos(kx)e^{-ax}dx = \frac{e^{-ax}(k\sin(kx) -\cos(kx))} {a^{2}+k^{2}}\Bigr|_{0}^{\infty}
$$
But how do you continue ...

On page 90 of Spivak's Calculus on manifolds
4-8 Theorem (3) $f^*(g.w)=(g\circ f)\cdot f^*w $.
Here I assume $g $ is a function $$f^*(g.w)(p)(v_p)=(g.w)(f(p))(f_*v)=g(f(p))(f_*v)\cdot w(f(p))(...

Let $f \colon [0,1] \to [0,1]$ be a function of class $C^1$ such that $f(0)=f(1)=1$ and $f'$ is nondecreasing, i.e., $f$ is convex. Show that the length of the curve defined by the graph of $f$ is ...

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