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

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2
votes
1answer
150 views

Cauchy Schwarz with integrals of integrable functions

I was reading and doing problems from Spivak's Calculus on Manifolds. Q1-6 (a) stumped me a little. Let $f$, $g$ be integrable on $[a,b]$. Prove that $$\left| \int_a^b f\cdot g \; \right | \leq \left(...
2
votes
1answer
87 views

Books and sources concerning the mathematics of Leibniz and the feud with Newton

I am trying to find books and other sources concerning the mathematical history of Leibniz, including the controversy due to the independent discoveries of calculus by both Newton and Leibniz. I can't ...
3
votes
3answers
52 views

Question about limits $\lim_{x\to\infty}\frac{x-2}{e^{1/x}\cdot x}$

How to calculate this: $$\lim_{x\to\infty}\frac{x-2}{e^{1/x}\cdot x}$$
3
votes
3answers
198 views

Possible alternative way of expressing continuity of a function?

In Calculus or Real Analysis the usual form of definition of continuity of a function is $\epsilon- \delta$ def. From a rigorous point of view, is it possible to say this way? and if so, why?: $f(\...
0
votes
1answer
355 views

How to find singular points of a function without knowing the graph?

Problem: Let $f(x) = (x-1)^{2/3} - (x+1)^{2/3}$. Locate and classify all local extreme values of this function. Determine whether any of these extreme values are absolute. Attempt at solution: We ...
-1
votes
2answers
47 views

Continuous derivative vs Continuous partial derivatives

Firstly, suppose $f:\mathbb R^n\to\mathbb R^m$ has all continuous partial derivative. I believe I have proved that this imply continuous derivative. Please tell me if this is actually true. For its ...
2
votes
1answer
75 views

Solving $x - a \log(x)=b$

Let $a>0$ and $b \in \mathbb{R}$: Assume there exists an $x >0 $ s.t. $$x - a\log(x) = b$$ holds. How can it be determined in closed-form?
1
vote
3answers
91 views

Why should I use derivatives and calculus?

I know that this question maybe sounds pretty generic, but it's a curiosity that I have and I didn't found any answer yet. I recently started studying calculus using this material where is said that "...
3
votes
2answers
88 views

Finding $F(x)$ from $F(kx),$ where $F(x)$ is the antiderivative of the function $f(x)$.

I have that $F(e^{x}x) = e^{x}x^{2} - e^{x}x + e^{x} - 1$, and I would like to find $F(x)$. Attempt Since $F(e^{x}x) = e^{x}x^{2} - e^{x}x + e^{x} - 1,$ $F(t) = \alpha_{1}t^{\beta_{1}} + \alpha_{2}t^...
0
votes
3answers
107 views

Derivation of formula for gradient in spherical coordinates

If we have a function $f=f(r, \theta, \phi)$, where $(r, \theta, \phi)$ are spherical coordinates on $\mathbb{R}^3$, how do we compute the gradient $\nabla f$ by using the formula $$\nabla f \cdot d\...
1
vote
1answer
76 views

About 'Marcinkiewicz–Zygmund inequality'

Marcinkiewicz–Zygmund inequality gives gives relations between moments of a collection of independent random variables. The statement of this inequality can be seen in Wiki https://en.wikipedia.org/...
2
votes
1answer
40 views

How to find this kind of function?

I am trying to find a function $f(x,y)$ with $f:\mathbb{R}^{2} \rightarrow [a, b]$ where $[a, b]$ is equal to $[-1, 1]$, $[0, 1]$, or some other small interval (open intervals are fine as well). The ...
4
votes
3answers
170 views

Calculate in closed form $\sum_{n=1}^{\infty} \frac{\arctan(1/n) H_n}{n}$

Playing with Taylor series is not helpful enough. What else would you try out? $$\sum_{n=1}^{\infty} \frac{\arctan(1/n) H_n}{n}$$ $$\approx 2.1496160413898356727147400526167103602143301206321$$ It's ...
0
votes
3answers
164 views

How do i find a closed form expression for $\sum_{k=0}^n \frac{(x-1)^k}{k+1}$?

How do I Find a closed form expression for : $$\sum_{k=0}^n \frac{(x-1)^k}{k+1}$$ Note :I have no idea how to do that, I am bad at evaluating series when we cannot use some standard series to do it. ...
2
votes
3answers
79 views

Prove any slope is obtained at some point

We are given $f:$$R\rightarrow$$R$ differentiable in $[a,b]$ such that $f'(a)<f'(b)$. We need to prove that $\forall\beta\in$$[f'(a),f'(b)]$$\exists$$x\in$$[a,b]$ such that $f'(x)=\beta$. This ...
5
votes
1answer
128 views

A limit of a sequence statistifying $S_{n} = \frac{1}{2}(a_{n}+\frac{1}{a_{n}})=a_1+a_2+…+a_n$

Sequence $\{a_n\}$ is a positive sequence and satisfies $S_{n} = \frac{1}{2}(a_{n}+\frac{1}{a_{n}})$ where $S_n = a_1+a_2+...+a_n$. Find $\lim_{n\to \infty} S_{n+1}*(S_{n}-S_{n-1})$
0
votes
1answer
34 views

Proof of change in position vector in spherical coordinates

I have found it hard to proof that ${d\vec r=dr\hat r+rd\theta\hat \theta}$ in spherical coordinates. Also it would be great if somebody can explain what ${d\vec r}$ is because I read different things ...
1
vote
0answers
23 views

A question about the differentiability conditions for a fractional power

Now consider the fractional power $$f(x)=z^{m/n}=|z|^{m/n}(\cos \frac{mArgz}{n}+i \sin \frac{mArgz}{n})$$ Since $f(x)$ satisfies the Cauchy-Riemann equations, and is therefore differentiable, we can ...
3
votes
2answers
190 views

Tangent to the curve

What is the equation of the tangent to the curve $$y = x^{1/3}$$ at the point $(0,0)$ ? This is a homework question. I tried solving it. The derivative comes out to be infinite at the given point. So,...
2
votes
1answer
237 views

log normalizer - exponential family

i am studying the exponential family- and read that, for $p(x|\mu)=h(x)exp(\eta^T t(x)-a(\eta))$, that $a(\eta)$ is the log normalizer, which ensures that the probability distribution integrates to ...
1
vote
4answers
71 views

Convergent sequence rigorous definition

I know what a convergent sequence is and how it works and everything but whenever I look at the definition my mind cant suddenly make the link with my intuition part of the brain so I realise that I ...
2
votes
0answers
39 views

How to find derivative of $\left\Vert x-a\right\Vert ^{p}:\mathbb{R}^{n}\rightarrow\mathbb{R}$?

Here is what I've tried: $\alpha\left(x\right)=\sum_{i=1}^{n}\left(x_{i}-a_{i}\right)^{2},\ \ \beta\left(y\right)=\sqrt{y}, \ \ \ \gamma\left(z\right)=z^{p}$. Clearly: $\left\Vert x-a\right\Vert ^{p}=...
1
vote
3answers
150 views

Arc length of $f(x)=x^2-\ln x$ over [1,e]

This is how I learned to solve for arc length: $$ \frac d{dx} f(x)=2x - \frac 1x$$ $$ \left(\frac d{dx}f(x)\right)^2 = \left(2x - \frac 1x\right)^2 = 4x^2-4+\frac1{x^2} $$ $$ 1+\left(\frac d{dx}f(x)\...
2
votes
1answer
65 views

Integration by partial fractions, reducing denominator through substitution

In the following case, suppose $b$ is a number and $b≠0$. Find $$ \int{1\over(x^2+b^2)^n}\mathrm{d}x$$ The textbook uses the following substitution: $$ x=bz$$ $$ \mathrm{d}x=b\mathrm{d}z$$ And ...
0
votes
4answers
85 views

Question about an infinite sum in which every term is infinite

Given the sequence $x_k=\lim_{n \rightarrow 0} \frac{1}{kn^2}$ where $k\in\Bbb{N}$, define an infinite sum $S=x_1-x_1+x_2-x_2+x_3-x_3+...$ Now every $x_k$ is infinite, but does $S=0$? I know we ...
0
votes
1answer
28 views

Question regarding differentiability of function at x=0

I removed the absolute signs ,but in D part how do i remove the outer absolute value symbol
4
votes
5answers
2k views

Is there a convergent, alternating series that fails the AST?

The alternating series test (AST) says, briefly, that if $a_k>0$ $a_k \geq a_{k+1}$, and $a_k \to 0$ as $k \to \infty$ then $\sum_k (-1)^k a_k$ converges. This seems to be a one-way test (that ...
1
vote
2answers
95 views

The derivative of a recurrence relation of functions

I am unsure of how to take the derivative of a recurrence relation of functions. For example consider the following recurrence relation: \begin{equation} \left\{ \begin{array}{cl} f_n(x) &= a_n\...
1
vote
2answers
428 views

why the standard deviation is not as the same as online calculator

I need to calculate the standard deviation for these numbrs: -12 -3 0 -13 8 -6 0 -22 -1 7 -7 1 -2 -13 -4 0 -6 -4 -10 3 I did everything, but still my answer is ...
3
votes
2answers
76 views

Distance involving 3D lines and vectors.

In this problem, a = \begin{pmatrix} 5 \\ -3 \\ -4 \end{pmatrix} and b = \begin{pmatrix} -11 \\ 1 \\ 28 \end{pmatrix} Vectors p and d exist such that the line containing a and b can be expressed in ...
1
vote
2answers
77 views

Using sandwich theorem to show convergence

I have arbitrary functions $f(x)$ and $g(x)$ that have a following property. $$x>f(x)>g(x)$$ And I know that as $x \rightarrow \infty $,$g(x) \rightarrow \infty $, thus by sandwich theorem,i ...
0
votes
1answer
20 views

Curvature at a point in a vector valued function

I am trying to determine the curvature when $t=2$ of the function $r(t)=<t^3,3t^2,8t>$ So I found $v(t)=<3t^2,6t,8>$ and $a(t)=<6t,6,0>$. So now that I have these two functions, I ...
1
vote
2answers
112 views

3D line in a 3D plane. Find the intersection of the two.

(I'm new to Math.StackExchange, so if you see any errors, please comment below!) $\mathcal{P}$ is the plane containing the three points $(-3,4,-2)$, $(1,4,0)$, and $(3,2,-1)$. $\ell$ is the line ...
1
vote
3answers
98 views

Find particular path on a plane

I'm working on the following question: Let $T(x,y) = 1-x^2-2y^2$ be the temperature at each point $P(x,y)$ in the plane. A heat-loving bug is placed in the plane at the point $P(-1,1)$. Find the path ...
1
vote
1answer
35 views

what would be power series of $x_t = e^{\beta_t} $ if $\beta_t$ is a Brownian motion process?

In general the power series of $e^x =1+x/1!+x^2/2!+x^3/3!+...$ but because the process is random we can't apply the direct differentiation than how can i write it's power series.In the book stochastic ...
1
vote
2answers
70 views

Faulhaber Formula Identity

I have to show the following identity: $$ S_n^p := 1^p+2^p+...+n^p $$ $$ (p+1)S_n^p+\binom{p+1}{2}S_n^{p-1}+\binom{p+1}{3}S_n^{p-2}+...+S_n^0=(n+1)^{p+1}-1 $$ What I did first is to use the binomial ...
3
votes
2answers
120 views

What is the derivative of the inner product norm on $L^2$ space?

Let $f \in L^2(X)$ such that $f$ is generated by some arbitrary constant; that is, $f = g(a)$ with $g: \mathbb{R} \to L^2(X)$. Then what can be said about the derivative with respect to some arbitrary ...
5
votes
4answers
397 views

Integration of $\frac{\sin x}{\sin 4x}$

Question: Solve the following integral: $$\int \frac{\sin x}{\sin4x}dx$$ Attempt: Using trigonometric identities to expand $\sin4x$, I obtained the integral: $$\int \frac{1}{4\cos x \cos2x}dx$$ ...
39
votes
2answers
1k views

Evaluate $ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}dx $

I need the method to evaluate this integral (the closed-form if possible). $$ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}\,dx $$ I used the relationship between $\tan x$ and $\tanh x$ but it didn't ...
1
vote
3answers
90 views

Limits. Finding positive value [duplicate]

Find a real number k such that the limit $$\lim_{n\to\infty}\ \left(\frac{1^4 + 2^4 + 3^4 +....+ n^4}{n^k}\right)$$ has as positive value. If I am not mistaken every even $k$ can be the answer. But ...
23
votes
2answers
372 views

Can you define arc length using a piece of string?

In calculus, how we calculate the arc length of a curve is by approximating the curve with a series of line segments, and then we take the limit as the number of line segments goes to infinity. This ...
1
vote
2answers
54 views

Maclaurin series for $f(x) =\pi x^8 e^{-x^3}$

Find the Maclaurin series for: $$f(x) =\pi x^8 e^{-x^3}$$ What I have : $e^x = $ $\sum_{n=0}^\infty {x^n\over n!}$ THEREFORE $x^8 e^{-x^3}= $ $$\sum_{n=0}^\infty {-x^{3n+8}\over n!}$$ Now I don't know ...
3
votes
3answers
164 views

How can I prove the following equality [duplicate]

I have the following equality : $$I_1=-\frac{ab}{2\pi}\int_0^\pi \frac{\cos(2t)}{a^2\sin^2(t)+b^2\cos^2(t)}dt=\frac{a}{a+b}-\frac{1}{2}$$ $$I_2=-\frac{ab}{2\pi}\int_0^\pi \frac{\sin(2t)}{a^2\sin^2(t)+...
4
votes
2answers
93 views

Are there some functions that cannot be optimized using calculus?

I've been working on a project to maximize a functions output using a genetic algorithm. However, from the limited calculus I know I thought there were methods to find the maximum of a mathematical ...
3
votes
1answer
63 views

An example of Lindelöf Theorem in Real Analysis

In any elementary real analysis books Lindelöf Theorem stated as follows: "Let $C$ be a collection of open sets of real numbers. Then there is a countable sub-collection of the elements of $C$ such ...
2
votes
3answers
89 views

Convergence of $\frac{\ln(n)}{n^2}$

During a Calc exam, I needed to state whether $$\sum_{n=1}^{∞} \frac{\ln n}{n^2}$$ converged or diverged. I was going to try a strict comparison test (with $\sum_{n=3}^{∞} \frac{\ln n}{n^2}$ to $\sum_{...
1
vote
1answer
81 views

Need to prove continuous periodic function of $\varphi (x) \equiv \psi(x)$

Question: Let two $\varphi(x) $ and $\psi(x)$ periodic and continous functions such that $$ \lim_{ x\to\infty}(\varphi(x)-\psi(x))=0, \quad x\in \mathbb{R}. $$ Prove that $$ \varphi(x)\...
1
vote
2answers
82 views

Two ideas for proving irrationality

I want to "construct" a proof showing that $\sqrt{n}$ is irrational and two ideas entered my mind. But i have doubts regarding the soundness of the subsequent reasoning that may spring forth based on ...
4
votes
1answer
328 views

Proof that the solution to cosx = x, is the limit of a recursive sequence.

So I've got this question. Exists a sequence $a_n$ such that: $$a_0 = \frac \pi4, a_n=\cos\left(a_{n-1}\right)$$ Prove that $\lim_{n\rightarrow\infty} a_n = \alpha$ Where $\alpha$ is the solution to $...
-3
votes
3answers
76 views

Differentiate $y=x^{\sqrt{x}}$

$y=x^{\sqrt{x}}$ 1) Take natural logarithms of both sides of the equation $y=f(x)$ and use the Laws of Logarithm to simplify $\ln\ y=\sqrt{x}\cdot \ln x$ 2) Differentiate implicitly to respect to x ...