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Questions tagged [calculus]

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

0
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1answer
16 views

Second derivative of itself

I know the $ \frac{d^2x}{dx^2}= 0 $, since $dx/dx = 1 ...$ But by playing with some equations it is easy to get that $d^2f/dx^2=f''(x)$, so $d^2f=f''(x)dx^2$ and $df=f'(x)dx$, so $df^2=f'(x)^2dx^2$. ...
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0answers
9 views

Question on Function of function polynomial

If $f(x)=x^3-12x^2+Ax+B>0$ $f(f(f(3)))=3$, $f(f(f(f(4))))=4$ then what is the value of $f(7)$
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4answers
33 views

$\int_{0}^{2\pi }\cos^{2}(nx)dx$

$$\int_{0}^{2\pi }\cos^{2}(nx)dx$$ Why If I solve the integral by parts I get an answer and if I solve by using formula I get another answer. What's wrong ? By parts: I take $f(x)=\cos^2(nx)$ so $f'...
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4answers
15 views

Finding the fundamental period of $-6\cos(5\pi x)$

I am trying to find the fundamental period of $ f(x) =-6\cos(5\pi x)$. I know a periodic function satisfies $f(x)=f(x+p)$. I know that $\cos(x)$'s periodicity is $2\pi$ as $\cos(x+2\pi)=\cos(x)$ so ...
1
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1answer
13 views

Proving a specific statement involving limits related to a function with some given properties.

Let a function $f(x)$ be bounded in every interval $(x_0, x_1)$. The domain of $f(x)$ is $x\in (x_0, +\infty)$. Prove that if the following limit exists and is either finite or infinite: $$ \lim_{x\...
1
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1answer
21 views

Existence of a function with $||grad f||>\epsilon$

I want to construct a function $f$ on the unit ball $B$ of $\mathbb{R}^n$, such that it is negative on a closed subset of the boundary $\partial'B\subsetneqq\partial B$, zero on a given point $p\in B$,...
2
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3answers
39 views

Derivative rule for a function raised to the power of another function eg $(3x^2+5)^{\arctan{x}}$

I am working some older final exams for my calc 1 class at university. There are several "find the derivative" questions, and while I am normally quite good at these, I got the wrong answer on one, ...
2
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1answer
18 views

If function has positve integral on domain then there exist interval and positve number such that function is bounded below by that number on Interval

Suppose $\int_a^b f$ exist and positive Prove that there exist interval [c,d] and m>0 such that $f(x)\geq m$. I was thinking to prove by contradiction . Suppose there is no interval with above ...
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0answers
7 views

Dirichlet's improper singular integral

If $f: \mathbb{R}\to\mathbb{R}$ is Lipschitz continuous and if $\int_{-\infty}^{+\infty} |f(t)|dt$ converges, then $$ \lim_{\lambda\to +\infty}\int_{-\infty}^{+\infty} f \Delta_{\lambda} = f(0).$$ $\...
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1answer
30 views

Understanding the domain of the triple integral for $f(x,y,z)=x+2y+z^2$

So, I am having trouble (again) with the domain for a triple integral of a function, bounded by the paraboloid $2y^2=x$ and the $x+2y+z=4$ and $z=0$ planes I have tried to guess the bounds for x,y ...
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2answers
15 views

Trignometric rearrangement

So there is this equation: $y=\tan \left(e^{x}+c\right)$ And in the next step it's differentiated w.r.t x $\frac{d y}{d x}=\sec ^{2}\left(e^{x}+c\right) \cdot e^{x}$ Upto this point I follow what ...
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1answer
30 views

Does $\lim_{x\to x_0}f(x) = a$ and $\lim_{t\to t_0}g(t) = x_0$ imply $\lim_{t\to t_0}f(g(t)) = a$?

Let: $$ \begin{align*} &\lim_{x\to x_0}f(x) = a \tag1 \\ &\lim_{t\to t_0}g(t) = x_0 \tag2 \end{align*} $$ Does $(1)$ and $(2)$ imply the following: $$ \lim_{t\to t_0}f(g(t)) = a\tag 3 $$ ...
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2answers
33 views

Finding where a curve has a given slope

Find the value of $x$ at which the slope of the tangent to the curve $y=8+2x-x^2$ is $6$. I have looked all over the internet an through my notes and cant find anything comparable
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0answers
13 views

Path Integral equals zero on non conservative field

I was doing some excercises and I was asked to compute the line integral along certain path. I used greens formula to calculate the work. When computing the integral I had to divide the domain in two ...
2
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2answers
34 views

What does interval of convergence for power series show?

For the simplest case, $f(x)=\dfrac1{1-x}$. This can be represented by $\sum\limits_{i=1}^nx^{n-1}$ or $\sum\limits_{i=0}^nx^n$. The series is only equal to that value for $|x|< 1$, so the interval ...
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1answer
25 views

How would the Dirichlet Test for convergence prove that $\sum_{n=1}^{\infty}\frac{\cos{n}}{n}$ does in fact converge?

I've been looking for a way to determine whether $\sum_{n=1}^{\infty}\frac{\cos{n}}{n}$ converges, and the test that I've most often seen recommended seen is the Dirichlet test for convergence. ...
0
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1answer
17 views

Use taylors theorem with integral reminder to prove log(1+x) uniformly converge

Use taylors theorem with integral reminder to prove $log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}+...$ uniformly converge for on every $[-a,a]$ where $0<a<1$. i have computed the $R_n=\frac{1}{n!}\...
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2answers
24 views

Given that $f(x)=(2x+1)^3$, find $\int (\lim_{h \to 0} \frac{f(x+h)-f(x)}{8h})\,dx$

I thought this was as simple as: $$ \int \left (\lim_{h \to 0} \frac{f(x+h)-f(x)}{8h}\right)\,dx = \frac{1}{8}\int f'(x)\, dx=\frac{f(x)}{8} + C $$ But the answer is supposed to be: $$ \left (\frac{...
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0answers
13 views

Partial differential Equation uniqueness

Let $\Omega\in\mathbb{R}^{n}$ be a bounded connected open set. I have the following partial differential Equation; \begin{align} \nabla\cdot\left(-D(x)\nabla \psi\right)&=F\quad \text{in}\quad \...
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0answers
9 views

Wave cone of the curl operator

How can one compute the wave cone $\Lambda_{\mathcal A}$, defined as \begin{equation*} \Lambda_{\mathcal A}:=\bigcup_{|\xi|=1} \ker \mathbb A^k(\xi) \qquad\textrm{with}\qquad \mathbb A^k(\xi)= (2\pi ...
2
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1answer
40 views

CAS integrals of discontinuous functions

Background This post is motivated by my interest in the performance of symbolic integrators in computer algebra systems (CAS's), such as Mathematica (MMA). I've found that, when an integrand has ...
1
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4answers
51 views

Integrating $\int \tan^3(x)$ in two different ways gives two different answers

I was trying to find the antiderivative of a function $$\int \tan^3(x)$$ However, due to substitution differences, my book has a answer of $$\frac12\tan^2(x)+\ln(\cos x)+C$$ while I got an answer $...
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0answers
15 views

Maximum likelihood estimation for 2 gaussian curve

math in general is not exactly my strong suit, i recently found myself staring at this equation. $y = p(\frac{1}{\sigma_1\sqrt{2\pi}})exp (-\frac{(x-\mu_1)^2}{2\sigma_1^2}) +(1-p) (\frac{1}{\sigma_2\...
1
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1answer
35 views

Find a and b and the position vector of the point of intersection of C of $l_1$ and $l_2$

I asked a similar question to this yesterday, and I think I managed, however, I have a similar question but a bit different, if I understand this I think I'll manage to confirm the other one as well, ...
6
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1answer
92 views

Prove that : $\frac{ab^2(b+c)}{c^2}+\frac{bc^2(a+c)}{a^2}+\frac{ca^2(b+a)}{b^2}≥2(ab+ac+bc)$ where $a,b,c>0$

Show that $$\frac{b^2(b+c)a}{c^2}+\frac{c^2(a+c)b}{a^2}+\frac{a^2(b+a)c}{b^2}≥2(ab+ac+bc)$$ where $a,b,c>0$ Can AM-GM work here? I need someone help me or hinting me please. Thanks!
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2answers
41 views

Does $x^n$ converges uniformly on $ [0,1)?$

Doing the limit we can see that in the open interval it converges pointwise to the constant function $f(x) = 0$. In the closed interval it doesn't converge uniformly because in $x=1$ $f(x) =1$ and ...
0
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1answer
29 views

calculus - Finding the global maximum and minimum points of this two variables function

$f(x, y) = 4-y$ in $D:=(x, y, z) \in R^3: x^2+y^2=8, x+y+z=1$ I used Lagrange for this function, but I got a bit confused after I solved the system of equations. $\mathcal{L}(x, y, z, \lambda, \rho)...
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0answers
44 views

Taylor series for $e^{-(x-t)^2}$

I can't seem to see why the Taylor series for $e^{-(x-t)^2}$ is as follows $$ e^{-(x-t)^2} = \sum_{n=0}^{\infty} \frac{(-t)^n}{n!} \left( \frac{d}{dx} \right)^n e^{-x^2} $$
3
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1answer
243 views

Fourier Transform of Airy Equation

I am trying to find $Y(k)$ of the equation $y''(x)-xy(x)=0$ and hence show that $$y(x)=\sqrt{\frac{2}{\pi}}\int_0^{\infty}\cos\left(\frac{k^3}{3}+kx\right) \ dk,$$ given $Y(0)=1$. Here, we use the ...
1
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1answer
38 views

Partial Derivative Disambiguation

There are at least two substantially different meanings to $\frac{\partial}{\partial x}f(x,\ y,\ z(x))$. The $\partial x$ could mean "with respect to $x$ the independent variable," or it could mean "...
4
votes
1answer
73 views

Intriguing Limit

Prove that: $$L=\lim_{n\to\infty} \frac {\sqrt 2 n^{\left(n-\frac 12\right)}}{n!}\left(\frac {(2\sqrt[n] {n} -1)^n}{n^2}\right)^{ \frac {n\left(n-\frac 12\right)}{\ln^2 n}}=\sqrt {\frac {e}{\pi}}$$...
4
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0answers
51 views

Is there a way to preform this integral such that the answer is $e^{-|y|}$?

Consider the function $f(y)=e^{-|y|}e^{y}$ I am trying to integrate this function with respect to another variable (such as $x$) so that the result from the integration is $e^{-|y|}$? The function $...
4
votes
1answer
37 views

Cartan homotopy formula and curl

In Topological Methods in Hydrodynamics, V. I. Arnol'd writes that the following expression $$curl(\mathbf a \times \mathbf b)=[\mathbf a, \mathbf b]+ \mathbf a \ div \ \mathbf b - \mathbf b \ div \ \...
0
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1answer
38 views

Computing partial derivatives of $f(a,b)= \int_{0}^{1}(ax+b+\frac{1}{1+x^2})^{2}dx$ using chain rule.

Let $f(a,b)= \int_{0}^{1}(ax+b+\frac{1}{1+x^2})^{2}dx$. I want to compute $\frac{\partial{f(a,b)}}{\partial{a}}$ and $\frac{\partial{f}(a,b)}{\partial{b}}$. I was told in the text that $$\frac{\...
-1
votes
1answer
34 views

Let $F(x)= \int^x_0 e^{sint}dt,x \in \mathbb{R}$, then $F'(0)$ equals: [on hold]

$-2$ $-1$ $0$ $1$ $2$ How can I solve this integral?
2
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1answer
37 views

Trying to prove an integration by parts formula

Denote pde operator $$ Lu = - div \cdot (p \nabla u ) + qu$$ where $x \in D$ and $p=p(x) > 0$ and q=q(x) are continuous on $\bar{D}$ an p has continous first partial derivatives on $\bar{D}$. I ...
1
vote
1answer
37 views

Find a and b given that $l_1$ and $l_2$ intersect at the midpoint of the line segment $\vec{AB}$

A past examination paper had the following question that I found somewhat difficult. I tried having a go at it but haven't come around with any solutions. How would one go about tackling it? The ...
1
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0answers
7 views

Policy gradient base line function

On the bottom of page ten of the following paper on probabilistic reinforcement learning, there are 3 equations where is author manipulates the policy gradient $\nabla_\theta J(\theta)$. Can someone ...
2
votes
1answer
61 views

Derivatives without limits

Update: H/t David K for pointing out that my assumption that I can force $a^2+b^2=r^2$ is wrong. This led to analyzing a cubic equation which is now moot, but I think the bulk of the question remains ...
0
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0answers
12 views

How to show a smooth function is an extremum of an integral function under constraints

I am taking a class and one of the problems in the book asks: Let $L$ be a function of $(y,y')$ in $R^2$. For any fixed $y_1$ in $R$ show that a smooth $y^*(t)$ is an extremum of $$ \int_0^1{L(y(t),...
0
votes
1answer
23 views

Understanding the domain of the triple integral for $f(x,y,z)=x^2+y^2$

So, I am having trouble with the domain for the triple integral of $f(x,y,z)=x^2+y^2$, bounded by the paraboloid $x^2+y^2=2z$ and the $z=4$ plane I am currently trying to project it on the XY axis, ...
1
vote
1answer
27 views

What's wrong with this proof? / uniqueness of least upper bound

this is a proof by contradiction let y and z be least upper bounds of a set A, such that y != z so, according to a theorem, L - ε < x,for all x in A. where L is the least upper bound and ε is a ...
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0answers
14 views

2nd O Differential equations with missing variable

enter image description here solve the following differential equation by (equation dependent or independent variable missing)
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1answer
30 views

Minimum number of zeros of an equation have $3$rd derivative.

Let $y=f(x)$ be thrice differentiable function defined on $\mathbb{R}$ such that $f(x)=0$ has at least $5$ distinct zeros? Then find minimum no. of zeros of $f(x)+6f'(x)+12f''(x)+8f'''(x)=0$ is ...
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0answers
25 views
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0answers
63 views

Step in the proof of the continuity of the Gauss Gamma function

Define the Gamma function as \begin{align} \Gamma(z):= \frac{1}{G(z)} \quad \forall \, z\in \mathbb{C}\setminus \{0, -1, -2, ... \} \end{align} where \begin{align} G: \mathbb{C} \rightarrow \mathbb{C}...
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0answers
19 views

Let $\lim_{t\to t_0}\phi(t) = a$. Prove that $f(x) = \mathcal{o}(g(x)) \implies f(\phi(t)) = \mathcal{o}(g(\phi(t)))$

Let: $$ \lim_{t\to t_0}\phi(t) = a $$ where $\phi(t)\ne a$ and $t\ne t_0$ in the neighbourhood of $t_0$. Prove that: $$ \begin{align*} f(x) \stackrel{x\to x_0}{=} \mathcal{o}(g(x)) &\implies ...
0
votes
0answers
30 views

Fourier Series - Integration Limits help

So I am doing separation of variables and I reach the stage $$ \sum_{n=1}^{\infty} Q_n \cos(P_nx) = 2 $$ where $$f(x) = 2. $$ Now this is similar to the Fourier series where the $Q_n$s are the ...
1
vote
1answer
24 views

Evaluate $\iint_{\Sigma} \vec{F}\cdot d\vec{\sigma}$:

$\vec{F}(x,y,z)=(x+1,y-2,z)$ and $\Sigma$ the part of the curved surface of the cylinder $x^2+y^2=2x\,(y\ge 0)$ bounded by the plane $z=0$ and the conical surface $x^2+y^2=z^2\,(z\ge 0)$. The normal ...
0
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0answers
11 views

Line integral over intersection of a plane and cylinder

Calculate $\int_{\Gamma}\vec{F}\cdot d\vec{s}$, with $\vec{F}(x,y,z)=(x^2,y^3,x^2)$ and Γ the intersection curve of $x^2+y^4=1\,(x\ge0)$ and the plane $x+y+z=1$, oriented from $(0,1,0)$ to $(1,0,0)$. ...