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

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26 views

Is there a diffeomorphism $f$ such that $V(B) < V(f(B)) <V(B)+V(B)^2$

Simple to understand calculus question that involved change of variable theorem in integration. Suppose $B$ is some open ball in $\mathbb R^n$, and $f: \mathbb R^n \to \mathbb R^n$ is a ...
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38 views

Laplace transform of $\sin(x)$

I am confused with Laplace transform of $\sin(\theta)$. For example, what is the LT of $A \sin(x(t))=Bx''(t)$ ($x$ is second order), $A,B$ are constants.
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35 views

Study functions at $-\infty$, 0 and $+\infty$

The functions are: $f(x) = \frac{(x-\sin(x))\log(1+x^4)}{x^7}$ and $g(x) = \frac{x^3-\arctan(x)\log(1+x^2)}{x^5}$ I know that $\lim_{x \rightarrow 0} f(x) = \frac{1}{6}$, $\lim_{x \rightarrow 0} ...
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23 views

Where can I find the TOC of “Calculus and analytic geometry” by George B. Thomas 4th ed?

I am currently following the course Calculus Revisited, by Prof. Herbert Gross. In his lecture notes he makes references to the book mentioned in the title, by section number, so far I found a copy ...
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51 views

Maximum and Minimum Value

Let $f : [0, 2\pi] \to \mathbb{R} $ be defined by $f(x) :=e^x\sin(x)$. Find the maximum and minimum value of $f$ over the interval $[0, 2\pi]$. You must fully justify your answer. Sorry for the bad ...
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1answer
28 views

derivation of transport equation

The amount of pollutant in the interval at time $t$ is $M=\int_0^bu(x,t)dx$ .At later time t+h,the same molecules of pollutant have moved to the right by $ch$ centimetres. ...
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78 views

Poles of a function involing Gamma- and Beta function

I am interested in the poles of following function of $s$ where $0\leq x\leq1$ and $0\leq \delta < \infty$: $$M(s) = \frac{B(x;\delta+s-1,\delta)}{ \frac{\Gamma(2\delta)}{2 \Gamma(\delta)^2} + ...
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1answer
63 views

How to compute $\sum_{n=0}^\infty \frac{4^{n-1}}{(\pi -1)^{2n}}$?

I wrote as $\displaystyle\sum_{n=0}^\infty \frac{4^{n-1}}{(\pi ...
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72 views

ODE with constraints

Given the ODE system $$\dot{x} = y \\ \dot{y} = \frac{1}{\alpha} (z - y)$$ where $\alpha > 0$ is a constant. How can I find a bound for $z$ depending on $x$ such that $\forall t ~x(t) \geq 0$ under ...
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1answer
47 views

Area inside curve and ellipse using integrals.

i've done some progress with this one but i get stucked right before the integration part: $A)$ Use the right double integral and replacing $x=ar$cos$(\theta)$, $y=br$sin$(\theta)$ in order to find ...
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1answer
30 views

Limit of vector-valued function is equal to the limit of its components

Let $f: \Bbb R^m \to \Bbb R^n$. Express $f(x)$ in terms of components: $$f(x)=(f_1(x), f_2(x), ... , f_n(x))$$ I need to prove that $f$ is continuous at $a$ if and only if each $f_i$ is continuous ...
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1answer
23 views

will locally extremal continuous function be constant

Let $f:[0, 1]\rightarrow \mathbb{R}$ be continuous function. Let us suppose that each point on the segment is either local maximum or local minimum for this function. Is this function a constant?
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1answer
83 views

Evaluate $\int_{0}^{\infty}\sqrt{\frac{\sqrt{(a^2-y^2)^2+4y^2}+a^2-y^2}{(a^2-y^2)^2+4y^2}}dy=\sqrt{2}\pi$

Prove or disprove that$$\int_{0}^{\infty}\sqrt{\frac{\sqrt{(a^2-y^2)^2+4y^2}+a^2-y^2}{(a^2-y^2)^2+4y^2}}dy=\sqrt{2}\pi$$ for any $a>1$. I came across with this integral evaluating inverse ...
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27 views

Definition of standard functions [duplicate]

In many texts and books about calculus we see There are functions $f$ for which the anti-derivative cannot be expressed in terms of standard functions or there are many integrals that cannot ...
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29 views

If $f$ is a continuous function $f:[0,1]\to[0,1]$ then there exists $x_0\in [0,1]$ such that $f(x_0)=2\sin x_0$

Prove/disprove: if a continuous function $f:[0,1]\to[0,1]$ then there exists $x_0\in [0,1]$ such that $f(x_0)=2\sin x_0$. Define: $g(x)=f(x)-2\sin x$ $g(\frac {\pi} 4)=f(\frac {\pi} 4)-2\sin\frac ...
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1answer
57 views

I need help on this calculus homework question.

How would I go about solving this with a step by step explanation, all help appreciated! http://tinypic.com/r/xckr9z/8 Find the instantaneous rate of change of the given function when $x=a$. ...
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1answer
26 views

differential inequality of continuous functions

Let $u:[0,+\infty)\to (0,+\infty)$ be a continuous function such that $\int_0^\infty u(x)dx<\infty$. Suppose there exist $a,b>0$ such that $\frac{du}{dx}\leq u(a+bu)$. Prove that ...
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71 views

Intersection of ellipse and hyperbola at a right angle

Need to show that two functions intersect at a right angle. Show that the ellipse $$ \frac{x^2}{a^2} +\frac{y^2}{b^2} = 1 $$ and the hyperbola $$ \frac{x^2}{α^2} −\frac{y^2}{β^2} = 1 $$ will ...
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57 views

prove this step in poisson equation

$\phi(x)$ is harmonic in $C^2$ in proof for solution of poisson's equation($- \Delta u=f$) ,this step came it maybe very fundamental but I am not getting it $u(x)= \int_{R^n}\phi(x-y)f(y)\mathrm dy ...
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1answer
32 views

Find volume of region bound by $y=x^2$, $y=4-x^2$

Find volume of region bound by $y=x^2$, $y=4-x^2$ I graphed this and it's a graph of two parabolas A) About the $x$-axis. I'm not sure if I use a washer method using $$V=\int{\pi(4-x^2)^2}dx - ...
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32 views

Can a third degree'Riccati' like differential equation be re-written as a third order linear ODE

Consider the equation $$ y' = a_0(x) + a_1(x)y + a_2(x)y^2 + a_3(x)y^3 $$ Does there exist a substition $y = f(a_0, a_1, a_2, a_3, u, u', u'')$ such that after simplifying from this subsitution we ...
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1answer
69 views

Stoke's Theorem Example - Help?

From Stoke's Theorem: \begin{equation*} \oint_c \textbf{F}\centerdot d\textbf{r}= \int\int_S (\nabla \times\textbf{F})\centerdot d \textbf{S}\end{equation*} Evaluate $\oint _C \textbf{F}\centerdot ...
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39 views

Arc length integration

Find the length of the arc formed by $x^2=10y^3$ from point A to point B, where A=(0,0) and B=(100,10). My attempt: $\int_0^{100} \! \sqrt{1+(\frac{2}{3x})^2} \, \mathrm{d}x. $ However this integral ...
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1answer
43 views

Prove that $\sum_{k=-\infty}^\infty e^{-j2\pi f k T}=\sum_{k=-\infty}^\infty\delta(f-\frac{k}{T})$

This is part of a proof itself. $\sum_{k=-\infty}^\infty e^{-j2\pi f k T}=\sum_{k=-\infty}^\infty\delta(f-\frac{k}{T})$ $\delta$ is Dirac function. It's been a while I am thinking about this part ...
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1answer
30 views

Implicit partial derivative of a spherical cap

Consider a spherical cap, for which the base radius is $a$ and the height is $h$. Then, the surface area and volume is (these equations can be found on Wolfram Mathworld) $A(a,h) = \pi(a^2 +h^2)$, ...
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1answer
56 views

Total differential of a function $F(x, y)$

I found this definition of total variation of a function $F(x, y)$ \begin{eqnarray} dF(x, y) &\equiv& F(x+dx, y+dy) -F(x, y)\\ & = & [F(x+dx, y+dy) - F(x, y+dy)] + [F(x, y+dy) - F(x, ...
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55 views

Can this integral be expressed in terms of elementary functions? $\int\frac{\ln(2+x^2)}{1+x^2}dx.$

If the following primitive can be computed, can someone show me the way ? $$\int\frac{\ln(2+x^2)}{1+x^2}dx.$$ I tried substituting $x$ with $\tan(t)$ but the integral didn't get much simpler.
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1answer
27 views

An example of a function which is (locally) above its tangents but is not convex

I am looking for an example of a function $f:[a,b]\rightarrow\mathbb{R}$, continuous in $[a,b]$, differentaible in $(a,b)$, such that for any $a<t<b$ there is some open interval $t\in I\subseteq ...
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43 views

How does one calculate $\int_0^1 \frac {\arcsin(x)}{x}dx$? [duplicate]

How can I evaluate the following? $$\int_0^1 \frac {\arcsin(x)}{x}dx.$$ Could not find a primitive, so I went for some other methods like arranging it as a double integral or introducing a ...
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1answer
34 views

simple dumb vector question

so I've started vector calc otherwise known as calc 3 and simple question for the image above does the vector $\parallel v \parallel$ or $\overrightarrow{PQ}$ change as the initial point moves ...
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41 views

Function with bounded derivative as ODE

Given a function $x(t)$, I am looking for a function $y(t)$ which closely follows $x(t)$ except that its derivative must be bounded by a constant $c$, i.e. $\dot{y} \leq c$. Is there a way to describe ...
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27 views

Additive like function representation [duplicate]

Let's $f:R\rightarrow R,~r\gt0.$ And we have that for every $x,y\in R\Rightarrow |f(x+y)-f(x)-f(y)|\le r.$ Prove that there are $h:R \rightarrow R$ additive and $g:R \rightarrow [-r,r]$ functions such ...
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1answer
32 views

Differentiation question find the normal to the curve

Hi I was struggling to this question, can anyone please help me :P The curve $C$ has equation $2x^2+y^2=18$. Determine the coordinates of the four points on $C$ at which the normal passes through the ...
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26 views

Local extrema given the graph of a function's derivative

I am given a graph of the derivative of a function and answered most of the questions, but am still stuck at answering where the local extrema are. I had a sample question to reference from and it ...
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33 views

Determine all points where 2 variable function is differentiable: when to use definition?

My function is: $$f(x,y) = \frac{x\sin^2 y}{x^2+y^2}\text{ when }(x,y) \ne (0,0)$$ and $0$ when $(x,y) = (0,0)$ When I use the definition (limit as $h$ approaches $0$), the limit for both partiales ...
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1answer
40 views

A question about extending solutions of an ODE

Suppose I have two functions $f_1(x)$ and $f_2(x)$, which are related by the differential equation: $$\cos(x)f_1'(x)=−\sin(x)f_2'(x)$$ I would like to find a solution over $x\in[\frac\pi6,\pi]$ such ...
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33 views

Jacobian elliptic functions regarded as functions of the modulus.

Could anyone provide any references where the properties of Jacobian elliptic functions (and their derivatives) regarded as functions of the modulus are discussed? Especially it would be interesting ...
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23 views

Cap-Independence

I'm just trying to figure out something regarding Cap-Independence. The problem reads $\partial S:= r(t)=(\cos t,\sin t,\sin 2t)$, $0\le t \le 2\pi;$ $\phi=z\,dzdx-6y^2dxdy$ ($\partial S $ ...
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24 views

Uniform Convergence of integrals of sequences of functions

Just looking for feedback on if I am thinking about this correctly: Let {$f_n$} be a sequence in $\mathscr{R}[a,b]$ (the set of Riemann integrable functions on $[a,b]$, not sure if this is standard ...
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2answers
78 views

I tried to develop the solution of this challenging integration but I never made it. Perhaps you could..

Integrating this term looked simpler but solving it stormed my mind. I found it rather difficult to integrate $\sqrt{\text{sin}x}$. When I solve it , an infinte sequence arrived and I got trapped.
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23 views

Change of variables to flatten the boundary

It is known that one can perform a change of variables to flatten a $C^2$ domain $\Omega$, that is, for any point $x \in \partial \Omega $, there is a $C^2$ diffeomorphism $\psi$ which maps a ...
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42 views

a step function 2

Let $f$ be a function defined by $f: [0,1]\longrightarrow \mathbb{R},\quad t \longmapsto \begin{cases} \lfloor \frac{1}{t} \rfloor & t\in (0,1] \\ 0 & t=0 \end{cases} $ $f$ is it a step ...
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14 views

Differential calculus on locally convex spaces

For real finite dimensional vector spaces $V,W,Z$, i know that a map $f:V \times W \to Z$ is smooth if the maps $f(v,.)$ and $f(.,w)$ are for every $v\in V, w \in W$. Does the same thing hold for ...
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2answers
57 views

Question on asymptotes

Consider a function $f: \mathbb{R} \to \mathbb{R}$ that has an asymptote at $- \infty$ of the type $y=\lambda x + \beta$. According to trigonometry $\lambda=\tan{\theta}$ for a very small value of x ...
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1answer
26 views

Determinant of the Hessian matrix where all second derivatives have no variables

first time posting. I've come across a little problem. For, $f(x) = 3x^2+2y^2+3x-1$ I have found the following partial derivatives: $f_x = 6x + 3 $ $f_y = 4y$ $f_{xx} = 6$ $f_{yy} = 4$ $f_{xy} ...
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1answer
58 views

How find the minimum of $a$ ,if $f(x)=-\frac{\ln{x}}{x}+e^{ax-1}-a,x>0$

Question: let $$f(x)=-\dfrac{\ln{x}}{x}+e^{ax-1}-a,x>0$$ if such $$f(x)_{min}=0,\forall x>0$$ **Question: Find the $a$ minimum of the value. My idea: this problem equivalent to ...
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33 views

Proof Validation Function From Integers to Rationals is Continuous

I am teaching myself real analysis, so any help is greatly appreciated. Let the function be defined as $F : Z \rightarrow Q$ where $Z$ is the set of integers and $Q$ is the set of rational numbers, ...
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42 views

Abel's Functional Equation for $L(x) = \sum x^{n}/n^{2}$

In "Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work" Hardy talks of Abel's Functional Equation $$L(x) + L(y) + L(xy) + L\left(\frac{x(1 - y)}{1 - xy}\right) + L\left(\frac{y(1 - ...
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1answer
37 views

Help with limit equation

The question is: $$\lim_{x \to + \infty} \frac{3x^4 - 11x^3 + 42x}{-15x^4 + 3x^2 +101}$$ So far my answer: $$\lim_{x \to + \infty} \frac{x^4}{x^4} \frac{3 - \frac{11x^3}{x^4}+\frac{42x}{x^4}}{-15 + ...
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44 views

Calculate the rate of change of volume, $V$ (in $m^{3}$), of water in a tank

$V=t^3-5t^2+6t+2$, where $t$ is the time in hours measured from a particular instant and $0 \leq t \leq 4$. Find the rate of change of volume with respect to time after: (i) $0.5$ hours My attempt: ...