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

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2
votes
2answers
99 views

Is this proof (using Rolle's Theorem) correct?

Let $\ f:]a,b[\rightarrow\mathbb{R}$ be a differentiable function on $]a,b[$ so that : $$\lim_{x\rightarrow a}_{>}\ f(x) = \lim_{x\rightarrow b}_{<}\ f(x) = +\infty$$ Show that there exists ...
2
votes
1answer
87 views

Which Test for Divergence/Conv should be used with a sin(n) in the numerator?

I see that quite a few questions on infinite series have been asked recently and figured why not continue the trend! going through my lectures and textbooks, I understand $\sin^2n$ is bounded by 0 ...
2
votes
0answers
380 views

Multivariable Inverse Function problem

Consider the system of equations $$\left\{\begin{align*} &x^5 v^2 + 2y^3 u = 3\\ &3yu - xu v^3 = 2\;. \end{align*}\right.$$ Show that near the point $(x,y,u,v) = (1,1,1,1)$, this system ...
2
votes
1answer
208 views

Equivalence of two norms

Define two norms as following: $$ \left\Vert f\right\Vert _{1}={ \max_{0\leq x\leq1}\left|f\left(x\right)\right|} , \quad\text{ and }\quad \left\Vert f\right\Vert ...
2
votes
1answer
276 views

How to calculate the Poisson integration?

When solving Laplacian equation, I need to integrate the following integration: $$ \int_0^{2\pi}\frac{1+3\text{sin}\theta}{a^2+r^2-2ar\text{cos}(\theta-\phi)}\text{d}\theta $$ How to work it out?
2
votes
1answer
363 views

Applied Math for economics question: mostly algebra help

I am teaching myself the calulus component necessary to get thorugh an econ based stats and applied math class. My algebra is killing me please help - the practive problem is given $y = –x^3 + 7x – ...
2
votes
3answers
90 views

Derivative of $\csc(f(x))$

Could someone explain how can I find the derivative: $$ \frac{d}{dx}\csc[f(x)] = ? $$
2
votes
0answers
90 views

is the positive part of derivative…?

If I have a very nice function as $f\in \mathcal{C}_0^\infty$, (infinitely many times differentiable with all derivatives bounded and continuous and with compact support). Can I apply integration by ...
2
votes
1answer
248 views

Reducing to exact form by integrating factor

$$(6xy)dx = (4y+9x^2)dy$$ to find out if its exact $$M= 6xy, N =4y+9x^2 $$ $$ \frac {dM}{dy} = 6x, \frac {dN}{dx} = 18x$$ Hence its not exact. Please correct me if i did something wrong and help me to ...
2
votes
1answer
93 views

What does brace below the equation mean?

An example of what I am trying to understand is found on this page, at Eq. 3. There are two braces under the equation... What is the definition of the brace(s) and how does it relate to Sp(t) and ...
2
votes
0answers
73 views

Proving Symmetrized Kullback-Leibler divergence

Kullback in his "Information theory and statistics" gives the symmetrized divergence as follows $J(1,2)=\iint(f(x,y)-g(x)h(y))log{\frac{f(x,y)}{g(x)h(y)}}$ Later (p.8), he states that symmetrized ...
2
votes
0answers
312 views

$\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$

I'm trying to solve the integral $\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$, where $s$, $r$ and $m$>1 are positive integers. My question is whether a closed form ...
2
votes
0answers
172 views

Normalization constant

This is probably a very obvious, but I am slightly confused. Suppose $f(\xi)$ is such that when $f^2(\xi)$ is integrated from $\xi=-\infty $ to $\xi=+\infty$ equals to $1$ and correspondingly for ...
2
votes
0answers
138 views

uniformly convergent

Define functions $f_n\colon [0,1]\to\Bbb R$ by $f_n(x)=n^px\exp(-n^qx)$ where $p$, $q>0$ and $f_n\to 0$ pointwise on $[0,1]$ as $n\to \infty$ and $\sup|f_n(x)|=(n^{p-q})/e$. Assume that ...
2
votes
1answer
3k views

When will the ball hit the ground.

A girl is throwing a ball. If I have an equation $h= -4.9*t^2+20t+1.2$, where $t$ is number of second after the girl has thrown the ball, and $h$ is the height above ground. I want to find out when ...
2
votes
1answer
140 views

Yet another Integral involving $e^{ax} +1$ and $e^{bx} + 1$.

A variation on: Another Integral involving $e^{ax} +1$ and $e^{bx} + 1$ Evaluate the integral $$I(a,b)=\int_{0}^{1}\frac{(e^{ax})(e^{bx})}{\left(e^{ax}+1\right)\left(e^{bx}+1\right)}dx$$ for ...
2
votes
0answers
66 views

Laplacian of a function implies the function cannot have max or min.

If $\bigtriangledown^2f = 0 $ in some region in the space, then $f$ cannot have maximum or minimum on that region. My approach was to assume $f$ has a maximum and then use the second derivative test ...
2
votes
0answers
139 views

Find the height of the centre of mass of an annular hemisphere

So as stated above I need to find the centre of mass of a annular hemisphere. the outer radius is $R$ and the inner radius is $a$. In a pкevious part to the question I found the volume of the annular ...
2
votes
2answers
61 views

Applicability of derivatives and integrals

Why do derivatives and integrals work? I understand the concept on how to apply them, but what makes it possible for them to be used? why does taking the derivative of volume equal an object's surface ...
2
votes
0answers
84 views

Is it possible for a function to be differentiable in the complex plane but not in the real plane?

I asked the question in the title. Is it possible for a function to be differentiable in the complex plane but not in the real plane? Could you help me find some examples or explain how it is ...
2
votes
1answer
182 views

Derivative of composition

I need to express the derivative of $f^{(n)}(x)$ in terms of $f'(x)$, meaning $f$ composed with itself n times. I was able to express $f(f(x))=f'(f(x))f'(x)$ Is the derivative of the composition ...
2
votes
0answers
389 views

On the geometric arguments used in Newton's *Principia Mathematica Naturalis Philosophae*

When one reads Newton's Principia Mathematica, one is immediately aware of the complexity of the synthetic geometry that he uses to prove his propositions. This I understand because all of the ...
2
votes
1answer
309 views

Help proving a condition where the greatest lower bound of a set is equal to the least upper bound of another.

So I'm just not sure if the proof I have given for this claim is acceptable, and I was wondering if anyone could help me smooth it out. Thanks: Let E be a nonempty subset of real numbers which is ...
2
votes
1answer
461 views

Taylor Expansion of $f(x)=\sin x$

The Taylor Expansion of $f(x)=\sin x$ with a Lagrange remainder is: $\sin x = x-{x{3}\over 3!}+{x^{5}\over5!}+\cdots+{(-1)^{m-1}x^{2m-1}\over(2m-1)!}+{(-1)^{m}x^{2m+1}cos \theta x\over(2m+1)!}, ...
2
votes
0answers
58 views

Can a rate be proportional to a shape?

This question may be a little vague, but it has a point. I woke up this morning with an idea. Let's say I wanted to design a projectile that has a velocity proportional to its 'shape'. When the ...
2
votes
0answers
52 views

linear DE - Where did I go wrong?

I'm trying to find the general solution for $(x+2)y' = 3-\frac{2y}{x}$ This is what I've done so far: $y'+\frac{2y}{x(x+2)}=\frac{3}{x+2}$ $(\frac{x}{x+2}y)'=\frac{3x}{(x+2)^2}$ ...
2
votes
0answers
45 views

How do we use Galois theory to show that an integral has no closed form? [duplicate]

Possible Duplicate: How can you prove that a function has no closed form integral? How do we use Galois theory to show that an integral has no closed form ? I know this is called ...
2
votes
0answers
73 views

Stereographic projection in $\mathbb{R}^n$

I'm trying to verify that if $\phi(x_1, \cdots x_n, x_{n+1}) = \left(\frac{x_1}{1-x_{n+1}}, \cdots \frac{x_n}{1-x_{n+1}} \right) $ then $$\phi^{-1}(\zeta_1 ,\cdots \zeta_n) = ...
2
votes
0answers
89 views

laplacian for functions problem with the integral on manifolds

I'm following the proof of the local expression for the Laplacian on a compact manifold and I'm having problems understanding how the integral on a manifold translates into an integral in $R^n$, in ...
2
votes
0answers
58 views

Is exponent of discrete-analytic function also discrete-analytic?

Lets define a discrete analytic function such a function that is equal to its Newton series: $$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)$$ Is function $g(x)=e^{f(x)}$ also ...
2
votes
0answers
122 views

How can it be proven that the Gaussian function has no “trivial” primitives? [duplicate]

Possible Duplicate: How can you prove that a function has no closed form integral? Why can't erf be expressed in terms of elementary functions? By trivial primitives I mean ...
2
votes
0answers
464 views

Gateaux and Frechet derivatives and boundedness

Suppose $f:X \to Y$ is map between Banach spaces. We know that the Frechet derivative of $f$ at $x$ is a bounded map by definition if it exists (satisfying certain properties that I won't write) ...
2
votes
1answer
128 views

Help in understanding integration by changing the variable

I need help better understanding how, and why integration by changing the variable works (I've seen it's related to the derivative of a composite function $f(g(x))$), and generally tips and tricks, an ...
2
votes
0answers
72 views

Compute: $I=\int_{0}^{\infty} \frac{e^{-a x} - e^{-b x}}{x} dx$ [duplicate]

Possible Duplicate: Best way to integrate $ \int_0^\infty \frac{e^{-at} - e^{-bt}}{t} \text{d}t $ Compute the improper integral: $$I=\int_{0}^{\infty} \frac{e^{-a x} - e^{-b x}}{x} dx$$
2
votes
2answers
482 views

Does multiplying by $dt$ have any meaning?

Consider, for example, the equation $x'=x$, then it is usually solved by writing $\frac{dx}{dt}=x\implies\frac{dx}{x}=dt\implies\int\frac{dx}{x}=\int dt$ ... I know that there is a theorem in ODE ...
2
votes
1answer
278 views

Derivative of an indicator inside an integral

I have a fairly basic question that relates to understanding a particular derivation. I have the following function $Q(x) = E\left[I(F(x+\varepsilon)>c)\right]$, where $x \in R$, $\varepsilon ...
2
votes
0answers
455 views

Adjoint of the infinitesimal generator of a stochastic process

I need help seeing that $$ \mathcal{L}^* g = -\frac{\partial (bg)}{\partial x} + \frac{1}{2}\frac{\partial^2(\sigma^2g)}{\partial x^2} $$ is the adjoint operator of $$ \mathcal{L} = b\frac{\partial ...
2
votes
2answers
192 views

Differential equation $d^n/dx^n f(x)=\pm k^2f(x)$

How to solve this differential equation: $$\frac{d^nf(x)}{dx^n}=\pm k^2f(x)$$ For $n=1,2,3$ and $\forall n\in\mathbb{N}$, and both signs, if this is possible.
2
votes
0answers
149 views

Taylor expansion: $\lim\limits_{x\to 0}{\frac{\sqrt{x}\sin{\sqrt{x}}+\log(1-x)}{x-\tan{x}}}$

Could someone tell me if my proceeding is correct? $$\lim_{x\to 0}{\dfrac{\sqrt{x}\sin{\sqrt{x}}+\log(1-x)}{x-\tan{x}}} =$$ $$= \lim_{x\to ...
2
votes
0answers
118 views

References on the equivalence of different definitions of integrability

While writing a chapter of a book about mathematical analysis, I decided to compare some definitions of integrability that are usually taught to sophomore students, in Italy. I briefly collect four ...
2
votes
2answers
182 views

Surface Integral Equations

I am studying for a Calculus exam, and one of the topics I should know about is surface integrals. Now, I am using Stewart 6e, and in there I have found several equations for computing surface ...
2
votes
0answers
74 views

polar form of a double integral

given the following region $R=\lbrace m,n \geq0$, $1 \geq m+n \geq 2\rbrace$ where $(m,n) \in \mathbb{R}^2$.write in polar coordinates $(r, \theta)$ the following double integral $\int\int_R m \,dA$
2
votes
1answer
139 views

Volume by integration

Find the volume generated by revolving the area bounded by $y^2=x^3$ , $x=4$ about the line $x=1$. I can't understand how the area will revolve about a line lying on the area. Many thanks in ...
2
votes
0answers
85 views

The number of roots of an equation

Suppose that $f$ is a twice differentiable function such that $$f(a) = 0,\, f(b) = 2,\, f(c) = − 1,\, f(d) = 2,\, f(e) = 0,$$ where $a < b < c < d < e$. What is the minimum number of ...
2
votes
0answers
60 views

$f(x) \rightarrow \infty $ when $x\rightarrow 1^+$

I want to prove that $f(x) \rightarrow \infty $ when $x\rightarrow 1^+$. My tactic is to prove that no matter how big you choose a $N\in \mathbb{R}$, you can always find a $\delta>0$ so the ...
2
votes
0answers
143 views

compute this integration

$$\int_{x_1}^{x_2}\frac{\sqrt{\frac{1}{3}x^3+a}}{(1-x)\sqrt{x}\sqrt{-\frac{4}{3}x^3+x^2-a}}dx$$ where $a\in(0,\frac{1}{12})$ is a constant. In this case, $-\frac{4}{3}x^3+x^2-a=0$ has exactly two ...
2
votes
3answers
373 views

The concept of gradient, related to lagrange multipliers, surface areas, tangent hyper planes

As we all know, gradient is always perpendicular to the level curve. On the other hand, $\nabla f(a,b) \dot\ h$ where $h=(x-a\ \ \ \ \ \ y-b)^T$, give a tangent hyper plane which is tangent to ...
2
votes
0answers
120 views

Calculus equation from paper

I'm trying to understand a paper but it has an equation in it that is lost on me.. I don't think it's very advanced but unfortunately I've never had much experience with this kind of thing. (From The ...
2
votes
0answers
914 views

Two Disk/Washer Method Problems (given a diagram)

Given a diagram from Calculus of a Single Variable by Larson and Edward (9th edition): I am interested in finding the volume of various regions when rotated about various lines. Specifically, I am ...
2
votes
0answers
249 views

Motivating questions for some topics in undergraduate calculus

Being a grad student I'm going to teach a whole class for the first time the coming summer and I'm looking for some motivational problems which I could use to introduce different topics. In other ...