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

learn more… | top users | synonyms

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
119 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 ...
2
votes
0answers
42 views

How to prove the only solution to f(x)=f'(x) is ce^x? [duplicate]

Possible Duplicate: Proof that $\exp(x)$ is the only function for which $f(x) = f'(x)$ I've learned from calculus that $Ce^x$ is a solution to the equation $f(x)=f'(x)$, where $C$ is a ...
2
votes
0answers
60 views

Integration with multiple derivatives

Suppose I have that: $$F(x)= \frac{d H(Z)}{dZ}\cdot\frac{d^2 Z}{d x \,d y}$$ Now I want to find $$\int F(x) \, dx $$ So can I say that $$\int F(x) \, dx =\int \frac{d H(Z)}{dY} \, dZ\ ?$$ Any ...
2
votes
1answer
423 views

Stokes's Theorem and Divergence Question

Just a theory question. Does a surface necessary have to be closed for Stokes's theorem to apply? I know for it is true for Green's theorem and it is supposed to be a baby version of Stokes's theorem. ...
2
votes
0answers
294 views

Lagrange Multiplier for discrete variables

I haven't formulated my problem formally yet. But, the goal is to minimize the squared error i.e. $ (L - \sum_{i=1}^{N}{w_{i}l_{i}})^2$ where $l_{i}$ are constants, $w_{i}$ are weights to be ...
2
votes
0answers
116 views

Identity show the following, divergence.

Calculate $\nabla\cdot F$ and $\nabla\wedge$$F$ for the vector function $F(x,y,z)=(a\wedge r)\wedge r$ where $r=xi+yj+zk$ and $a$ is a constant vector. So little stuck here, I have used the fact ...
2
votes
0answers
65 views

solve $\int_{-\infty}^{\infty}\frac{e^{qiy -K(\sqrt{\lambda -a-iy}-\sqrt{\lambda})}}{a+iy}$?

Is there any way to solve this integral? $$\int_{-\infty}^{\infty}\frac{e^{qiy -K(\sqrt{\lambda -a-iy}-\sqrt{\lambda})}}{a+iy} dy$$ where $K,\lambda, a$ and $q$ are real numbers and $K>0$, ...
2
votes
0answers
520 views

Showing that a curve is not rectifiable if its arc length is not a continuous function

This is a (translated) proof from a textbook of the fact that arc length of a rectifiable curve is a continuous function. Let $\phi:[T_0,T_1]\rightarrow\mathbb C$ be a function whose real part and ...
2
votes
1answer
130 views

Condition for inequality for norms

Let $a$ be a vector in $R^m$. Under which condition the following inequality is always true: $$ \sqrt m \|a\|_{\infty}\leq \|a\|_2^2-\frac{\left(\sum_{i=1}^ma_i\right)^2}{m}. $$
2
votes
0answers
436 views

What is a hump function?

I have been in trouble with the hump function(s) What are them? Could you give me an explicit formula for "Hump"(not bump) function. Thanks
2
votes
1answer
329 views

Convergence of a sequence written as infinite products

Let $$a_n=\prod_{j\in\mathbb{Z}}\frac{1+\exp(-2n e^{-|j|/n})}{1+\exp(-(1+e^{-1/n})n e^{-|j|/n})}$$ Then each term in the product goes to $1$ as $n\to\infty$. Does $a_n\to 1$?
2
votes
0answers
204 views

I want to prove the following limit $ \lim_{x\rightarrow 0}\frac{\sin x}{x}=1 $ using the definition of limit. [duplicate]

I want to prove the following limit \begin{equation} \lim_{x\rightarrow 0}\frac{\sin x}{x}=1 \end{equation} using the definition of limit. Then: \begin{equation} \left| \frac{\sin x}{x}-1 ...
2
votes
0answers
322 views

Confused about notation and derivatives inside integrals

EDIT: To make what I am asking more clear. I've simplified it and have a more direct question. Let's say I am writing out an expression, and I want to write: $$\int_0^xF'(y)\,dy$$ However, for ...
2
votes
0answers
632 views

Solving the terminal velocity equation with a initial velocity.

I was trying to create a better answer for this Stack Overflow question. I wanted to give the person a example of code using air resistance however every example on the net I find shows the formula : ...
2
votes
2answers
140 views

Are there any places to get highly graphical/visual math videos, specifically for calculus?

I love watching National Geographic and Discovery channel pieces on the universe/outer space because they are so visually appealing, but if I had to read about the topics, I wouldn't have much ...
2
votes
0answers
220 views

How to prove that $e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)} = \lim_{n \to \infty} \sqrt[n]{n\#} $?

While reading this post, I stumbled across these definitions (Wiki_german) $$e = \lim_{n \to \infty} \sqrt[n]{n\#}$$ and $$e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)}.$$ The last one seems ...
2
votes
1answer
72 views

Finding min or max on $f(x) = a x e^{1+ax}$

I have the function $f(x) = a x e^{1+ax}$ and I want to find where it has a min or max value. To do this I calculate the derivative $f'(x) = a^{2}x e^{1+ax}$. This is equal to $0$ only if $a=0$ ...
2
votes
1answer
925 views

Curvature of a parametric curve in three-dimensional space

Given a parametric curve $$x=t\cos t, y=t\sin t, z=at$$ I try to calculate the curvature by using http://en.wikipedia.org/w/index.php?title=Curvature&section=8#Local_expressions_2 I checked the ...
2
votes
0answers
224 views

What is the constant $e$, fundamentally? [duplicate]

Possible Duplicate: Why is the number e so important in mathematics? Intuitive Understanding of the constant “e” The number $e$ is important in many respects. If you ask ...
2
votes
0answers
372 views

Preparing for reading Penrose's “Road to Reality”

I am reading Road to Reality by Roger Penrose and I although I know about calculus, complex analysis, differential equations I do not know about manifolds, Riemann surfaces and so on. Which books can ...
2
votes
0answers
120 views

How to solve this equation

How do I solve the equation given below for values for $\epsilon$? $$ \sum_{j=1}^{n-1} (2{\alpha_{j}}+1) ~~ \prod_{k=1,~k\neq j}^{n-1} \bigg(({\alpha_{k}}-1){\Big((2{\alpha_{k}}+3) + ...
2
votes
0answers
257 views

How to evaluate $I_1=\int\frac{\left(\frac{a}{y}-\frac{a}{b}\right)^{1/2}}{1-\frac{a}{y}} \mathrm {d} y$?

How to evaluate this integral? $$\displaystyle I_1=\int\frac{\left(\frac{a}{y}-\frac{a}{b}\right)^{1/2}}{1-\frac{a}{y}}\mathrm {d}y$$ Here $a$ and $b$ are real constants and $y$ real variable. It is ...
2
votes
0answers
48 views

Trouble grasping surface area formula variations

In the surface area formulas above, I understand that the f(x) will just be what f(x) (aka y) equal (e.g. for y=x+4, f(x) in that formula will be x+4) but I do not understand what h(y) will be. Will ...
2
votes
0answers
117 views

Is there an interpretation for this classic identity? [duplicate]

Possible Duplicate: Different methods to compute $\sum_{n=1}^\infty \frac{1}{n^2}$. $\sum \frac{1}{n^2}= \frac{\pi^2}{6}$ There are a few proofs for that fact but can anybody see why is ...
2
votes
2answers
669 views

find the extremal of a function

i need to find the extremal of the functional $\int I(y,y'') dt$. Could anyone tell me the concepts of finding the extremal so that i can go about solving this one? Update: So my function ...
2
votes
0answers
70 views

Relating product integrals to indefinite products

The product integral is the multiplicative version of standard integrals. Indefinite products are the discrete counterpart to this integral; they multiply iterations on a function $f(x)$ by each ...
2
votes
1answer
200 views

How to find $\int_{-\pi/2}^{\pi/2} \frac{\sin y}{\sqrt{\sin y+c}}\ \text{d}y$?

This is related to my previous asked question here. The integration can be simplified as (As pointed out by Sivaram, the integral diverges to $-\infty$ as $c\to1+$): $$\int_{-\pi/2}^{\pi/2} ...
2
votes
0answers
322 views

Uniformly continuous functions

Given a uniformly continuous function $f(x)$ on the real numbers $\Bbb R$, then by the definition of uniform continuity this means: for any $\epsilon>0$ there exists $\delta >0$ such that ...
2
votes
1answer
124 views

What structure is this?

I had hoped that this would be easier. I have power series that I'm using. I want to know what simple structure describes the operations that I'm performing on them. It would be better if I could ...
2
votes
0answers
218 views

Is this the right way to evaluate this integral?

I am trying to evaluate this integral or at least get bounds for its aboslute value. I have where $\tau \to \infty$ $$\int_{1}^{\infty} f(t) \frac{\tau \sin(\tau\log t)}{t^{\sigma+1}} dt$$ $f(t) = ...
2
votes
0answers
203 views

How to find a function which satisfies such functional equation?

How to find a function which satisfies: $$a^x=\lim_{h\to\infty} \left( f_a \left(f_a^{[-1]}(x)+\frac xh\right)\right)^{[h]}$$ where $f^{[n]}(x)$ is the number if iterations of a function (if n=-1 ...
2
votes
0answers
281 views

Sharp (Reverse) Harmonic-Arithmetic Mean Bounds

Let $\mathbf{x} =$ {$x_{i}$} be a set of $n$ positive reals. In every good book on inequalities, one finds the classical result \begin{eqnarray} AM(\mathbf{x}) \geq GM(\mathbf{x}) \geq ...
2
votes
1answer
72 views

Is there a way of solving integrals where the numerator is an integral of the denominator?

Is there a way of solving integrals where the numerator is an integral of the denominator? I was evaluating the integral $$\int \frac{x-\sin x}{1-\cos x}\mathrm{d}x$$. I separated the numerator into ...
2
votes
2answers
55 views

Asymptotic behavior of elliptic integral (first kind)

I came accross some obstacles in proving that the time $T(\delta)$ taken by a pendulum to travel from $\theta=\pi-\delta$ to a considerably distant angle $\theta=\theta_0\in(0,\pi/4)$ diverges ...
2
votes
1answer
63 views

Integral Evaluation.

How can we justify the fact that some integrals can't be evaluated? It's like we can't sum up a function within two bounds or we are unable to find the area under the curve of a function. How's that ...
2
votes
1answer
54 views

Derivative at the point of inflection

Is it always true that when a function changes concavity from concave-up to concave-down, its derivative at the point of inflection is undefined? And in the reverse order the derivative is Zero?
2
votes
3answers
119 views

One point following another moving in a straight line?

There is a plane with two points on it, let's say A and B. A starts at an arbitrary constant point, let's say $(0, 0)$, and $B$ at a point that needs to be tested, which we'll call $(c, d)$. A moves ...