Articles of calculus

What is the limit $\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$

If $\vartheta_{2}(q)$ is jacobi’s theta function, what is the limit $$\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$$ for the nome $q$. I would like to know whether the limit exists or not. If it does, please let me know and provide it’s evaluation

A Series Problem from Calculus-2 course

This problem is from calculus-2 course. The basic knowledge includes integral test and $p$-series test. Find an $N$ so that $$\sum_{n=1}^\infty {1\over n^4}$$ is between $$\sum_{n=1}^N {1\over n^4}$$ and $$\sum_{n=1}^N {1\over n^4} + 0.005$$ The Answer is $N=5$. How to solve it? Thanks!

What is the constant $e$, fundamentally?

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 anyone why it is important, you will get multiple answers. Some say the defining property is that the derivative of $e^x$ is $e^x$. Some say that it […]

The form $w_p(x,y) = \frac{(x-a)\,dy-(y-b)\,dx}{(x-a)^2+(y-b)^2}$ is exact $\iff$ $\exists$ angle function

Given $p=(a,b)\in\mathbb{R^2}$, show that the $1$-form $w_p:\mathbb{R^2}-\{p\}\to (\mathbb{R^2})^*$, defined by $$w_p(x,y) = \frac{(x-a)\,dy-(y-b)\,dx}{(x-a)^2+(y-b)^2}$$ is closed. Prove that this form is exact in an open $U\subset \mathbb{R^2}-\{p\}$ if and only if there exists a continuous function (necessairly $C^\infty$) $\theta_p:U\to\mathbb{R}$ such that $\cos \theta_p (z) = \frac{x-a}{|z-p|}$ and $\sin \theta_p (z) = \frac{y-b}{|z-p|}$ for all $z=(x,y)\in U$. […]

Understanding the graph for $x^y = y^x$

I graphed $x^y=y^x$ and it is a union of the line y=x, with some other curve. So my first question is, how do I derive that other curve? My next question is, why don’t I get the same graph when I plot $x^{1/x}=y^{1/y}$?

solve $\lim_{x\rightarrow \infty}\frac{x^3}{e^x}$ using L'Hospitals rule

I am having problems with this question. I know the answer is 0 but I keep getting infinity over infinity. I am using L’Hospitals rule. Any help would be much appreciated 🙂 edit* I only used L’Hospitals rule once. edit*** Now it works when I use it 3 times 🙂

Integral of gaussian distribution divided by square root of x

I, maxima and WolframAlpha are struggling to evaluate the following integral: $$ \int_{0}^{\infty} {x^{-\frac{1}{2}}\exp{\left(-\dfrac{(x-\mu)^2}{2\,\sigma^2}\right)}}dx $$ There should be a probability distribution with wich one could model this function and use its normalization for integration but I could not find it either.

Derivative change sign

If the function $f:\mathbb{R} \to \mathbb{R}$, $f$ has an extremum at the point $x$, and $f$ is differentiable in some neighborhood of $x$. Is it right that the derivative changes sign when passing through $x?$

Finding a limit with a cube

I am reviewing for a test and I can not figure this out. $$ \lim\limits_{h\to 0}\frac {(h-1)^3 + 1}{h} $$ I tried to multiply by the conjugate and that game me nothing sensible.

Simple algebra in a differential equation.

I have the differential equation: $$\frac{dy}{dx}=\sin (x-y).$$ Substituting $v=x-y$ and $dy=dx-dv$, I got down to the equation:$$\frac{dv}{1-\sin(v)}=dx.$$ Multiplying the LHS by $\dfrac{1+\sin (v)}{1+\sin(v)}$, I got:$$(\sec^2(v)+\tan (v)\sec (v))dv=dx.$$ This is an easy integral, and I got that $x=\tan(v)+\sec(v)$, with some constant of integration. Now, doing this in Maple gives the result: $$\frac{2}{1-\tan(\frac{v}{2})}=x,$$ where there should be […]