Articles of calculus

prove that the rose (in the polar plane) has $2n$ “petals” when $n$ is even

prove that the rose $r=\cos(n\theta)$ (in the polar plane) has $2n$ “petals” when $n$ is even. How can I start this demonstration? I would appreciate your help

Prove that a series converges if $|a_n|<1$

Given a succession of real numbers $\{a_n\}_n$ with $n \in \mathbb{N}$, I would like to prove, or find a counterexample, of the fact that if $|a_n|<1$ from a $n$ on then, then $\begin{aligned} \sum_{n=0}^{\infty} (a_n)^n \end{aligned}$ converges. Someone could give me a hand? Thank you!

Prove that the length of a curve is given by integral

I have found the following problem in “Introduction to Analysis” by Rosenlicht. I am not sure if my solution is correct and I highlighted my uncertainties. First we have to show that the set $$S=\Bigg\{\sum_{i=1}^N d(f(x_{i-1}),f(x_i)):x_0,x_1,\ldots , x_N \mbox{ is a partition of } [a,b]\Bigg\}$$ is bounded from above and that the sequence of the […]

Using the Mean Value Theorem, show that if $f'(x) > 0$ $\forall x \in (a, b)$ then $f$ is increasing on $(a, b)$

Using the Mean Value Theorem, show that if $f'(x) > 0$ $\forall x \in (a, b)$ then $f$ is increasing on $(a, b)$. The Mean Value Theorem states: a function $f$ which is continuous on the closed interval $[a, b] $ $^{\textbf{(1)}}$ and differentiable on the open interval $(a, b)$ $^\textbf{(2)}$ has at least one […]

Describing asymptotic behaviour of a function

For question B! x^2+x+1/x^2 = 1+ [x+1/x^2] shouldnt the answer be asymptote at x=0 and y=1 ?? i dont understand the textbook solution

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 $x^{1/x}=y^{1/y}$?