Articles of calculus

R as a union of a zero measure set and a meager set

Let $ \left\{ {r_i } \right\}_{i = 1}^\infty = \mathbb{Q}$ an enumeration of $\mathbb{Q}$. Let $ J_{n,i} = \left( {r_i – \frac{1} {{2^{n + i} }},r_i + \frac{1} {{2^{n + i} }}} \right) $ $ \forall \left( {n,i} \right) \in {\Bbb N}^2 $ Then define $$ A_n = \bigcup\limits_{i \in {\Bbb N}} {J_{n,i} } $$ […]

Order preserve after taking expectation “piecewisely”

There are four continuous functions $\mathbb{R^+}\rightarrow \mathbb{R^+}$, $g_1(x),g_2(x),g_3(x),g_4(x)$, and they satisfy $g_1(x)g_2(x)<g_3(x)g_4(x)$ for $\forall x$, I’m wondering under what conditions about $g$ or their relationship will we have $E[g_1(x)]E[g_2(x)]<E[g_3(x)]E[g_4(x)]$ for an arbitrary probability distribution.

How would I undo a gradient function?

If we are given a vector, how can we tell if that is a gradient of a vector? And how would we find the original function? I was assigned this problem, and I know how to get a gradient of a function, but not how to go backwards. $$\bigl(6\cos(x^2+4y^2) – 12x^2 \sin(x^2+4y^2)\bigr)\vec \imath + \bigl(-48xy\sin(x^2+4y^2)\bigr)\vec\jmath$$ […]

$f(x)$ monotonic integrable function and $\lim_{x\to \infty}\frac{1}{x}\int_{0}^{x}f(t)dt=a$, Prove: $\lim_{x\to \infty}f(x)=a$

Let $f(x)$ be a monotonic increasing function on $[0,\infty)$ and for every $x>0$ it is integrable in $[0,x]$, so that $\lim_{x\to \infty}\frac{1}{x}\int_{0}^{x}f(t)dt=a$. I need to prove that $\lim_{x\to \infty}f(x)=a$. I tried to use the limit definition: $|\frac{1}{x}\int_{0}^{x}f(t)dt -a|<\epsilon$ and to use the fact that $\int_{0}^{x}f(t)dt\geq x\inf f(x)$, but I can’t see how does it help […]

$\int_0^{a} x^\frac{1}{n}dx$ without antiderivative for $n>0$

My exercise is to find $\int_0^{a} x^\frac{1}{n}dx$ without antiderivatives for $n>0$. The first thing I did is plot of some of the $x^\frac{1}{n}$ for the first twenty n. This is what I got. I wanted to make sure that the functions were non-negative and monotonic and I am sure they are now. So I know […]

Cox derivation of the laws of probability

I have read Jaynes’ Probability Theory: The Logic of Science a while ago, but mostly skimmed over parts of his derivations that I didn’t immediately understand. Now I’m trying to really understand it, but it appears he mostly skips over steps he sees as obvious or trivial. Now, I’ve been able to construct most of […]

How to prove that $\int_0^b\Big(\int_0^xf(x,y)\;dy\Big)\;dx=\int_0^b\Big(\int_y^bf(x,y)\;dx\Big)\;dy$?

Problem. Let $f:[0,b]\times[0,b]\to\mathbb{R}$ be continuous. Prove that $$\int_0^b\left(\int_0^xf(x,y)\;dy\right)\;dx=\int_0^b\left(\int_y^bf(x,y)\;dx\right)\;dy.\tag{1}$$ My first thought was to use Fubini’s theorem: the left hand side of $(1)$ equals the double integral $$\iint_{D_1} f(x,y)\;dA\tag{2}$$ and the right-hand side equals $$\iint_{D_2} f(x,y)\;dA,\tag{3}$$ where $D_1=\{(x,y);\;0\leq x\leq b,\;0\leq y\leq x\}$ and $D_2=\{(x,y);\;0\leq y\leq b,\;y\leq x\leq b\}$. Since $D_1=D_2$, the integrals $(2)$ and $(3)$ are […]

Prove that the sum of convex functions is again convex.

I must to prove that the sum of convex functions is again convex. I know the definition of convex function: $f(tx_1+(1-t)x_2)\leq f(x_1)+(1-t)f(x_2)$ – this the first convex function, then I have the second one $g(tx_1+(1-t)x_2)\leq g(x_1)+(1-t)g(x_2)$ What should I do next? Thank you for your help and time.

Why is $x^{1/n}$ continuous?

Why is $x^{1/n}$ continuous for positive $x,n$ where $n$ is an integer? I can’t see how it follows from the definition of limit. And I don’t see any suitable inequalities so is this an application of Bernoulli’s or Jensen’s inequality?

Which one is bigger? $e^{\pi} $ or $\pi^e$

This question already has an answer here: Comparing $\pi^e$ and $e^\pi$ without calculating them 9 answers