Articles of calculus

proof of$\frac{\partial^2 f(x,y)}{\partial x\partial y}$=$\frac{\partial^2 f(x,y)}{\partial y\partial x}$

I was at my physics class(electrodynamics).I saw a relation which frequently uses in my course.Relation is that $$\frac{\partial^2 f(x,y)}{\partial x\partial y}=\frac{\partial^2 f(x,y)}{\partial y\partial x}$$.I saw this relations frequently used in the derivation of different formula for multiple gradient,curl etc. which is fundamental for many other important theorems.My doubt is that is it true for all […]

Limit $\lim\limits_{x\to\infty} \frac{5x^2}{\sqrt{7x^2-3}}$

Evaluate the following limit: $$\lim_{x\to\infty} \frac{5x^2}{\sqrt{7x^2-3}}$$ I’m not really sure what to do when there is a square root for an infinity limit. Please Help!

What is the limit of $\lim\limits_{x→∞}\frac{\sin x}{x}$

This question already has an answer here: Evaluating and proving $\lim_{x\to\infty}\frac{\sin x}x$ 6 answers

A bound on the derivative of a concave function via another concave function

Let $f:\mathbb{R}^+\to\mathbb{R}^+$ and $g:\mathbb{R}^+\to\mathbb{R}^+$ be two strictly concave, strictly increasing, twice differentiable functions, such that $f(x)=O(g(x))$ as $x\to\infty$, i.e. there exists $M>0$ and $x_0$ such that $$f(x)\leq Mg(x)\qquad \forall x\geq x_0.$$ Is it true that $f'(x)=O(g'(x))$ as $x\to\infty$? (this is an extensions of this question)

For $0 \le x \le 1 , p\gt 1$, prove $1 \over2^{p-1}$ $\le x^p +(1-x)^p \le 1$.

For $0 \le x \le 1 , p\gt 1$, prove $1 \over2^{p-1}$ $\le x^p +(1-x)^p \le 1$.

$\int_{1}^{\infty} h(x)\ dx$ converges $\Rightarrow$ $h$ is bounded in $[1, \infty)$

Let $h:[1, \infty)\rightarrow \mathbb R$ a continuous non-negative function, such that $\int_{1}^{\infty} h(x)\ dx$ converges. does $h$ must be bounded in $[1, \infty)$? I tried to prove it by showing that if $\int_{1}^{\infty} h(x)\ dx$ converges, then by the definition the $\lim_{b \to \infty}\int_{1}^{b} h(x)\ dx$ exists. Can I conclude that from the existence of […]

The range of a continuous function on the order topology is convex

Let $(W, \leq)$ be a linear order, and let $f : [0, 1] \rightarrow W$ be a continuous function (where [0, 1] has the usual topology and W has its order topology). Show that the range of f is convex. Let $Y \subseteq X$. The subspace $Y$ is a convex set if for each pair […]

asymptote problem?

I was curious about this exercise, because I thought it could be a valuable tool to use the theorem dell’asintoto … I do not think that is the way, does anyone have any idea? ! Let $ h $ is a function defined on $(a, + \infty)$ and limited all intervals $(a, b)$ $a <$ […]

Computing a Double Limit

How would one compute $\lim_{\delta \rightarrow 0, k\rightarrow\infty} (1+\delta)^{ak}$, where $a$ is some positive constant? I am finding a lower-bound of the Hausdorff Dimension on a Cantor-like set and this expression appeared in my formula. Here’s what I have, even though I’m not sure if I can use L’Hopital in this case (where $k, \delta$ […]

Lacking information on a Calculus Problem, have $f(x)$ need ${{dg} \over {dx}}$?

I am a senior in high school taking an AP Calc AB class. The textbook I am using is Calculus Concepts and Calculators (second edition) distributed by Venture Publishing. I am utterly stuck on question 18. Let f(x) = int(x). a) Explain why g’(2.5) = 0. b) Explain why g’(2) does not exist. I am […]