Articles of calculus

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 […]

What's $\limsup_{(h_x,h_y)\to(0,0)} \frac{\left|\frac{x+h_x}{y+h_y}-\frac{x}y\right|} {\sqrt{{h_x}^2+{h_y}^2}}$?

This originally comes from $f_1(x,y)=\frac{x}{y}$, where $X=\mathbb{R}^{n}, Y=\mathbb{R}^m, x \in X, f: X \rightarrow Y, x \neq 0, f(x) \neq 0$ $$\limsup_{(h_x,h_y)\to(0,0)} \frac{\left|\frac{x+h_x}{y+h_y}-\frac{x}y\right|} {\sqrt{{h_x}^2+{h_y}^2}}$$ If I understand it correctly, $x$ and $y$ can be anything but zero, and $h_x,h_y$ go towards zero. Moreover, both numerator and denominator cannot be negative. But since $x,y$ could be […]

Absolute and Conditional Convergence of the integral $\frac{\sin(x)}{x^p}$ for real values of $p$

This question already has an answer here: Convergence $I=\int_0^\infty \frac{\sin x}{x^s}dx$ 5 answers

Differentiate the Function: $y=x^x$

$y=x^x$ Use $\frac{d}{dx}(a^x)=a^x \ln a$ My answer is: $x^x \ln x$ The book has the answer as $x^x\ (1+ \ln\ x)$ Am I missing a step?