Articles of calculus

Help on differential equation $y''-2\sin y'+3y=\cos x$

$y”-2\sin y’+3y=\cos x$ I’m trying to solve it by power series, but I just can’t find the way to get $\sin y’$. Is there any special way to find it?

Showing summation is bounded

I’m currently taking a Comp Sci class that is reviewing Calculus 2. I have a question: Show that the summation $\sum_{i=1}^{n}\frac{1}{i^2}$ is bounded above by a constant I realize that this question is already answered here Showing that the sum $\sum_{k=1}^n \frac1{k^2}$ is bounded by a constant Could anyone explain it to me further? I […]

$\frac{d\Phi^{-1}(y)}{dy} = \frac{1}{\frac{d}{dy}}$?

If $\Phi(y)$ is a monotonic decreasing function is true that $$\frac{d\Phi^{-1}(y)}{dy} = \frac{1}{\Phi'(\Phi^{-1}(y))}$$ If so, how? It works for $y = \Phi(x) = e^{-x}, \quad \Phi^{-1}(y) = -log(y), \quad \frac{d\Phi^{-1}(y)}{dy} = \frac{-1}{y}, \quad $

How to solve $f'(x)=f'(\frac{x}{2})$

How do we solve this given $f'(0)=-1$. It does not look separable. I can integrate both sides but end up with a functional equation with is not helpful.

Is $\sqrt{\left(\operatorname{Si}(x)-\frac\pi2\right)^2+\operatorname{Ci}(x)^2}$ monotonic?

Recall the definitions of the sine and cosine integrals: $$\operatorname{Si}(x)=\int_0^x\frac{\sin t}t dt,\quad\operatorname{Ci}(x)=-\int_x^\infty\frac{\cos t}t dt.$$ Both functions are oscillating, with a countably infinite number of minima and maxima. Note that $$\lim_{x\to\infty}\operatorname{Si}(x)=\frac\pi2,\quad\lim_{x\to\infty}\operatorname{Ci}(x)=0.$$ Consider the following function: $$f(x)=\sqrt{\left(\operatorname{Si}(x)-\frac\pi2\right)^2+\operatorname{Ci}(x)^2}.$$ It appears that the function $f(x)$ and all its derivatives are monotonic for $x>0$. Specifically, the function itself and all […]

Integration with a quartic in the denominator $\frac{1}{(x^2-1)^2}$

I had to integrate $$\int\frac{x^2+1}{(x^2-1)^2} dx$$ Well my first approach was to write$\ (x^2+1)$ as $\ (x^2-1)+2$ so as to obtain fractions $$\frac{1}{(x^2-1)} + \frac{2}{(x^2-1)^2}$$ Now I know how to integrate the first part but how to integrate the second part i.e. a quartic (biquadratic) in the denominator? (I got the answer to the original […]

Calculate a limit $\lim_{x \to \pi/2} \frac{\sqrt{ \sin x} – \sqrt{ \sin x}}{\cos^2x}$

$$\lim_{x \to \pi/2} \frac{\sqrt[4]{ \sin x} – \sqrt[3]{ \sin x}}{\cos^2x}$$ I have an idea of replacing $\sin x$ to $n$ when $n \to 1$ but wolfram says that answer is $\frac{\pi}{48} $ so my suggestion is it’s had to use trigonometry simplifications which I do not know so well. Assuming that L’Hopital is forbidden but […]

For a matrix $A$, is $\|A\| \leq {\lambda}^{1/2}$ true?

In class I saw a proof that went something along these lines: Define $\|A\| = \sup \dfrac{\|Av\|}{\|v\|}$ for v in V, where the norm used is the standard (Does this even exist?) Euclidean norm in V. $\|Av\|^2 = <Av, Av> = <A^TAv,v>$ where $<,>$ denotes a dot product. Note that $A^TA$ is a non-negative matrix, […]

Proof of concavity of log function

Does anybody have a proof of the concavity of the $\log{x}$ that does not use calculus?

Longest pipe that fits around a corner.

This question already has an answer here: Intuitive explanation for formula of maximum length of a pipe moving around a corner? 3 answers