The derivative of the absolute value |x|
Derivative of $f(x)=|x|$
What is the function of this circuit? Just wondering if someone can give me an explanation for this capacitor layout - what is it's function? The chip to the right is an ADC.  Circled in blue.    I'm guessing it is well known but i don't know how to search for it. How can I recover contact information from a stolen phone? If $$\tan(\frac{\pi}{4}+\frac{\theta}{2})=\tan^3(\frac{\pi}{4}+\frac{\alpha}{2})$$, then prove that $$\sin \theta=\frac{3 \sin \alpha+\sin^3 \alpha}{1+3\sin^2 \alpha}$$  I tried using the fact that  $\frac{\cos A}{1-\sin A}=\tan(\frac{\pi}{4}+\frac{A}{2})$ but now not able to eliminate $\cos \alpha$ and $\cos \theta$. How should I proceed? What is the meaning of the term "thread-safe"? Does it mean that two threads can't change the underlying data simultaneously? Or does it mean that the given code segment will run with predictable results when multiple threads are executing that code segment? How tipe are this series?  $$\sum_{n=2}^\infty \frac 1 {n^2\ln n}$$ But $$\sum_{n=2}^\infty \frac 1 {n\ln n\ln(\ln n)}$$ I used Condensation Test, but I stack after that. Thanks! How does the current know how much to flow, before having seen the resistor? With the following circuits as examples :    and    How will the current I know how much to flow? Would any other wave travel first in the circuit and then come back and say so much current should flow? Accessing the web page's HTTP Headers in JavaScript PHP in_array() / array_search() odd behaviour The question is based on the following exercise in real analysis:     Assume that $A,B\subset{\Bbb R}$ are both bounded and $x&gt;0$ for all $x\in A\cup B$. Show that $$ \sup(AB)=\sup A\sup B $$   where    $$ AB:=\{ab\in{\Bbb R}:a\in A, b\in B\}. $$   Since $0&lt;a\leq\sup A$ and $0&lt;b\leq\sup B$ for all $a\in A$ and $b\in B$, we have $$ ab\leq\sup A\sup B $$ for all $ab\in AB$ which implies that $\sup AB\leq\sup A\sup B$. I have trouble with another direction: $$ \sup AB\geq\sup A\sup B $$ I was trying to show that for every $\epsilon &gt;0$, $\sup AB-\epsilon \geq \sup A\sup B$. If one uses the definition of supremum, one has the estimates that for every $\epsilon&gt;0$, $$ \sup A-\epsilon\leq a, \quad \sup B-\epsilon\leq b $$ for some $a\in A,\ b\in B$. It follows that $$ \sup A\sup B\leq (a+\epsilon)(b+\epsilon)=ab+\epsilon(a+b)+\epsilon^2\leq \sup AB+\epsilon (a+b)+\epsilon^2 $$ which seems quite close to what I want. How can I go on? I never dealt with Android devices before so I'm completely newbie at it, please comment if something is too simple.  Say I purchase , that has got Android Froyo 2.2 installed in it.  If there is a new version of Android released tomorrow or next month, will I be able to update it (like I will be able to update my iPod Touch 4 to iOS 5)? Should an adverb go before or after a verb? For example:     The word rarely turns up outside of those contexts.      The word turns up rarely outside of those contexts.   Which one is correct and why? The dock doesn't work correctly after I enter my password, either. I would like to find all ellipses which contain 2 given points and has one focus at origin (zero). All in 2D plane.   There are several possible approaches but I'm not sure which is the best - both looks quite difficult to solve algebraically.   using polar equation relative to focus with $(R_1,\phi_1),(R_2,\phi_2)$, being coordinates of points $$ R_1 = \frac{a(1-e^2)}{1-ecos(\phi_1-\theta)} $$ $$ R_2 = \frac{a(1-e^2)}{1-ecos(\phi_2-\theta)} $$ then for given $\theta$ solve for semimajor axis $a$ and eccentricity $e$ using deffinition of elipse as a set of points of the same distance from both foci. Given 2 points of cartesian coordinates $(x_1,y_1),(x_2,y_2)$ and one focus in origin $(0,0)$. For each given distance parameter $L$ solve for coordinates of second focus $(x_f,y_f)$, $$ L = \sqrt{x_1^2 + y_2^2} + \sqrt{(x_1 - x_f)^2 + (y_1 - y_f)^2} $$ $$ L = \sqrt{x_2^2 + y_2^2} + \sqrt{(x_2 - x_f)^2 + (y_2 - y_f)^2} $$ I can also first rotate the coordinate system (or my input points) by given angle (which is my arbitrary parameter) and then use some simplified equation of ellipse which has major axis parallel to x-axis which has just 2 degrees of freedom. But even after this rotation I don't see much simplification of algebraic solution.    Or is there any better way?    The resulting equations are quite difficult to solve. I wonder if there is some trick to simplify it and make it more elegant.  Also, because I would later to implement it into computational code where it would be computed manytimes per second, I would prefer solution inf form which is fast to evaluate numerically ( prefferably without goniometric or any transcedental functions )    Why I'm interested ? - I need it for some optimalization of orbital transfers for spaceship orbiting the Sun. I asked also in physics.stackexchange, and people recomand me that I should rather go to math section. Why do electrons move towards anode in the Photoelectric effect experiment? Consider a Photoelectric effect experiment apparatus as shown in Figure 11.1    The variation of photocurrent with the voltage applied across $\mathrm{A}$ and $\mathrm{C}$ is as shown in Figure 11.3.    My question –     Why is the current non-zero even if the voltage is zero?    Consider a situation in which the voltage across the plates is zero. Photons of sufficient frequency strike the outermost surface of the cathode, emitter plate $\mathrm{C}$. Electrons become free from the attraction of nucleus as their net energy becomes positive.  Why I think that there should be no net current?  As soon as photons strike the outermost surface of the cathode(assumed to be $1$ to $10$ atomic diameters thick), electrons become free. Now, as the electrons have kinetic energy, they'll come out of that outermost surface. Now, the electrons(with kinetic energy) should have an equal probability of going in every direction of the surface. So, around half of the electrons should go in the right direction towards the anode and strike it and half should go to the left, trying to penetrate the inner surfaces of the cathode plate.   Now, the electrons which will go to the right will face some field due to the space charge present in the tube but nevertheless, some electrons will definitely reach the anode plate. After they strike with the anode plate, they'll face some resistance but still, some electrons will reach the low resistance copper wires connected to the anode plate and milliammeter.  Now, consider the electrons which go towards left. They'll definitely face resistance from cathode plate but some will still reach the copper wire.  Now, as there is no potential difference applied between $\mathrm{A}$ and $\mathrm{C}$, there is no need of commutator and voltmeter. So, the ammeter is connected in series with the anode plates and cathode plates with copper wires.  Now, if there is steady state current in the circuit, electrons emitted to any one of the sides should dominate over the other. But the number of electrons emitted on both the sides is same, and all the electrons should face the same resistance as the current(electrons) should flow in a closed loop. But if the number and velocity of electrons coming from both the sides are same, there should be no net current. But there is. Why?  Edit –  From the answers of anna v and mmainville, it can be seen that the emission of electrons is dependent on the angle of emission of light.  But still, the question remains. If the electrons emitted towards anode have enough energy to complete the circuit and again reach the holes in cathode despite the resistance in between, the electrons emitted to the left should have enough energy to come to anode plate via copper wires and again get ejected through anode plate by the virtue of their kinetic energy and reach the cathode plate. In this case too, the current should be zero which is not true. Why? How do I fix the left mouse opening the menu instead of breaking blocks? How do I get HTML5 h.264 Video working on Firefox? According the the , h.264 video should now be working in Linux.  It does also say the appropriate gstreamer plug-ins need to be installed.  Does anyone know what these are?    I used this  and the  to verify that h.264 support works in Google Chrome on my Ubuntu 12.04 64-bit, but does not work on Firefox 26. Q: How many arrangements of a,a,a,b,b,b,c,c,c are there such that  $\hspace{5mm}$ (i). no three consecutive letters are the same?  $\hspace{5mm}$ (ii). no two consecutive letters are the same?  A:(i). 1314. ${\hspace{5mm}}$ (ii). 174.  I thought of using the General Principle of Inclusion and Exclusion along with letting $p_i$ denote a property that.. , and doing so, I will evaluate $E(0)$, which gives us the number of arrangements without any of the properties.   How do I go about doing that? I am trying to use the method stated above in solving this question, but I am unable to generalize the properties $p_i$. I am trying to show that $a^{12} \equiv 1 \mod 32760$ for each $a \in \mathbb{Z}$ coprime to $32760$. I have found the prime decomposition $32760 = 2^3 \cdot 3^3 \cdot 5 \cdot 7 \cdot 13$ and determined $\phi(32760)$, where $\phi$ is the Euler's totient function. This gives me that $\phi (32760) = 8640$.   Using Euler's congruence, this gives that $a^{8640} \equiv 1 \mod 32760$, but this is still far from the result I want to show... I also know a consequence of Eulers congruence, which states that $e \equiv e' \mod \phi (n)$ implies that $a^e \equiv a^{e'} \mod n$, but I do not see how to use this...  Any hints would be appreciated.  $\textbf{EDIT:}$ I made a mistake in my primefactorization, it is edited now. Finding the limit of $\left(\frac{n}{n+1}\right)^n$ Find the limit of: $$\lim_{n\to\infty}\left(\frac{n}{n+1}\right)^n$$   I'm pretty sure it goes to zero since $(n+1)^n &gt; n^n$ but when I input large numbers it goes to $0.36$.  Also, when factoring: $$n^{1/n}\left(\frac{1}{1+\frac1n}\right)^n$$ it looks like it goes to $1$.   So how can I find this limit? Can't see a glass object through another glass object in Eevee