I'm recently learning about differentiation and I have no idea why its important. All I know is that has something to do with rate of change and acceleration over time? I understand how but I'm really struggling to understand why we do it, it doesn't make sense! Can someone give a real life example of how and why differentiation is important
Can any one explain me what is the use of derivatives in real life. When and where we use derivative, i know it can be used to find rate of change but why?. My logic was in  real life most of the things we do are not linear functions and derivatives helps to make a real life functions into linear.  eg converting parabola into a linear function $x^{2}\rightarrow 2x$  but then i find this; derivation of $\sin{x}\rightarrow \cos{x}$  why we cant use $\sin{x}$ itself to solve the equation. whats the purpose of using its derivative $\cos{x}$. Please forgive me if i have asked a stupid questions, i want to improve my fundamentals in calculus.
I have this directory /foo/bar/. Inside, I have a python file named myfile.py. I made it executable by adding (at the top):  #!/usr/bin/env python   Then, I made it executable by issuing   chmod +x myfile.py   I edited my path by issuing  sudo nano /etc/paths   I added /foo/bar/ to it.  The problem is that although I can run myfile.py by typing ./myfile.py while I am inside /foo/bar/, it doesn't work from any other location. What am I doing wrong?  Thank you. If $(a_n)\subset[0,\infty)$ is non-increasing and $\sum a_n<\infty$, then $\lim{n a_n} = 0$ I'm studying for qualifying exams and ran into this problem.  Show that if $\{a_n\}$ is a nonincreasing sequence of positive real numbers such that $\sum_n a_n$ converges, then $\lim_{n \rightarrow \infty} n a_n = 0$.  Using the definition of the limit, this is equivalent to showing \begin{equation} \forall \varepsilon &gt; 0 \; \exists n_0 \text{ such that } |n a_n| &lt; \varepsilon \; \forall n &gt; n_0 \end{equation} or \begin{equation} \forall \varepsilon &gt; 0 \; \exists n_0 \text{ such that } a_n &lt; \frac{\varepsilon}{n} \; \forall n &gt; n_0 \end{equation} Basically, the terms must be bounded by the harmonic series. Thanks, I'm really stuck on this seemingly simple problem! What is the difference between the displace socket and a bump map? Aside from the fact that you can control the bump mapping per shader with a bump node, are there any differences between these materials?      As far  as I can tell, they are the same. Is the displacement socket just a convenience thing? Is there a performance advantage? Multiplexing UART I'd like to interface my main microcontroller's hardware serial port to more than one device at the same time. My understanding is that UART on my microcontrollers is based on an active-low 5V TTL.  (I know what I'm explaining here can be technically achieved by using more suited protocols, such as SPI and I2C, but my project requires to be only implemented using UART.)  Here's a pseudo-schematics chart of what I'm trying to achieve:         &ndash; Schematic created using   Please note that:   I don't want to use a second UART interface on the master, or any of the slaves. Only one slave at a time is going to transmit or send data on the line to the master. (e.g. If the slave 1 is transmitting data, slave 2 and 3 should remain idle.) Whatever the master is transmitting can, and should be received by all of the slaves. All of the slaves should be hot-pluggable, meaning they could be removed or inserted at any time. The RX and TX wires to the master are parallel between each slave, so no daisy chaining.   So, my questions are as follows:   Assuming the correct impedance is considered, I shouldn't need an additional circuit to transmit data from the master's TX line to any of the slave's RX line simultaneously. Is this correct? I understand that without current limiting resistors between slaves' TX line and the master's RX line, if any of the devices begin to broadcast data, the line between the devices will short circuit and I risk damaging them. Is this correct? Assuming the items 1 and 2 are correct:  a) Would just some current limiting resistors be enough to use?  b) Can I use 74-series logic gates instead of resistors? Between options a and b above, what route should I choose? For a reliable 115200 baud rate connection, will the switching frequency of a 74HC08 AND gate suffice to be incorporated?   EDIT: I'm well aware of other well-suited protocols such as SPI and I2C. There's a reason I'm using UART, and that's because the optiboot bootloader that is used on the master's mcu communicates over UART, albeit UART's obvious limitations.  Also, while daisy-chaining is definitely a clever approach, due to breaking the 4th requirement, it also breaks bootloader functionality.  The devices could also be USB-to-Serial chips, and using SPI-to-USB or I2C-to-USB would not be feasible in this project.  So, I'd appreciate if you could focus on the 6 questions listed here instead, and share your UART knowledge. Must $k$-subalgebra of $k[x]$ be finitely generated? What determines the number of comments that are displayed? Let $(\mathcal{H}, (\cdot, \cdot))$ be a complex Hilbert space, and $A :  \mathcal{H} \to \mathcal{H}$ a positive, bounded operator ($A$ being positive means $(Ax,x) \ge 0$ for all $x \in \mathcal{H}$). Prove that $A$ is self-adjoint. That is, prove that $(Ax,y) = (x, Ay)$ for all $x,y \in \mathcal{H}$. Here's what I have so far. Because $A$ is positive we have $\mathbb{R} \ni (Ax,x) = \overline{(x,Ax)} = (x,Ax)$, all $x \in \mathcal{H}$. Next, I have seen some hints that tell me to apply the polarization identity: $$(x,y) = \frac{1}{4}((\lVert x+y \rVert^2 + \lVert x-y \rVert^2) - i(\lVert x + iy \rVert^2 - \lVert x - iy \rVert^2)),$$ where of course the norm is defined by $\lVert \cdot \rVert^2 = (\cdot, \cdot)$. So my guess is that I need to start with the expressions: $$(Ax,y) = \frac{1}{4}((\lVert Ax+y \rVert^2 + \lVert Ax-y \rVert^2) - i(\lVert Ax + iy \rVert^2 - \lVert Ax - iy \rVert^2)),$$ $$(x,Ay) = \frac{1}{4}((\lVert x+Ay \rVert^2 + \lVert x-Ay \rVert^2) - i(\lVert x + iAy \rVert^2 - \lVert x - iAy \rVert^2)),$$ and somehow show they are equal. But here is where I have gotten stuck. Hints or solutions are greatly appreciated. Why are random walks intercorrelated? Often floats may land quite a bit later in the document than the point they are created, sometimes after a section break.  Is there a way to force a new section to start on a new page, after any unplaced floats? I want to say:  Please, check out our handbook/knowledge base.  ...but it can be interpreted in two ways:  Please, check out our (handbook/knowledge) base. — as if there is a &quot;handbook base&quot; and a knowledge base. Please, check out our handbook/(knowledge base). — the meaning I'm after.  I thought that maybe I should put spaces around the slash to better separate the two choices:  Please, check out our handbook / knowledge base.  ...but I'm not sure that does it. I checked out  on the topic and it didn't answer my question.  Is it possible to convey the right meaning with a slash, or should I just avoid it altogether? At this moment there are 4 meta-site questions on the HNQ list:    (World Building Meta)  (Meta StackOverflow)  (Workplace Meta)  (Mathematica Meta)   When I first saw it, I only saw the SO Moderator question, so I assumed that this was a one-off special thing for the election; but now we've got four questions crept in. Was this an intentional change, or a bug? I am not talking about mirrors, just a plain window made of glass like material. Would it be possible to allow light pass only in one direction but not the other? How to remove duplicate lines inside a text file? A huge (up to 2 GiB) text file of mine contains about 100 exact duplicates of every line in it (useless in my case, as the file is a CSV-like data table).  What I need is to remove all the repetitions while (preferably, but this can be sacrificed for a significant performance boost) maintaining the original sequence order. In the result each line is to be unique. If there were 100 equal lines (usually the duplicates are spread across the file and won't be neighbours) there is to be only one of the kind left.  I have written a program in Scala (consider it Java if you don't know about Scala) to implement this. But maybe there are faster C-written native tools able to do this faster?  UPDATE: the awk '!seen[$0]++' filename solution seemed working just fine for me as long as the files were near 2 GiB or smaller but now as I am to clean-up a 8 GiB file it doesn't work any more. It seems taking infinity on a Mac with 4 GiB RAM and a 64-bit Windows 7 PC with 4 GiB RAM and 6 GiB swap just runs out of memory. And I don't feel enthusiastic about trying it on Linux with 4 GiB RAM given this experience. Joint density for exponential distribution Let $X_1$ and $X_2$ be independent random variables each having a exponential distribution with mean $\lambda = 1$.  (a) Find the joint density of $Y_1 = X_1$ and $Y_2 = X_1 + X_2$.  (b) Get the marginal density of $f_1(y_1)$ and $f_2(y_2)$.  $f(x_1)=e^{-x_1}$, $f(x_2)=e^{-x_2}$  Since $X_1$ and $X_2$ are independent, $f(x_1, x_2)=f(x_1)f(x_2)=e^{-x_1-x_2}$  However, how could I find the joint density?  I mean, it is quite easy to find marginal ones from the joint one, but how can I do the opposite? How can I achieve the correct garden tap orientation when threading it in? I'm installing an external tap using a wall plate like this one:    And a tap like this one:    This might be a stupid question, but if I screw the tap into the wall plate it will end up at a certain angle. I can install the wall plate with that angle and the tap will end up with the correct orientation (i.e. pointing down). But what if I want to put another tap in later that has its thread cut slightly differently? How do I prevent it ending up upside-down or whatever? Are all the threads cut exactly the same in these taps?  This seems like a stupid design. What is the best way to flatten a linked list with lists as values? Subscribe to tags I am interested in? How do I subscribe to the tags that I am interested in?  I like to get notified via email or so, when someone asks new questions or if someone responds to my questions. Proof of $\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s)$ Remove spaces from std::string in C++ What is the preferred way to remove spaces from a string in C++? I could loop through all the characters and build a new string, but is there a better way? I added two planes in Cycles with Image Texture in png with some transparency. I covered one by the other but transparency doesn't work, I mean it is impossible to see the one which is under despite transparency in the one which is above. What could be the problem?