derivative of $e^x$ using the limit definition
Using the Limit definition to find the derivative of $e^x$
What Is Energy? Where did it come from? My 3-month-old son has started sleeping through the night, with a vengeance. He sleeps for hours. And when he finally wakes up and I expect him to be starving, he's often still ready to happily play for a bit - say 1/2 hour to 45 min. Now, I know this isn't such normal behavior for a 3-month-old, and I'm definitely appreciating having my nights back, but I'm also a bit concerned - how long is the longest I should let him go without eating?  Edit (more details):  I used to try to stretch him and make his last feeding right before I went to sleep, so that I could take full advantage of his sleep-time. So we used to put him to bed about 12:30-1:00, and then head to sleep ourselves. He started off going about 6-7 hours... But a couple of weeks ago he started sleeping longer, and longer, and longer... Now he goes nearly 12 hours. He'll go to sleep at 12:30 AM and wake up at 12 PM or later. And he'll then happily play for another half hour, making about 12 hours of no-eating. I generally trust that I can rely on him to tell me when he's hungry, but I was wondering if there's a limit to how long I can let him go without eating... Suppose that $X$ is a topological space equipped with a delta-complex structure. Prove that $X$ is Hausdorff.  I can actually "see" why it should be Hausdorff after the hint from Hatcher asks to "think of a delta-complex X as a quotient space of a collection of disjoint simplices".  I would appreciate if somebody can write out a full formal argument showing this proof. I'm trying to start from the definition of Hausdorff but I don't know how to carry on. I found the following proof arguing for the irrationality of $\sqrt{2}$.     Suppose for the sake of contradiction that $\sqrt{2}$ is rational, and choose the least integer $q &gt; 0$ such that $(\sqrt{2} - 1)q$ is a nonnegative integer. Let $q'::= (\sqrt{2} - 1)q$. Clearly $0 &lt; q' &lt; q$. But an easy computation shows that $(\sqrt{2} - 1)q'$ is a nonnegative integer, contradicting the minimality of $q$.   Clearly, this is a vague proof in that it expects the reader to work out the math involved on his own. I took that as an exercise and worked out the following:  Assuming $\sqrt{2}$ to be rational, let $$\sqrt{2} = \frac{p}{q}$$ where $p$ and $q$ are least possible integers.  Clearly then, $\sqrt{2} - 1$ would also be rational, $$\sqrt{2} - 1 = \frac{p-q}{q}$$ where both $(p-q)$ and $q$ are also integers.  Multiplying both sides by $q$, we get $$(\sqrt{2} - 1)q = (p-q) = q'$$ where $q'$ is an integer. Using, $2&gt;\sqrt{2} &gt; 1$ , we get $$1&gt;\sqrt{2}-1&gt;0$$ proving that $q'$ is a non-negative integer.  As stated in the proof, computing $(\sqrt{2}-1)q'$ we get, $$(\sqrt{2}-1)q' = \frac{(p-q)^2}{q}$$which is not necessarily an integer (in fact necessarily not an integer). However, the proof claims it to be.  Am I missing something or something is wrong with how I proceeded?  Here is a link to the textbook where I found this proof: , (p 32, problem 1.14).  P.S. This is my first post here and my first time with TeX. I'd also like to know if there is anything wrong with the way my question is presented. From injective map to continuous map I have a problem called "The Prison Problem" that I need to explain to my 9-year-old cousin. I would think that he has just started learning about divisors and perfect squares, and as such, I have a proposed solution. Any input from you guys would be welcome, as to what is the best way to go about this.  Problem:  There was a jail with 100 cells in it, all in a long row. The warden was feeling very jolly one night and told his assistant that he wanted to give all the prisoners a wonderful surprise. While they were sleeping, he wanted the assistant to unlock all the cells. This should be done, he told the assistant, by putting the key in each lock, turning it once. The assistant did this, then came back to report that the job was done. Meanwhile, however, the warden has second thoughts. "Maybe I shouldn't let all the prisoners free," he said. "Go back and leave the first cell open, but lock the second one (by putting the key in and turning it once). Then leave the third open, but lock the fourth, and continue in this way for the entire row." The assistant wasn't very surprised at this request as the warden often changed his mind. After finishing this task, the assistant returned, and again the warden had other thoughts. "Here's what I really want you to do," he said. "Go back down the row. Leave the first two cells as they are, and put your key in the third cell and turn it once. Then leave the fourth and fifth cells and turn the key in the sixth. Continue down the row this way." The assistant again did as instructed. Fortunately, the prisoners were still asleep. As a matter of fact, the assistant was getting pretty sleepy, but there was no chance for rest yet. The warden changed his mind again, and the assistant had to go back again and turn the lock in the fourth cell and in every fourth cell down the row. This continued all through the night, next turning the lock in every fifth cell, and then in every sixth, and on and on, until on the last trip, the assistant just had to turn the key in the hundredth cell. When the prisoners finally woke up, which ones could walk out of their cells?  Proposed solution:  Ten cells are left open after this process. Every cell that is a perfect square will remain open (1, 4, 9, 16, 25, 36, 49, 64, 81, and 100). If a number is not a perfect square, then it has an even number of divisors, therefore it will be "toggled" an even number of times and end up where it started (closed). Perfect squares have an odd number of divisors, so they will end up the opposite of where they started (open). How to use multiple Effect Strips in Blender Video Sequence Editor? "Error: Invalid signature" when configuring dual boot Ubuntu 14.04 and Windows 10 Difference between Session and HttpContext.Current.Session Why is uniform prior on log(x) equal to 1/x prior on x? I'm trying to understand Jeffreys prior. One application is for 'scale' variables like the standard deviation $\sigma$ (or its square, the variance $\sigma^2$) of Gaussian distributions. It is often said that using a uniform prior over $\sigma$ is not really non-informative and instead one should either:   Use instead $\ln \sigma$ as the free parameter, with a uniform prior (this is often called a log-uniform prior) Or keep using $\sigma$ as the free parameter but use $1/\sigma$ as the prior (which is not uniform).   Why are the above two methods/priors equivalent? I feel it has something to do with the fact that the derivative of ln $\sigma$ is $1/\sigma$ but I can't take the next step.   Also, why does this even matter, in simple language with minimal jargon? I see all these complicated explanations online involving the Fisher information matrix but in the end all I see is that the above log-uniform or $1/\sigma$ priors preferentially weight lower values of $\sigma$ more highly. Why? If possible, a simple analytic example or python snippet would be very helpful. Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$. What happens when a caster targets an object that looks like a creature with a spell that targets only creatures? Powershell Rename Multiple Files with LastWriteTime Prefix I am trying to put together a Powershell script that will:  -Search recursively from a root directory -Exclude files with existing "yyyyMMdd " file name prefix -Rename the files not excluded above with ""yyyyMMdd " file name prefix based on their specific LastWriteTime  Note: ideally, this script should include all file type extensions  I have tried myself, but it would seem to be above my head, as my attempts are not going well.  Thanks, in advance.    #StampMe.ps1 param( [string] $fileName)  # Check the file exists if (-not(Test-Path $fileName)) {break}  # Display the original name "Original filename: $fileName"  $fileObj = get-item $fileName  # Get the date $DateStamp = get-date -uformat "%Y-%m-%d@%H-%M-%S"  $extOnly = $fileObj.extension  if ($extOnly.length -eq 0) {    $nameOnly = $fileObj.Name    rename-item "$fileObj" "$nameOnly-$DateStamp"    } else {    $nameOnly = $fileObj.Name.Replace( $fileObj.Extension,'')    rename-item "$fileName" "$nameOnly-$DateStamp$extOnly"    }  # Display the new name "New filename: $nameOnly-$DateStamp$extOnly"   I am hoping to change the line:  $DateStamp = get-date -uformat "%Y-%m-%d@%H-%M-%S"   To be LastWriteTime instead of get-date, but I have no idea how to make this happen.  This code (from another Super User question):  Get-ChildItem "Test.pdf" | Rename-Item -newname {$_.LastWriteTime.toString("yyyyMMdd") + " Test.pdf"}   ...successfully renames a single file, but I can't seem to integrate this into the larger script.    With regards to running the above recursively, I have tried this (also from ):  foreach ($file in get-ChildItem *.* -recurse) { ./StampMe.ps1 $file.name }   Which succeeds to apply the PS script to files in the root directory, but fails to apply to any files in the sub-folder tree that propagates from the root directory. Where do the American and Japanese Transformers franchises meet? The original Transformers TV series was created by the American company Hasbro, and was adapted into Japanese as well. At some point, the Japanese franchise split off into its own continuity, launching its own set of sequels and spin offs.  The questions is: What part of the American franchise is considered canon by the Japanese one? Minor details such as name changes are not to be taken into consideration. What's the canonical way to check for type in Python? What is the best way to check whether a given object is of a given type? How about checking whether the object inherits from a given type?  Let's say I have an object o. How do I check whether it's a str? What happens when Wild Shape/Polymorph runs out in a space that's too small? Why is Minecraft giving an OpenGL error? I recently got Minecraft but not from the website, I paid for a code and everything seemed ok. When I tried to play the game a thing came up saying it crashed and then a message popped up saying The game crashed whilst initializing game     Error: java.lang.IllegalStateException: GLFW error 65542: WGL: The driver does not appear to support OpenGL. How do I close a connection early? Is it possible to make an EXACT ISO of my system to put on other computers? How to yield from parallel tasks in .NET 4.5