Elementary proof that $2^x$ is derivable
Show $\lim_{h\to 0} \frac{(a^h-1)}{h}$ exists without l'Hôpital or even referencing $e$ or natural log
Best way to start investing, for a young person just starting their career? I am young, just started my career, and looking to begin investing so I can make my money work for me. I'm still living like I did in college (occasional purchase on myself, but most of it goes into savings) and I would like to take the extra money and invest it.   What would you suggest are the best options for me to begin? I was considering getting a personal investor [a financial advisor] for perhaps a year or so, but I've heard that they really are not worth the money. I wanted to get the input of the great minds here. How can I emulate 1.7.10's adventure mode in 1.8? Use second argument for optional first argument if not provided in macro I'm trying to construct a command with one optional and one mandatory argument and I'd like the default if the optional argument is not provided to be #2.   Why does this not work?  (error: Illegal parameter number ...)  \documentclass{article}  \newcommand{\mycommand}[2][#2]{\label{#1}#2} \begin{document} \mycommand{mytext} \end{document}   And what would be a way around it?   EDIT: For clarification of my specific use case: I'm trying to define a command that passes a variable on as an option to an environment along those lines, where myenv can have options like optionone=X or optiontwo=Y so the default for the optional #1 should be optionone=#2 (not literally #2).   \newcommand{\mycommand}[2][optionone=#2]{\begin{myenv}[#1]\input{#2}\end{myenv}}   Maybe there's generally a better approach?   (The environment definition I'm using is from .) Writing letters of recommendation for more than one student to the same department Suppose multiple students have asked me for letters of recommendation to the same graduate department in the US. Am I expected to explicitly compare the students to one another? Am I supposed to do this in both letters (assuming they haven't been submitted yet)? Do I need to explicitly state a preference for one over the other, or can I just get away with listing relative strengths and weaknesses (for instance, "X is stronger academically, but Y has more research experience and is a stronger programmer")? Schengen visa validity start date My Chinese in-laws intend to visit us in one month from now. Their visa is valid from Nov 23rd. Would it be possible for them to take the airplane on November 22nd in China and arrive in Schengen area on November 23rd?  Or does the visa already have to be valid upon departure?  In our particular case, they still need to transfer in Beijing first. The actual flight to Schengen zone effectively only departs on the 23rd so I guess that should be completely fine then? SQL Delete clears the table instead of erroring If $f'(x)\le g'(x)$, prove $f(x)\le g(x)$ I have to do the following exercise:     Let $f$ and $g$ two differentiable functions such that $f(0)=g(0)$ and $f'(x)\leq g'(x)$ for all $x$ in $\mathbb{R}$. Prove that $f(x)\leq g(x)$ for any $x\geq0$.   Now, I know this is true because the first derivative of a function is the angular coefficient of the function in a point $x$. So, $f'(x)\leq g'(x)$ means, in other words, that the function $g(x)$ grows faster than $f(x)$. I think this is the base for a more formal proof, could someone help me to figure out a more formal proof? Real World Example of the Strategy Pattern How to find the index of an element in an array in Java? Graph partition that span a third of edges Given a graph G is easy to see that we have a partition $V=V_1 \cup V_2$ so that  $$e(G[V_1])+e(G[V_2])\leq e(G)/2$$. How can we improve this result showing that we can choose $V_i$ such that $e(G[V_i])\leq e(G)/3$ for $i=1,2.$ Regular Expressions: Is there an AND operator? Obviously, you can use the | (pipe?) to represent OR, but is there a way to represent AND as well?  Specifically, I'd like to match paragraphs of text that contain ALL of a certain phrase, but in no particular order. past simple or present perfect Please have a look at this sentence:     We haven't gotten a lot of Finnish groups that we liked here at Bananas but I'm pretty happy with this four tracker from Helsinki, apparently recorded in the same place where the Ramones played in the 70's.     The first verb is in present perfect (up to now) and the second is in simple past tense. I don't understand the simple past for the second verb.  Why "like" is not in its present perfect form as the first verb is? Before today, I would visit Stack Overflow every day and click somewhere in the upper-left corner (I've forgotten what it said, and it's gone now!) and if I didn't have any notifications I would see a list of interesting questions from various sites in the Stack Exchange network which - I believe - were classified as "hot".  Now this new top bar has appeared, and I have no idea how to get this list any more. I'm not really sure what it was called so my searches have been useless as well. Should laptops remain plugged in when their battery is 100% charged? When your laptop's battery is 100% charged, should you leave it plugged in so any battery power doesn't get used, or will that cause overcharging, overheating, etc.? Should the laptop be unplugged when the battery level is 100%?  I'm asking because my laptop's screen tends to get dim when unplugged, so I don't like to run it on battery. Copy all files created after a specific hour in command line (Windows)? I would like to easily copy a batch of files from one very big directory to another based on the hour of creation (every hour i want to copy the files from last hour).  I have used Xcopy /D function but you can only put the date but not the hour.  Is there another command?  the directory is really big and it's very long to test each file for it's creation date in a Script. Xcopy is great but lacks the hour. I created the shiftcommandn shortcut for Firefox in Syst Prefs to match other browsers' behavior.     It registers in the Firefox Menubar.     But pressing the shortcut opens a new non-private window, and it loads there the first item in my "Top Sites".   I imagine I could create another shortcut to override this behavior, but what is this behavior? I don't see anything highlighted in the Firefox Menubar when I hit the shortcut. Or might there be another solution (other than using a different shortcut to open new private windows)?   Problem exists on both Mojave and High Sierra. Default Firefox shortcut (shiftcommandp) still opens a new private window. Problem with Non English(Farsi, ...) subtitle This might be subjective, I don't know. However, it's a serious problem for me. My electrical bill is outrageous. It's three times as large as my next-door neighbor, and four or five times the size of my neighbor's across the street. It's worse in the summer (I live in central TX), but I've had the air conditioner inspected, and while the house is right at the maximum capacity for our unit, the unit should still be able to take care of the load. That also doesn't explain why our winter bills are larger than the comparables.  What I would like to do is check the amount of current drawn by each running appliance and calculate where all my money is going. I just don't know how to go about doing this. Many browsers and Internet companies have recently claimed that SSL Certificates with a signature algorithm of SHA1 will imminently no longer be considered secure.  Most notably, .  Yet I've read that since the SSL/TLS negotiation and Cipher Suite don't make use of plain SHA1 (or even MD5), and instead use SHA1 within an ; that using a Cipher Suite with SHA1 is still an acceptable security practice.  Is this the case?    While we're on it, is it also the case of using MD5 in a cipher suite? How do I get Perl to read the contents of a given directory into an array?   can do it, but is there some method using 'scandir' or a similar term?