Suppose $u$ and $v$ are continuous on $[a,b]$ and differentiable on $(a,b)$, and assume that for all $t\in(a,b)$, at least one of $u'(t)$ and $v'(t)$ is nonzero. Let $C$ be the curve given by $(u(t),v(t))$ for $t \in [a,b]$. Let $A = (u(a),v(a))$ and $B = (u(b),v(b))$ be the endpoints of the curve, and assume $A\ne B$. Show that there is some point $c \in (a,b)$ such that the tangent line to $C$ at $(u(c),v(c))$ is parallel to $\overline{AB}$.  I've been stuck on this question for a while now without any idea on how to get started. Any whatsoever help is appreciated!
I know and understand the mean value theorem. But at the moment I don't have the intuition to understand the generalized mean value theorem     If $f$ and $g$ are continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, then there exists a point $c\in(a,b)$ where$$[f(b)-f(a)]g'(c)=[g(b)-g(a)]f'(c).$$If $g'$ is never zero on $(a,b)$, then the conclusion can be stated as$$\frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}.$$   What is the intuition? And how can you prove this (generally) ?
How can I test two static methods (one invokes the other) with PowerMockito? After doing a sudo apt-get install , where does the application get stored to? After installing apps via apt-get install -app name-. Where does the files get copied to? Are the copied to just one location or multiple locations? And if so how do I find this out?  Here's an example: Identifying genes from a list of genes I have a gene list file. Some thing like this      SWT21     SSA1     NRP1     EFB1     TFC3     MDM10   I have another file which also contains the names of these genes in my list along with other essential information about them. The second file looks like this:  chrI    147593  151166  YAL001C -   TFC3 chrI    143706  147531  YAL002W +   VPS8 chrI    142173  143160  YAL003W +   EFB1 chrI    140759  141407  YAL004W +   YAL004W chrI    139502  141431  YAL005C -   SSA1 chrI    137697  138345  YAL007C -   ERP2 chrI    136913  137510  YAL008W +   FUN14 chrI    135853  136633  YAL009W +   SPO7 chrI    134183  135665  YAL010C -   MDM10   I want to extract those lines in the 2nd file which have gene names as are present in first file. Why are there humans in the Star Wars Universe? Kind of brought to my attention by .  The poster mentions non-humans, which implies that there are humans in the Star Wars universe.  However, since it happened "A long time ago, in a galaxy far far away", how could there be any humans in the Star Wars universe?  Are there "humans" in Star Wars there just because it was easier to make the movies that way?  Other shows such as Star Trek are about the future of humans, so it's easy to understand the existence of humans in those storylines.  However for Star Wars it just doesn't make sense.  It also seems odd that they'd go around creating all these different species and then just leave the main characters as humans. I guess another possibility is that the humans in star Wars eventually populated Earth, but then what happened to all the technology? Two figures side by side How to change account profile picture? I can't believe I have to ask this...but, how do I change my account profile picture in Ubuntu 12.04 Desktop? One would think it would be under the "User Accounts" section. Can we re-request features that were declined a sufficient while ago? Faction Membership Limits in Fallout 4 Are there any limitations to the faction memberships you can have in Fallout 4?  I'm brand-new to the Fallout universe, and just got to the point of being invited to the Brotherhood of Steel. However, I'm a bit concerned about their politics and how it may affect my standing with other factions I may encounter.  If I join the Brotherhood, will there be other factions that I can't join? Are there any other factions that have such limitations?  Note: I'm still not very far into the game, and not very familiar with the Fallout 4 universe in general, so please make liberal use of spoiler markdown in your answers. Essentially, the only things that need to be un-spoilered are Yes/No answers for the questions asked above. Why are my sauces splitting? Recently all my attempts at sauces are splitting and I'm not sure what's going wrong. Tonight I tried a balsamic vinegar reduction / chicken stock / pan juices recipe to go with my lamb chops, but once again the dreaded split occurred. My wife is patient and forgiving, but I feel like I must be missing something fundamental to get this wrong so often.  Two questions.   What could I be doing wrong to cause these sauces to split? What can I do to recover when it happens again?   UPDATE: The Pan juices contained a tbsp or two of olive oil, and some of the fresh herb rub I'd coated the lamb chops in (thyme, rosemary, and mint). I did not drain the pan before starting the sauce. It has crossed my mind that perhaps there was too much oil / fats in the pan at that point?  For completeness, I should point out that I also sauteed some chopped shallots in the pan juices immediately before adding the vinegar. We  that there's a difference in meaning between try used with the infinitive and  try used with the gerund:     I tried sending her flowers. (Meaning: An experiment: I sent her flowers to see what will happen).   I tried to send her flowers. (Meaning: I tried to go through the procedure necessary to send flowers to her. The procedure might have been difficult.)   (per Michael Swan's PEU, Unit 299.6)   Will the same distinction hold if we use trying:     A man is trying setting up a password on his computer. (Is he experimenting, wanting to see what will happen?)   A man is trying to set up a password on his computer. (Is he trying to go through all the necessary steps, probably difficult, of setting up a password?) I have a 4GB DDR3 800Mhz recently I installed another 4GB. It turns out it's  DDR3 667MHz. Does it really matter I should own both rams 800MHz?    Slot1 shows recent RAM installation. ALSO I don't know if this is 1600MHz (it shows 800x2 and 667x2 for other) or just 800MHz ram. Preprocess categorical variables with many values I have a dataset that consists of only categorical variables and a target variable. I want to predict the (binary) target variable with the categorical variables.  I am trying to do this in Python and sklearn.  The categorical variables have many different values. I was thinking to create dummy variables for each value in all the categorical variables. However, in the end this results in a sparse data set with thousands of variables.  How would you go about to reduce the number of dummy variables? Would you use Chi2 to select useful features?  Or would you not make dummy variables at all? I am debating whether or not to get the Potato Sack (yes, I realize I've been debating for several weeks now) bundle through steam.  I already own: Defense Grid: The Awakening, Super Meat Boy, and Audiosurf, all of which are part of the sack pack. What happens if I buy the Potato Sack?  Do I get 2nd copies of the three above mentioned games to gift elsewhere?  Is the price of the bundle reduced by the games I already have?  Do I get nothing special for already owning those 3 games? Immutability, Association and typing Close Vote Review Audits are too obvious The other day, I got an obviously good question in a Close Vote Review.  Even if I didn't read the question,  is a bit of a giveaway.  Today I got one where   Also, on sites other than SO (where I understand the queues are huge), the number of reviews needed is often low enough (single digits) that you usually notice when you get an extra one.  I don't mind - I'm quite happy to get easy-to-pass audits - but maybe they could be a little harder... How do I return the response from an asynchronous call? Let $R$ be a commutative ring with identity, and $S$ a multiplicative subset of $R$ not containing $0$. Which step in my reasoning is wrong?   Every element of $S$ is a unit in $S^{-1}R$. Every element of $S$ that's a zero divisor in $R$ is a zero divisor in $S^{-1}R$. But zero divisors can't be units. Therefore it's not possible to localize $R$ at $S$. Error: Declaration of MyClass::start_lvl() should be compatible with that of Walker_Nav_Menu::start_lvl() Prove: $\gcd(a,b) = \gcd(a, b + at)$. I know that $\gcd(a,b)$ divides $a$ and $b$, and must also then divide $(a)(t)$ ($t$ being some integer).  This makes sense to me, but how do I prove it?  It seems that the addition of $(a)(t)$ is a continuation of the linear combination of $\gcd(a,b) = av + bu$ for some $v$, $u$ being elements of $\mathbb{Z}$.    Any help? Deep learning model (LSTM) with temporal and non temporal attributes