Finding the tenth derivative of $f(x) = e^x\sin x$ at $x=0$
$n$th derivative of $e^x \sin x$
I'm having trouble understanding the ROC curve.  Is there any advantage / improvement in area under the ROC curve if I build different models from each unique subset of the training set and use it to produce a probability? For example, if $y$ has values of $\{a, a, a, a, b, b, b, b\}$, and I build model $A$ by using $a$ from 1st-4th values of $y$ and 8th-9th values of $y$ and build model $B$ by using remained train data. Finally, generate probability. Any thoughts / comments will be much appreciated.   Here is r code for better explanation for my question:  Y    = factor(0,0,0,0,1,1,1,1) X    = matirx(rnorm(16,8,2)) ind  = c(1,4,8,9) ind2 = -ind  mod_A    = rpart(Y[ind]~X[ind,]) mod_B    = rpart(Y[-ind]~X[-ind,]) mod_full = rpart(Y~X)  pred = numeric(8) pred_combine[ind]  = predict(mod_A,type='prob') pred_combine[-ind] = predict(mod_B,type='prob') pred_full          = predict(mod_full, type='prob')   So my question is, area under ROC curve of pred_combine vs pred_full. Calculate mean of ordinal variable I've read in a number of places that calculating the mean of an ordinal variable is inappropriate. I'm trying to get an intuition for why it might be inappropriate. I think it is because, in general, an ordinal variable is not normally distributed and so calculating the mean will give an inaccurate representation. Could somebody give more detailed reasoning for why calculating the mean of an ordinal variable might be inappropriate? Mountain Bike on Paved Roads I'm an occasional biker. I bought a mountain bike (nothing fancy) with the idea that I might use it off-road or on mountain trails (of which there are many here in Calgary). However, since I've bought it I've done nothing but bike in the city on streets and paved bike trails.  Is there anything worth doing to my bike to convert it to something more suited to urban biking? In particular, the tires have me wondering; they have large rubber off-roading lugs that surely aren't doing me any favors on pavement. I'm not looking to spend a tons of money on a retrofit, but will do so if its worthwhile. What does the tilde before a function name mean in C#? I want to create a 2D game but I'd like to have the character to wear tons of different equipments on different combinations, like a RPG or something.  So let's say the user changes his/her shoulders and pants, this must be shown to the player and all these different equipments must follow the different characters action (hitting, being hit, spelling magic, etc.).  I see a few problems that I'd like to know the best approach/algorithm/architecture to solve.  1 - Sprites or Animations  Should I create different sprites for each equipment in each action animation?  Is it better to just have the equipment sprite and animate it directly on code like rotating and translating (using something like tween on Flash)?  Is there any other better option? (I really don't like the ideas above)  2 - Positions  Let's say during one of the character's move he starts looking directly in front to the player's view but ends his movement showing his right side to the player (like if he swings his sword from one side to the other).  If we consider one part of the user, like his head for example, it starts in front and then turns left.  It means at least 3 different positions for each head equipment (helmet, cap, whatever).  This surely influence the answer of #1 question. How is the best way to achieve that?  3 - Layers  Let's consider a character move where he spins 360 degrees with his arms open. At the beginning of the animation, his right hand is close to the user's view and then, at the middle of the movement, this is probably behind the character's body on the animation.  Whatever is the option on #1 question, you surely needs to use some kind of layer model to make this sprite or animation to starts closer to the player's view and later change to far away from player's view.  Is there any good way to do that?  I know the question is quite long and difficult to understand. Let me know if you think it's better to have some drawings to try to explain which one and I'll try to do that. Individual ColorFunction with RGBColor Verification the integral identity $\int_0^\pi \ln(1+\alpha\cos(x))dx=\pi\ln\left(\frac{1+\sqrt{1-\alpha^2}}{2}\right)$. I have to verify that     $$\int_0^\pi \ln(1+\alpha\cos(x))dx=\pi\ln\left(\frac{1+\sqrt{1-\alpha^2}}{2}\right)$$   with $|\alpha|&lt;1$. It is my homework and don't know where to begin. Oven temperature for multiple ducks? Do I need to adjust the oven temperature in a roast duck recipe if I want to put more than one duck in the oven at a time.   I suspect I should keep the same temperature but keep them in longer.  Should I cover each duck in foil individually or together?  Does it matter if they are on the same pan? $\DeclareMathOperator{\Char}{char}$$\DeclareMathOperator{\Aut}{Aut}$ Let $G$ be a finite group, $H$ a subgroup, such that $H \Char G$. If $\phi \in \Aut(H)$ is there an automorphism $\widehat{\phi} \in \Aut(G)$ such that $\widehat{\phi}|_H$ = $\phi$?  In words, if $\phi$ is an automorphism of $H$, is there necessarily an automorphism $\widehat{\phi}$ of $G$ where $\widehat{\phi}$ maps the elements in $H$ to $H$ in exactly the same way?  I am thinking this is false (consider the non-finite example of $\bar{Q}$ and $\mathbb{C}$), but would like a counter example (for the finite case). I read something that suggests that if $I$ is the $n$-by-$n$ identity matrix, $v$ is an $n$-dimensional real column vector with $\|v\| = 1$ (standard Euclidean norm), and $t &gt; 0$, then $$\det(I + t v v^T) = 1 + t$$ Can anyone prove this or provide a reference? More generally, is there also an (easy) formula for calculating $\det(A + wv^T)$ for $v,w \in \mathbb{K}^{d \times 1}$ and some (invertible) matrix $A \in \Bbb{K}^{d \times d}$? How do you compile OpenSSL for x64? After following the instructions in INSTALL.W64 I have two problems:   The code is still written to the "out32" folder. I need to be able to link to both 32-bit and 64-bit versions of the library on my workstation, so I don't want the 64-bit versions to clobber the 32-bit libs. The output is still 32-bit! This means that I get "unresolved external symbol" errors when trying to link to the libraries from an x64 app. Is $\{0\}$ a field? Consider the set $F$ consisting of the single element $I$.  Define addition and multiplication such that $I+I=I$ and $I \times I=I$ .  This ring satisfies the field axioms:   Closure under addition.  If $x, y \in F$, then $x = y = I$, so $x + y = I + I = I \in F$. Closure under multiplication.  $x \times y = I \times I = I \in F$ Existence of additive identity.  $\forall x \in F$ (i.e., for $x=I$), $x + I = x$, so $I$ is the additive identity. Existence of mulitiplicative identity. $\forall x \in F, x \times I = x$, so $I$ is the multiplicative identity. Additive inverse. $\forall x \in F, \exists y = I \in F: x + y = I$ Multiplicative inverse. $\forall x \in F, \exists y = I \in F: x \times y = I$.  However, because the additive identity need not have a multiplicative inverse, this is a vacuous truth. Commutativity of addition.  $\forall x, y \in F, x + y = I = y + x$ Commutativity of multiplication.  $\forall x, y \in F, x \times y = I = y \times x$ Associativity of addition.  $\forall x, y, z \in F$, $(x + y) + z = I + I = I$ and $x + (y + z) = I + I = I$, so $(x + y) + z = x + (y + z)$ Associativity of multiplication.  $\forall x, y, z \in F$, $(x \times y) \times z = I \times I = I$ and $x \times (y \times z) = I \times I = I$, so $(x \times y) \times z = x \times (y \times z)$ Distributivity of multiplication over addition. $I \times (I + I) = I$ and $I \times I + I \times I = I$, so $\forall x,y,z \in F, x(y+z) = xy+xz$   Based on the above, $\{I\}$ seems to qualify as a field.  If $I$ is assumed to be a real number, then the unique solution of $I + I = I$ and $I \times I = I$ is, of course, $I = 0$.  So, is {0} a field, or is there generally considered to be an additional field axiom which would exclude it?  Specifically, is it required for the multiplicative identity to be distinct from the additive identity? What's the 1 inch square knob sticking out of the wall? I have a question:     Suppose $f$ is continuous and even on $[-a,a]$, $a&gt;0$ then prove that   $$\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$$   How can I do this? Don't know how to start. I need a step-by-step, simple and easy way to configure swappiness. What program can be used to package software into one executable? Building a wormhole We regularly get questions about wormholes on this site. See for example  and . Various wormhole solutions are known, of which my favourite is  because it's closest to what every schoolboy (including myself many decades ago) thinks of as the archetypal wormhole.  The trouble is that Visser has pulled the same trick as  of starting with the required (local) geometry and working out what stress-energy tensor is required to create it. So Visser can tell us that if we arrange exotic string along the edges of a cube the spacetime geometry will locally look like a wormhole, but we know nothing about what two regions of spacetime are connected.  My question is this: suppose I construct a Visser wormhole by starting in Minkowksi spacetime with arbitrarily low densities of exotic matter and gradually assembling them into the edges of a cube, how would the spacetime curvature evolve as I did so?  I'm guessing that I would end up with something like Wheeler's bag of gold spacetime. So even though I would locally have something that looked like a wormhole it wouldn't lead anywhere interesting - just to the inside of the bag. I'm also guessing that my question has no answer because it's too hard to do any remotely rigorous calculation. Still, if anyone does know of such calculations or can point me to references I would be most interested. How do I use the conditional (ternary) operator? I've always wondered how to write the "A ? B : C" syntax in a C++ compatible language.  I think it works something like: (Pseudo code)  If A &gt; B    C = A Else    C = B   Will any veteran C++ programmer please help me out? I'm very new to Blender and I'm trying to make a water vortex like this.(may be not the easyest first project for a newbie)     I watched several tutorial on liquids particle etc but I'm not sure what is the best aproach. What I tried  until now:   I made a particle system with a vortex force field. It looks good, but I'm not able to add the right material.      I played around with the fluid physics, the water looks good but if I add a vortex force field it does nothing.   I would be very grateful if somebody could point me in the right direction because I'm definitely stocked. Famously:     "Try not. Do or do not. There is no try."           -- Yoda   Thing is, this is from a movie. A writer wrote it and a voice actor spoke it. Is there a correct form of attribution for a case like this? If so, what?