Consider the surface $S$ in $\Bbb R^3$ given by the graph of $f(x,y) = xy^2 − 2y^2 + e^x$.  Calculate the intersection point of the tangent planes to $S$ above $(x, y) = (0, 0)$, $(0, 1)$, and $(0, 2)$.  I am confused about what this question is asking. Would I have to choose a tangent plane above that point and see where it intersects $S$? Thanks
Consider the surface $S$ in $\mathbb{R}^3$ given by the graph of $f(x, y) = xy^2 − 2y^2 + e^x$ . Calculate the intersection point of the tangent planes to $S$ above $(x, y) = (0, 0), (0, 1)$, and $(0, 2)$.  Can someone guide me through solving this problem? I am confused on where to start because we didn't cover this in class. Thanks so much
Find the maximum value of $ f(x) = \int^{x}_{0} \sqrt{(x^2-x)^2+y^4}~dy$ The Wikipedia article on  mentions two methods of sampling "small" polynomials namely uniform sampling and discrete Gaussian sampling. Uniform sampling is clearly the simplest, involving simply uniformly selecting the coefficients from the set of small coefficients, thereby guaranteeing they will be small. However, the articles mention that security proofs rely on discrete Gaussian sampling.  However, I have a bit of difficulty wrapping my head around    Why discrete Gaussian sampling would be better than uniform, security-wise?   I mean, usually uniform is preferable - it maximizes the entropy/information in each sample, making the outcome more unpredictable. I could understand it if the Gaussian sampling technique would somehow serve to avoid (at least with high probability) "weak" parts of the space of outcomes but I have found no information/justification that this is the case.  Another reason I can see why discrete Gaussian might be better is that there is, after all, a chance (albeit a small one) one would select a non-small coefficient, so the space of outcomes is actually larger. However, given that this probability of hitting this extra space is (by design) quite small but also that one would likely immediately reject such a sample (because the algorithm calls for a small polynomial), this seems to be irrelevant.  It is not because I am against discrete Gaussian sampling I just want to understand why it is more secure. Also, I want to understand if it really IS more secure or it is simply easier to prove security in that model. I would actually consider that very interesting if it is the case because it raises the question of why is it easier to prove the security under conditions that are intuitively worse (i.e. discrete Gaussian vs the intuitively better uniform case)? If it is so intuitively clear that uniform sampling is better, surely it must be possible to transform that intuition into a security proof of RWE also under uniform sampling. Otherwise, I would fear the intuition must with high likelihood be wrong or at least dubious. How can I deduce/identify an app from a notification icon? random access file in java I have the following fields:   Inventory control (16 byte record)   Product ID code (int – 4 bytes) Quantity in stock (int – 4 bytes) Price (double – 8 bytes)    How do I create a fixed length random access file using the above lengths? I tried some examples online, but I either get an EOF exception or random address values when I try to access them.  I tried some more examples and couldn't understand the concept very well. I'm trying a project with it and will try to explore more on it.  Here is some example data. There might be holes in the data where No. in stock could be 23 == 023.            Quantity ID. No.   In Stock   Price  -------   --------   ------  1001       476      $28.35  1002       240      $32.56  1003       517      $51.27  1004       284      $23.75  1005       165      $32.25   Thanks for the help. Regex for quoted string with escaping quotes How can I use awk to (sometimes) remove the first character of a column? That's a lot of ___ vs. those are a lot of ___ That's a lot of cars!   Those are a lot of cars!      The context is when a someone sees many objects (in this example, cars) and makes the exclamation.   According to , lot can handle either singular or plural verbs.  I agree with the author who states that that's sounds better, but the post doesn't seem definitive either way.     Are both grammatically correct and do they have the same meaning? Using Windows DLL from Linux For which values of $a$ does $a-2 \mid a^3+4$ Why $SU(3)$ and not $U(3)$? Highlight elements in the matrix I have a matrix and need to highlight some elements in the matrix. I have found how to highlight a single element (I want to put the cell in the square):  $$ A = \begin{bmatrix} \fbox{0} &amp; \fbox{0} &amp; 0 \\  \fbox{0} &amp; \fbox{0} &amp; 0 \\ 1 &amp; 1 &amp; 1 \\  \end{bmatrix}$$   But how can I outline elements of the submatrix like in the picture below? I need just a frame where some elements are located. How to uninstall Intel Linux Graphics Installer and revert to default drivers I’m a Filipino citizen. My flight goes from Vancouver to Manila via Tokyo Haneda. I will arrive in Japan at 6pm then depart at 9am the next day. I'm not planning to go out of the airport during my layover, so I don't see a need to have a transit visa. However, I called the Consulate-General of Japan here in Canada, and they asked me to get a transit visa.  Then I called the Ministry of Foreign Affairs in Japan and asked them the same question.. at first, they told me I don’t need a visa as long as I stay inside the airport, but when I told them that the Consulate-General of Japan in Canada asked me to get a visa, they also asked me to get one.  I'm so confused. I would like to know your knowledge or experience regarding this. What's the difference between "cardinal" and "ordinal" as adjectives? Difference between $:=$ and $=$ I am sorry if this is quite elementary question. But I always think, that why we use $:=$ at some places, instead of $=$. Is there any fundamental difference between these two? Before reading Terry Tao's blog (4 months ago), I had never seen a symbol like this (:=). "ORA-01950: no privileges on tablespace 'SYSTEM' " : What error is it? Earlier I created a user :  SQL&gt;create user suhail identified by password SQL&gt;User created. SQL&gt;Grant create session,create table to suhail; SQL&gt;Grant Succeeded.   but when I login as suhail and try to create a table I get an error saying ORA-01950: no privileges on tablespace 'SYSTEM' .  What error is it ? Why does it occur ?    SQL&gt; connect suhail Enter password: Connected. SQL&gt; create table suhail_table(name char(10)); create table suhail_table(name char(10)) * ERROR at line 1: ORA-01950: no privileges on tablespace 'SYSTEM'   I am using Oracle Database 11g Express Edition Release 11.2.0.2.0 Showing that $\Bbb R[x] / \langle x^2 + 1 \rangle$ is isomorphic to $\Bbb C$ question I was wondering this today, and my algebra professor didn't know the answer.      Are subgroups of finitely generated groups also finitely generated?    I suppose it is necessarily true for finitely generated abelian groups, but is it true in general?  And if not, is there a simple example of a finitely generated group with a non-finitely generated subgroup?  NOTE:  This question has been merged with another question, asked by an undergraduate.  For an example not involving free groups, please see Andreas Caranti's answer, which was the accepted answer on the merged question. Use Cauchy-Schwarz inequality to show series convergence How to get Intel Dual Band Wireless-AC 3168 work on Ubuntu 16.04?