# Tag Info

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Generally speaking, it is impossible to prove every theorem and solve every problem in all the books you read. Many books cover much more than necessary, so they can cope with various needs. And even in the chapters laying foundations, there will probably be a few theorems forgotten later (which may call for a review). How exactly you read a book depends ...

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please be aware that this answers stems from personal experience and is meant more to apply when referring to books such as "calculus year one" or other more generic books that are intended to be large pools of practice problems in early college/high school caclulus/precalculus textbooks. Obviously there are books that are intended to be thouroughly read (...

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Overall answer: no. I struggled with the same problem as you for quite a long time and, in hindsight, I think I could have spent my time more wisely. Here are my current general guidelines at the time of this post. They may or may not work for you. The overall philosophy I employ is that exercises are usually there to get you comfortable with the material: ...

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Mathematics is kind of a subject that is fun as well as scary. I always know the concept and do one or two problems per concept. As solving each & every problem is time consuming and is of no use. My advice is practice all the concepts and theorems, and do two or 3 different types of problems which based on same concept. It helps us to know which ...

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I would suggest you read every problem, and in your head if you can see the direction pretty clearly then no need doing that, generally big texts do have repetition, but concise books meant for only problem solving without any theory do try to make sure each problem is unique. As far as second part of your research goes, I believe your approach gives you ...

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I have two favorite arguments that we should have $\exp (i\theta)=\cos \theta +i\sin \theta$ for real $\theta$. The first is closely related to Mathologer's video e to the pi i for dummies, and the second is discussed in slightly more detail in II.2 “Moving Particle Argument” in Visual Complex Analysis. Finally, I conclude with a summary of how Euler did it, ...

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One way is to define the function $$f(x)=e^{-ix}\left(\cos x+i\sin x\right)$$ Differentiating by $x$ yields $$f'(x)=-ie^{-ix}\left(\cos x+i\sin x\right)+e^{-ix}\left(-\sin x+i\cos x\right)=0$$ (where we assume that $i$ acts like a real scalar in differentiation). That means that $f$ is constant, and of course $f(0)=1$. So $f(x)=1$ for all $x$, implying $e^{... 0 As the$x$and$y$are constant they are irrelevant. Putting the$Z$in order we have$2=>0, 4=>.5, 8=>1$. The$z$are increasing by factors of 2 whereas the answer are increasing linearly by steps of 1/2. We can with some fudging figure$2^{2*answer + 1} = Z$. Or$2*answer + 1 = \log_2 Z$Or$answer = (\log_2 Z - 1)/2$. But there are infinite ... 0 I tried to find the number of ways in which a number can be expressed in term of sum of two numbers and I ended up learning Partitions which showed me how everything can be expressed mathematically.... 0 You asked: What was the first bit of mathematics that made you realize that math is beautiful? For me, it was when I was 3 years old (possibly 4), contemplating my hands and fingers. I had the sudden epiphany that 5+5 absolutely had to equal 10 every time that you added them together -- not merely that they had done so repeatedly, mind you, but that ... 0 I found the following resources: http://www.educatorstechnology.com/2012/10/8-great-youtube-channels-for-math.html http://www.avatargeneration.com/2012/10/learning-math-with-youtube/ http://www.freetech4teachers.com/2012/04/seven-youtube-channels-not-named-khan.html#.V2pEdbh9600 TV show called Dara O'Braian's School of Hard Sums (highly recommended) http://... 2 I just got done teaching the class again and it went extremely well, so I wanted to post my methods here. This isn't technically an answer, but more of what I did which works as an answer. All credit for my fundamental idea goes to @dxiv and the link he posted in the comment to my question. I gave each kid half of a cored orange (essentially a hemisphere)... 1 Reading @NominalAnimal 's informative nonanswer suggests this almost answer. I would start thinking about spherical trigonometry with your fifth graders. Examples show that the sum of the angles of a triangle is always greater than$\pi$(radian measure, of course, which you've introduced if they don't know it). Then motivate Girard's Theorem - the area of ... 1 This is definitely not an answer, but if you intend to use the "four parts of an orange peel" analogy, this might be relevant. This might be entertaining for some, especially those who enjoy confusing people with numerical coincidences (like Randall Munroe's XKCD comics 217 and 1047). I was thinking about the regular simplex in 3D -- the regular ... 10 One way to proceed is to make use of the well-known (well, it should be well-known) property of a sphere: If you inscribe a unit sphere within a right cylinder, and slice them "horizontally" (i.e., perpendicular to the axis of the cylinder) the corresponding strips of the sphere and of the cylinder have equal areas. That this is true can be seen by ... 0 I think the idea of trying to do this without calculus is misguided. Instead, try to understand the steps in the calculus. The surface area formula is derived from the volume formula so maybe the question should be: can I get the volume formula from the formula for circle area$A=\pi r^2$? Consider a pile of disks, with the bottom one of diameter$2r$and ... 0 Assuming units of distance to be$km$and that of time to be$hrs$. Let the velocity of the river be$v$, in the direction from$A$to$B$, and let the speed of the boats be$u$. Then, the velocity of the boat leaving from location$A$(or boat$A$) will be$u+v$from$A$to$B$, and that of the boat leaving from location$B$(or boat$B$) will be$u-v$... 1 Its great that you raised this point as a question (+1)! The issue has regularly been discussed in comments to many questions/answers tagged limits but now there is hope that all the arguments of both types (for and against) can be found in one place. There is a very common misconception that limits can be evaluated via plugging and use of L'Hospital's Rule ... 1 To avoid confusion, let me write$f$instead of$x$. I do this because we can think of$f$(or$x$) as a function. I don't know the exact situation in your book, but this is not uncommon. If$f$is a measure of something, then$\Delta f$usually means a difference (or increase) in$f$. So$f$could be a height, voltage, velocity, etc. You can, often, think ... 2 I was pretty good in maths during my school life especially Euclidean Geometry & Calculus. I participated & cleared many Maths Olympiads/Competitions like RMO (USAMO of India) & I understand what you are trying to say. Since I solved all the problems alone I would often spiral down the thought chain that there are some super-human ... 4 Mathematics is too broad to declare enjoyable or not. Maybe I love reading J.K. Rowling but I don't enjoy Dostoevsky. My point here is that like many subjects, "Mathematics" is too broad of a subject to either find generally enjoyable or not. In fact, I bet I would really love reading Dostoevsky if I actually understood him more... I think it comes down ... 6 Keep running. Things may change with struggle . 1 I'm confused as to how exactly [mathematics] helps me become more intelligent. Well the hope is not to just gain more knowledge (including techniques or tricks), but also to gain a deeper understanding of both logical and mathematical structures. Since you have only graduated from high school, you'll not know much about these. Originally, mathematics was ... 8 Form of any quadratic Equation is$ax^2+bx+c=0$Here for you$ a=1 , b= \sqrt{2}$and$c = -\frac{1}{2}$, $$x^2 + \sqrt{2}x + \left(-\frac{1}{2} \right)=0$$ Solution is $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$ $$x=\frac{-\sqrt{2}\pm \sqrt{(\sqrt{2})^2-4\cdot 1 \cdot \left(-\frac{1}{2}\right)}}{2 \cdot 1}$$ $$x=\frac{-\sqrt{2}\pm \sqrt{2+2}}{2 }$$ $$x=\... 3 Hint: x^2+\sqrt{2}x=\frac{1}{2}\implies2x^2+2\sqrt{2}x-1=0 Now use the general way to solve an quadratic. 0 Although it is a bit messy, you may be interested in the following map: If anyone knows who made it, please let me know! 6 This answer consists of some impressions I formed of your question. Feats of memory are party tricks. While it is essential for success in any cognitive pursuit to have a strong memory of your interests, that comes with the intense focus that is even more essential for success. Behind these effortless demonstrations are years of training that you have ... 10 The answer to the question in the title is no, but eidetic memory is not the same as intelligence. That having been said, the ability in question is really about remembering the important parts of a proof and how they fit together; memorizing lists of arbitrary numbers and facility with arithmetic computation are irrelevant. It's similar to the way ... 0 Set up lines of latitude and longitude on the sphere so that the two points, call them A and B, are on the same meridian. Then any motion from A to B can be "tracked" along the meridian by taking a moving point on the meridian that has the same latitude as the point moving from A to B. The motion along the meridian (an arc of a great circle) ... 1 Use a ball. Note that the less curvy a line is, the straighter it is (and therefore shorter). Then note that cutting a slice through the middle of the ball gets you the straightest line available. Which is a good definition of a great circle. (And as other answerers have pointed out, if the ball is edible then you will be nourishing bodies as well as ... 3 There are two ways I tried with students. Case 1. Equator of ball On a plastic ball toy carefully tie a string around any great circle, ( use a smal cellulose tape/tab if needed, to prevent side slippage ,) for exactly one rotation. Make the string taut by pulling in opposite directions. The ball will be compressed, tension in taut string increases. ... 6 You can always rotate the sphere so that points A and B are both on the equator. The idea then is you reduce your distance from point B the fastest if you head in the direction of point B, and that direction is along the equator. 2 If you connect the two points by a rubber band in the shape of a meandering path on the sphere, it is intuitive the rubber band will snap into a great circle shape. Alternatively you can explain the geodesic as the path a magnetic marble would take if the sphere were a steel ball, and you let the marble roll along the surface of the sphere. It will roll in ... 2 I have found it helpful to replace the sphere by an apple and introduce an "internal" observer by placing an ant on the apple. The ant will crawl from point A to point B on the sphere by following the shortest path (the queen can't wait) which is always an arc of great circle. An additional point that students find illuminating is the phenomenon that a ... 3 You can apply the following logics: the units are denoted I, II, III, IV, V, VI, VII, VIII^*, IX; the tenths are denoted X, XX, XXX, XL, L, LX, LXX, LXXX, XC; the hundredths C, CC, CCC, CD, D, DC, DCC, DCCC, CM; the thousands, M, MM, MMM. numbers are written in thousands, hundredths, tenths and units from left to right. no other pattern is allowed. ... 17 When reading roman numerals, I prefer to think in the following way: Read from left to right, and if at any point the value of a character decreases, put a comma between the decrease. Then, add each block together. MCMXCVI \mapsto M,CM,XC,V,I \mapsto 1000+900+90+5+1=1996 MDCCCLXXIV \mapsto M,D,CCC,L,XX,IV \mapsto 1000+500+300+50+20+4=1874 ... 5 When you are reading Roman numerals, start from the left-most character. Read rightward until the value of the character increases. Then, section those two characters off, and repeat. That sounds really complicated, and I wrote it somewhat poorly, so here are some examples. In XIX, we start with the left X which is 10. Then we move to the I which ... 2 XIX is read left to right, the "I" is always applied to the final X. XIX = X + IX = 10 + 9 XXI = X + XI = 10 + 11 2 For a non-symmetric matrix take:$$A=\begin{pmatrix} 1& 1 \\ 2& 2 \end{pmatrix}. $$Then (1,2)^T is a eigenvector to eigenvalue 2 and (1,-1)^T is a eigenvector to eigenvalue 0 but they are not orthogonal. 0 I would disagree with your premise that l'Hopital's rule is almost never used. Actually it is important in building up the basic framework of the calculus. For example, if you wish to establish typical limits for transcendental functions, the rule is useful. I would agree with you that more advanced applications require more advanced estimates. 0 Maybe Simplify[SquaredEuclideanDistance[{x, y, z}, a], {x, y, z} ∈ Reals] == SquaredEuclideanDistance[a, n1] (-3 + x)^2 + (-3 + y)^2 + (-1 + z)^2 == 12 is what you are looking for. 1 Here are my thoughts. I am an undergraduate engineering/math double major, so I have the advantage of seeing calculus from both the "rigorous" side and the "engineering" side, and these are two wholly different things, as you are probably aware. Calculus, as it's taught to non-math majors, puts a very heavy emphasis on applicability. For instance, a ... 1 I knew that this was actually what I wanted to do for the next few years or even the rest of my life. That's great! I feel like my skills don't develop fast enough, and therefore I don't know in which direction I'm heading right now. It's okay. Just enjoy the journey. I always think that there is something wrong with my way of approaching ... 3 Your thoughts are certainly normal. In fact, one could generalise and say that no mathematician is ever a success in his own eyes. How can he be, given that so much of maths involves working for weeks on a problem which, once you have solved it, has a solution that can be understood in minutes? I strongly recommend that you read section VIII.6, "Advice to a ... 2 You might want to look at Basic Mathematics by the ever productive Serge Lang. The title is definitely understatement compared to other "college algebra" and precalculus texts, but it's quite accurate from the perspective of a mathematician. As a bonus, it's considerably cheaper than standard College Algebra / Precalculus texts (including the ones my ... -1 This is an interesting question, thoughtfully asked. I really like @SSS 's answer, and have upvoted it. You are right that mathematics is much more than being able to pass the tests you get in school. One way to learn what mathematics is in life after school is to learn more - but not just what's in the next course If you're curious and determined enough ... 0 If we take for granted that$$\lim_{t\rightarrow \infty} \frac{\ln 2 + t}{\ln 3 + t} = 1$$then what this means is by making t large enough, say greater than x_0, the fraction can be can be made within \epsilon of 1. Now, consider$$\frac{\ln 2 + \ln x}{\ln 3 + \ln x}$$When$\ln x$is greater than$x_0$, this fraction will be within$\epsilon\$ ...

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This is in no way a complete answer, but I want to touch on something really important that you said. The only thing that differentiated an easy problem from a difficult one was the fact that the person solving the problem did not know the trick it required to solve the problem. I've done a lot of competition math, and I've also taken the IIT/JEE exam ...

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Precalculus, Mathematics for Calculus, 4th edition by James Stewart, Lothar Redlin, and Saleem Watson. This edition is a little old so you might be interested in a newer one, but this is the book which I learned precalc from and it was challenging. At the end of some chapters it has "problem solving"sections which should give most precalc student a headache (...

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The key to excel in anything is sheer practice and hard work. Keep this thing in your mind whenever you suffer any setback at any stage of your life. Now for your advice on how to do well in mathematics, especially calculus. Well, since you say that you are a computer geek, I would recommend that the algorithm to follow is: First and foremost, take a ...

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