Sin⁻¹2x +cos⁻¹x = π/6 - inverse trig equations Please help? If F=f(x,y) and it is an exact function then why ∂/∂x(∂F/∂y)=∂/∂y(∂F/∂x)? why it is incorrect if it is not an exact function? From Wolfram MathWorld. Bayes Billiard Balls 10/28. Next: Beta Random Variable 10/28 Up: Conditional Distributions 10/27 Previous: Conditional Density 10/28 Bayes Billiard Balls 10/28 First argument: Suppose we throw 1 red billiard ball on the table and measure how far it goes on the scale from 0 to 1, call this value x, then throw n balls, what is the distribution of the number of balls to the left of the red ball?
Now what is x distribution? It is Uniform(0,1), and the overall distribution of the number of successes is the sum for all possible x's: Second way: Suppose I throw all the balls down first, and choose which is to be the red one, then the probability that the red one has k to the left of it is: . Which tells us that: Graph theory - What's the number of possible structures of alkanes $C_n H_{2n+2}$? Derivatives - Show that $\lim_{h\to 0}[1/(f(a+h)-f(a))-1/hf'(a)]=-f''(a)/2f'(a)^2$.
Determinant - How to prove that $\det\left[\pmatrix{u_1 & v_1\\ u_2 & v_2\\ u_3 & v_3}\pmatrix{s_1 & s_2 & s_3\\ t_1 & t_2 & t_3}\right]=0$? Matrices - Why does $A^2=I$ imply $nullity(A)=0$? Polynomials - Factoring $(a+b)(a+c)(b+c)=(a+b+c)(ab+bc+ca)-abc$ Integration - How to solve $\frac{d}{dx}\int_{\pi}^{x^2}\sin(t) \ \text{dt}$ using the Fundamental Theorem of Calculus? Real analysis - Question about differentiating under the integral sign. Real analysis - Stronger version of AMM problem 11145 (April 2005)? \text{e}^{-i\xi t} - \text{e}^{-i \eta t} \right. Hyperbolic Function. Combinatorics - An odd question about induction.
Power series for the rational function $(1+x)^3/(1-x)^3$ What do we know about the solution of this set of linear equations? Prove Local extrema are stationary. Type systems and logic. An important result in computer science and type theory is that a type system corresponds to a particular logic system.
How does this work? The basic idea is that of the Curry-Howard Correspondence. Discrete math study strategy - Tips and advice! Trying to self-learn calculus but stuck on this idea. Another proof of Wilson’s Theorem » mixedmath. While teaching a largely student-discovery style elementary number theory course to high schoolers at the Summer@Brown program, we were looking for instructive but interesting problems to challenge our students.
By we, I mean Alex Walker, my academic little brother, and me. After a bit of experimentation with generators and orders, we stumbled across a proof of Wilson’s Theorem, different than the standard proof. Wilson’s theorem is a classic result of elementary number theory, and is used in some elementary texts to prove Fermat’s Little Theorem, or to introduce primality testing algorithms that give no hint of the factorization. Theorem 1 (Wilson’s Theorem) For a prime number , we have The theorem is clear for , so we only consider proofs for “odd primes .”
The standard proof of Wilson’s Theorem included in almost every elementary number theory text starts with the factorial , the product of all the units mod . Now we present a different proof. Contest math - Olympiad Style Inequality. Reading Math «mixedmath mixedmath. First, a recent gem from MathStackExchange: Task: Calculate as quickly as you can with pencil and paper only.
Yes, this is just another cute problem that turns out to have a very pleasant solution. Here’s how this one goes. (If you’re interested – try it out. There’s really only a few ways to proceed at first – so give it a whirl and any idea that has any promise will probably be the only idea with promise). Chinese Remainder Theorem (SummerNT) «mixedmath mixedmath. This post picks up from the previous post on Summer@Brown number theory from 2013.
Now that we’d established ideas about solving the modular equation , solving the linear diophantine equation , and about general modular arithmetic, we began to explore systems of modular equations. That is, we began to look at equations like. A MSE Collection: A list of basic integrals «mixedmath mixedmath. The Math.Stackexchange (MSE) is an extraordinary source of great quality responses on almost any non-research level math question.
There was a recent question by the user belgi, called A list of basic integrals, that got me thinking a bit. It is not in the general habit of MSE to allow such big-list or soft questions. But it is an unfortunate habit that many very good tidbits get lost in the sea of questions (over 55000 questions now). So I decided to begin a post containing some of the gems on integration techniques that I come across.
I don’t mean this to be a catchall reference (For a generic integration reference, I again recommend Paul’s Online Math Notes and his Calculus Cheat Sheet). I don't get this at AAAAAAAAALLL! Help? I don't get this at AAAAAAAAALLL! Help? How Do Polar Bears Drink? Photo: trasroid (Flickr) Polar bears get their water from the chemical reaction that breaks down fat.
Polar Paradox You’ve heard the joke that at the North Pole you need a refrigerator to keep your water from freezing, right? What do polar bears do for water? They spend most of their life at sea, on drifting ice packs or on the shores of the Arctic. Eating snow to obtain water is metabolically expensive—it would take too much energy to melt down enough snow. High Fat Diet Polar bears get their water from the chemical reaction that breaks down fat. 求1,5,11,19,29,.......的nth term.
Calculus - If we know $x+y+z=1$, $x^2+y^2+z^2=2$, and $x^3+y^3+z^3=3$, how to find $x^4+y^4+z^4$? How find this limit $\lim_{n\to \infty} \left(\frac{(2n)!}{2^n\cdot n!}\right)^{\frac{1}{n}}\cdot \cdots$ How find this limit$\lim_{n\to\infty}\frac{1}{n}\left(\frac{n}{\frac{1}{2}+\frac{2}{3}+\cdots+\frac{n}{n+1}}\right)^n$ Calculus - Find a limit $\lim_{x \to - \infty} \left(\frac{4^{x+2}- 2\cdot3^{-x}}{4^{-x}+2\cdot3^{x+1}}\right)$
[??????] find this limit $\lim_{x\to0^{+}}\frac{\tan{(\tan{x})}-\tan{(\sin{x})}}{\tan{x}-\sin{x}}$ [??????????????????] Hard Olympiad Inequality. Improper integrals - Calculation of $\int_0^{\infty}\frac{\sin^2 x\,\cos^2 x}{x^2}dx=\frac{\pi}{4}$ Calculus - Find the maximum value of $ \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} $ Calculus - Evaluate: $\lim_{x \rightarrow 0}\left(\frac{\tan \left(\pi\cos^{2}x\right)}{x^2} \right)$ [v smart!!!!] polynomials - Solve $(4x+3)^2(2x+1)(x+1)=75$? Calculus - Find zeros of function $f(x)$ Calculus - How find this limit $\lim_{x\to 0}\frac{1}{x^4}\left(\frac{1}{x}\left(\frac{1}{\tanh{x}}-\frac{1}{\tan{x}}\right)-\frac{2}{3}\right)=？$ Calculus - Proving a limit using another limit. MATHS HELP NEEDED Prove that f(x)=1+x+x^2 is positive for all x?
What does this question mean? Give a geometric justification of the triangle inequality: /z1+z2/ ≤ /z1/+/z2/? Tn=((-1/3)^n)*t(n-1) (n-1) is a subscript of t and t1=1 this is in a) A.P b) G.P c) H.P d) None of these? Is Linear Algebra hard? 3 qualities of successful Ph.D. students: Perseverance, tenacity and cogency. What doesn't matter There's a ruinous misconception that a Ph.D. must be smart.
This can't be true. A smart person would know better than to get a Ph.D. "Smart" qualities like brilliance and quick-thinking are irrelevant in Ph.D. school. Students that have made it through so far on brilliance and quick-thinking alone wash out of Ph.D. programs with nagging predictability. Certainly, being smart helps. Moreover, as anyone going through Ph.D. school can tell you: people of less than first-class intelligence make it across the finish line and leave, Ph.D. in hand. As my advisor used to tell me, "Whenever I felt depressed in grad school--when I worried I wasn't going to finish my Ph.D.
Productivity hints, tips, hacks and tricks for graduate students and professors. Contents Jump to: My philosophy: Optimize transaction costs Distilled into empirically-wrought principles, my high-level advice is: Reduce transaction costs to engaging in productive behavior.
Trigonometry and Complex Exponentials. Amazingly, trig functions can also be expressed back in terms of the complex exponential.
Then everything involving trig functions can be transformed into something involving the exponential function. This is very surprising. In order to easily obtain trig identities like , let's write. Butterfly Effect. English Learning Strategies. Free course of Algebraic Fractions, www.easycoursesportal.com. Rationalizing Denominators Composed by Trinomials. Example: rationalize: Answer: 1st. We place 2 square roots of the denominator in parenthesis: Then, we write the remaining term with its corresponding sign: Actually, its the same as:, but it is easier for us to know the conjugate of the denominator: . Be careful now that the first term is composed by two addends (placed in parenthesis). 2nd. We multiply the numerator and denominator by the conjugate of the denominator. Be careful with the parenthesis: We have to bear in mind that the first term of the denominator is composed by two addends and when we multiply the sum of two numbers by their difference, we will get the difference of their squares.
[?????????????] For numbers a_1, ... , a_n, define p(x) = a_1x + a_2x^2 + ... + a_n x^n for all x. Suppose that (a_1)/2 + (a_2)/3 + ... + (a_n)/(n +1) = 0.? Evaluate the sum: ∑(k=0 to ∞) (-1)^k (4 arctan((3-sqrt(14))/5))^(1+2k) /((1+2k)!)? Very problematic integral from challenge problem (just need a hint, no need to evaluate for me)? Lim x=>0 ((ln(1-x^41))/(1-exp(pi*x^110)))? [??????] Complex number conversion? [???????] How do you find the slope of the slant Asymptote: 2x^2+1/x? [???????????] Let f : [0, 2] -> R be a continuous function with f(0) = f(2). Show there is some x ∈ [0, 1] such that f(x) = f(x+1).?
Using mathematical induction, prove that n^2>_ 2^n, n>_4 and n is a set of natural numbers? What is the laplace of t^(sqrt of 3)? The polynomial x^8 + 8x^7 - 28x^6 - 56x^5 + 70x^4 + 56x^3 - 28x^2 - 8x + 1 has tan(π/32) for one of its roots. What are the other roots? If p(a) = 0.68 p(a u b) = 0.91 and p(a ∩ b) = 0.35 then p(b) =? Intermediate Value Theorem? Prove by contradiction that every prime number greater than 3 is of the form 6n(+/-)1 where n is positive? [✓] Find the remainder when you divide x^4-3x^3+5x-1 by x^2? Magnetic field reduces beer foam and could make brews cheaper and less bitter too. Belgian researchers have used magnets to make beer less foamyApplying a magnetic field made antifoaming agents more effectiveThe field is applied when hops extract is added to beer's malt baseThis prevents it from attracting too much CO2 - which causes foamThe method takes just one minute and could make beer cheaper - as less hops extract is needed to prevent the formation of foam By Jonathan O'Callaghan for MailOnline Published: 14:25 GMT, 17 December 2014 | Updated: 18:10 GMT, 17 December 2014 It’s every beer drinker’s worst nightmare: You set your bottle down on a table and open it, only for the drink to foam up and come shooting out, drenching the table and most likely you as well.
But that nightmare might soon become a thing of the past thanks to an unlikely source: magnets. Researchers have found that applying a magnetic field to an antifoaming agent, they can reduce the foaming effect to a huge degree - and it could bring the cost of beer down in future. 港人太忙 愈專業愈無得收工. Music and math: The genius of Beethoven - Natalya St. Clair. Natalya would like to acknowledge the amazing support of her friends Wendy Cho, Carolyn Meldgin, Antoinette Evans, Will McFaul, Aaron Williams, and her fantastic students and colleagues at Countryside School and Math Zoom, especially Chris Antonsen, Kim File, Harold Reiter, Jeffrey Huang, Kashyap Joshi, and Priscilla Wang. Frequency and Music Our abilities to recognize patterns in music using sine waves help to “see” innovative ways of problem-solving. For teachers, a great introduction to teaching frequency theory can be found here ( Students might enjoy discussing the activity found in NCTM Illuminations to explore more with the mathematics of music.
Sound travels through energy in the form of wavelengths, which can be described using the function of the form f(x) = A sin (B x). For a nice overview of sine waves, watch The Math of Music. Frequency theory has many interesting and unique properties, some of which lead to harmonic analysis in upper-level math classes. Proper Cloth Reference. Denim Shirting Fabric Overview While chambray (below) and denim (above) are often confused for one another, they’re not the same fabric.
Parenting: Should parents help their children with homework. School life can get tough for children sometimes. School provides activities, sport facilities and leisure facilities along with academic studies. Such activities come like a stress-buster for children who otherwise get tired and exhausted due to studies. It is not easy for children to carry on their academic studies all the time.
This makes them stressed out and completely exhausted. The most important part of studies which leave kid more worn out is the homework. Homework no doubt forms an important criterion of education. The Success of Failure: Sylvia Chang shares her stories of defeat on the road to victory. November 2014 There is nothing more instructive than listening to the stories of failure of celebrities.
7 Big Benefits Of Exercising Outside This Winter. Let's face it -- it's tough to find the motivation to exercise outside these days. During the work week, sometimes both legs of our daily commutes are completed in utter darkness. [✓] What are the equations of the lines through (8,-3) and passing at distance square root of 10 from (3,2)? [?] Linear Algebra question about orthogonal projection. Please see the attached image for the problem.?
[?] Is there a fast method to algebraically factor cubic equations into completed binomial cubes? [?] Line Integral Problem (Parameterize)? Why is it important to understand the continuity of a function? [✓] Cos55°cos40°+sin55°sin40° is equivalent to what? [✓] How to solve the first order differential equation in term of x? Given tan x dy/dx + y sec^2 x = cos x.? What is the hardest math question you know? Solving any of the following will net you a Millennium Prize of $1 million: P versus NP Hodge conjecture Riemann hypothesis Yang–Mills existence and mass gap Navier–Stokes existence and smoothness Birch and Swinnerton-Dyer conjecture. Here's a few others, divided by category that are unsolved: So lost right now, got a packet on diff eq's but don't understand how to solve them. Here's one: y'=e^(x+2y) I can't seem to separate it.?
Probability_cheatsheet_140718. Lim to 4 f(x) =8 and lim x to 4 f(x)=3 evaluation lim ->4 [g(x) -(fx)]/8{f(x)}? A team of 8 players is to be chosen from a group of 15 players. One of the 8 is then to be elected as Captain and another as Vice Captain.? Determine whether or not the following series converges or diverges: Σ [n = 1 to ∞] (-1)^n ((3n-1) / (2n+1))? PROVE (COT^A)-(TAN^2)A = 4COT2A + COSEC2A? If sinA =4/5 & casB=5/13 find cos((a-b)/2)? You're not going to tell me anything I don't already know. 3 Questions You Have to Ask Yourself Before Buying Anything. When you buy something new, do you feel happier?
Is the feeling different if you purchase "things" or "experiences"? Happiness researchers have studied the difference between buying material goods—earrings, a smartphone, a new car—and buying “life experiences"—dinner out, a trip to the theatre, a music class. The results: Multiple studies have suggested that most people do get a temporary happiness boost from material purchases, but that the happiness benefit quickly fades. By contrast, the joy of a new experience provides a greater happiness bonus and is more enduring. So says an overwhelming amount of research. But according to three recent studies that examined the buying habits of 675 people, about a third of these people were unhappy if they bought things and unhappy if they bought experiences. What kept this miserable minority from gaining happiness? There are many reasons why someone might be chronically unhappy—depression, anxiety, and worries about the future, just to name a few.
The 7 Laws of Impatience. Have you ever wanted to give up on a paper you were writing for school or had an interesting project turn frustrating when unexpected complications arose? 何謂 ＂離散數學＂ Easily distracted: why it's hard to focus, and what to do about it. My new book 'Your Brain at Work' is out this week, so I thought I would share one of the ideas from the book that's been having the biggest impact: how to manage distractions. If you are paid to answer emails or deal with customers all day then this post might not be for you. Ace Medical School Exams by Maximizing Study Time. More than anything else, you want to practice medicine and become a doctor. You jumped through some major hoops to get to medical school. But before you can work with patients, and get that coveted M.D. after your name, you have to get through a mountain of coursework in a short amount of time and score well on your exams, without losing your marbles in the process!
Will the skills you gained as an undergrad be enough to carry you through the marathon of medical school? Imad Zak - Photos de la publication de Imad Zak dans... Why Sugar Doesn't Spoil. Mark U. asks: Why doesn’t sugar ever seem to go bad? Two foods are left out on the counter – fresh tomatoes and a bowl of sugar. Within a week or so, one will develop black spots and the other remains pristine, albeit perhaps a little clumpy depending on the humidity of the air. The reason? Osmosis. Why Tattoos Don't Fade as Skin Regenerates. Facebook. 10 ways to make the most of your time as a PhD student.