Integration by Substitution (H2 Maths Tuition)

This is a 1 page article prepared by me for students to learn how to do Integration by Substitution, a very useful technique that can integrate many functions.

This is especially useful for students taking H2 Maths, as it is one of the two tools for integration. The other is Integration by Parts. SMU First Year Students also have to take a calculus course which includes Integration by Substitution too.

This is also my first time trying out embedding Scribd into WordPress, so that users can view the document on the website itself without downloading anything. 🙂

Print version: Integration by Substitution


Featured book:

Calculus: An Intuitive and Physical Approach (Second Edition) (Dover Books on Mathematics)

Application-oriented introduction relates the subject as closely as possible to science. In-depth explorations of the derivative, the differentiation and integration of the powers of x, and theorems on differentiation and antidifferentiation lead to a definition of the chain rule and examinations of trigonometric functions, logarithmic and exponential functions, techniques of integration, polar coordinates, much more. Clear-cut explanations, numerous drills, illustrative examples. 1967 edition. Solution guide available upon request.

Why We Use “X” as the Unknown in Math

Why do we use “x” in algebra? Why not “a”, “b” or even “z”?

Find out the answer here: http://gizmodo.com/why-we-use-x-as-the-unknown-in-math-1657254357

Excerpt:

For hundreds of years, x has been the go-to symbol for the unknown quantity in mathematical equations. So who started this practice?

Algebra was born in the Middle East, during the Golden Age of medieval Islamic civilization (750 to 1258 AD), and its early form can be seen in the work of Muhammad Al-Khwarizmi and his 9th century book, Kitab al-jabr wal-muqabala (al-jabr later morphing into algebra in English). During this heyday, Muslim rule and culture had expanded onto the Iberian Peninsula, where the Moors encouraged scholarship in the sciences and math.

So what does this have to do with the letter “x” in math? In a recent TED talk, the director of The Radius Foundation, Terry Moore, posited that the the use of “x” in this way began with the inability of Spanish scholars to translate certain Arabic sounds, including the letter sheen (or shin). According to Moore, the word for “unknown thing” in Arabic is al-shalan, and it appeared many times in early mathematical works. (For example, you might see “three unknown things equals 15,” with the “unknown thing” then being 5.)

But since Spanish scholars had no corresponding sound for “sh,” they went with the “ck” sound, which in classical Greek is written with the chi symbol, X. Moore theorizes, as many others before him have done, that when this was later translated into Latin, the chi (X) was replaced with the more common Latin x. This is similar to how Xmas, meaning Christmas, came about from the common practice of religious scholars using the Greek letter chi (X) as a shorthand for “Christ.”


Featured book:

Algebra Survival Guide: A Conversational Guide for the Thoroughly Befuddled

Theorem on friends and strangers

Suppose a party has six people. Consider any two of them. They might be meeting for the first time—in which case we will call them mutual strangers; or they might have met before—in which case we will call them mutual acquaintances. The theorem says:

In any party of six people either at least three of them are (pairwise) mutual strangers or at least three of them are (pairwise) mutual acquaintances.

 


 

Featured book:


Competition Math for Middle School

Very Inspirational Math Video

Just to share a video here:

Very Inspirational Math Video

It is a video of a girl who once did a math quiz and totally blanked out for the whole quiz. However, it turned out that her teacher did not actually ask for the quiz back, and gave her as much time as she wanted to complete the quiz. Under the relaxed circumstances, she completed the quiz and got a ‘C’. (big improvement from totally blank).

Then, she went to UCLA (very good school in US), and became a mathematics major, and wrote the book that is listed below the video!

Truly inspiring. For some kids, too much pressure may result in Math anxiety and totally blank out, while for other kids a little bit of pressure is needed to ensure that they do take studies seriously. Need to find the perfect balance for each child.

**The book the girl above wrote is: Math Doesn’t Suck: How to Survive Middle School Math Without Losing Your Mind or Breaking a Nail

The Monty Hall Problem

This is the clearest and most interesting explanation of the Monty Hall Problem I have ever seen:

What is the Monty Hall Problem? It is basically a game show with 3 doors. Behind one of the doors is a car, while behind the other two doors are two goats. Most people will want to get the car of course.

The player gets a chance to choose one of the doors. Then, the host will open a door which contains a goat. Now, the player is allowed two choices: either stick to his original choice, or switch to the other unopened door. Which choice is better?

Watch the video to find out!


Featured book:

The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brain Teaser

Mathematicians call it the Monty Hall Problem, and it is one of the most interesting mathematical brain teasers of recent times. Imagine that you face three doors, behind one of which is a prize. You choose one but do not open it. The host–call him Monty Hall–opens a different door, always choosing one he knows to be empty. Left with two doors, will you do better by sticking with your first choice, or by switching to the other remaining door? In this light-hearted yet ultimately serious book, Jason Rosenhouse explores the history of this fascinating puzzle. Using a minimum of mathematics (and none at all for much of the book), he shows how the problem has fascinated philosophers, psychologists, and many others, and examines the many variations that have appeared over the years. As Rosenhouse demonstrates, the Monty Hall Problem illuminates fundamental mathematical issues and has abiding philosophical implications. Perhaps most important, he writes, the problem opens a window on our cognitive difficulties in reasoning about uncertainty.

Math Handheld Computer Game

Featured Item:

Educational Insights Math Whiz

Is your child disinterested in Math? Looking for some fun and educational Math games?

Math Whiz plays like a video game and teaches like electronic flash cards. This portable ELA quizzes kids on addition, subtraction, multiplication and division, AND works as a full-function calculator at the press of a button. Problems are displayed on the LCD screen. Features eight skill levels, as well as lights and sounds for instant feedback. Two AAA batteries required (not included).

 

Chinese Lucky Numbers – Numberphile

8 and 6 are lucky but 4 is unlucky… if you’re Chinese!

Featuring Xiaohui Yuan from the University of Nottingham.

Website: http://www.numberphile.com/
Numberphile on Facebook: http://www.facebook.com/numberphile
Numberphile tweets: https://twitter.com/numberphile

Videos by Brady Haran

Brady John Haran is an Australian independent film-maker and video journalist who is known for his educational videos and documentary films produced for BBC News and for his YouTube channels. (http://en.wikipedia.org/wiki/Brady_Haran)

Highly recommended to subscribe to Numberphile on Youtube for fun and interesting Math videos!


Featured book:

Number: The Language of Science

Number is an eloquent, accessible tour de force that reveals how the concept of number evolved from prehistoric times through the twentieth century. Tobias Dantzig shows that the development of math—from the invention of counting to the discovery of infinity—is a profoundly human story that progressed by “trying and erring, by groping and stumbling.” He shows how commerce, war, and religion led to advances in math, and he recounts the stories of individuals whose breakthroughs expanded the concept of number and created the mathematics that we know today.

– Rated 4.5/5 on Amazon

News: Singapore Education Ranked Third in World

Singapore takes third spot in global education rankings
Straits Times
Teacher Anthony Tan conducting an English lesson with a class of Primary 6 pupils at Woodlands Primary School. Singapore’s education system has …
Singapore offers Saudi Arabia help in education
Arab News
PROPOSAL: Singapore Senior Minister of State Lee Yi Shyan with Mazen Batterjee, vice chairman of the JCCI, on Wednesday. (AN photo by Irfan …
In Singapore, Training Teachers for the ‘Classroom of the Future’
Education Week News
Welcome to the Classroom of the Future—a mock-up housed by Singapore’s National Institute of Education (NIE) to demonstrate what learning might …
Singapore Polytechnic Assists CDIO Implementation At Malaysia’s Polytechnic
Bernama
PUTRAJAYA, May 6 (Bernama) — Singapore Polytechnic is assisting Malaysia on the implemention of innovative engineering education framework …
Lift education standards: Linfox boss
The Australian
“Most of our graduates are now coming out of Thailand, Vietnam, Singapore and China because they are just so well educated,” he said. “I can get …
In search of education
The News International
Unless we start investing massively in education, science, technology and innovation, as was done by Singapore, Korea, Malaysia, China and others, …
Sultanate, Singapore and the Indian Ocean
Oman Daily Observer
These are thoughtful words from your education minister (Heng Swee Keat), … a pragmatism which incidentally I believe we share with Singapore.
Direct School Admission not meant to lower academic standards
TODAYonline
In Singapore, there is no compromising a good education. Having a talent does not give a student the licence not to pursue academic excellence.
NAFA inspires
The Hindu
The safe and comfortable cosmopolitan environment Nanyang Academy of Fine Arts, Singapore makes it the perfect destination for education abroad.
Japan’s Education Minister visits SMU
Perspectives@SMU
Singapore Management University (SMU) received a special guest on its campus on 3 May 2014 – Japan’s Minister of Education, Culture, Sports, …

Monster Group

Check out this Youtube video on the Monster Group (related to Group Theory, a branch in Mathematics)


In the mathematical field of group theory, the monster group M or F1 (also known as the FischerGriess monster, or the Friendly Giant) is a group of finite order. (See Wikipedia: http://en.wikipedia.org/wiki/Monster_group)


Featured book:

The Symmetries of Things

This book is written by John Conway, one of the mathematicians who worked on the Monster Group. Rated highly on Amazon.

Start with a single shape. Repeat it in some way—translation, reflection over a line, rotation around a point—and you have created symmetry.

Symmetry is a fundamental phenomenon in art, science, and nature that has been captured, described, and analyzed using mathematical concepts for a long time. Inspired by the geometric intuition of Bill Thurston and empowered by his own analytical skills, John Conway, with his coauthors, has developed a comprehensive mathematical theory of symmetry that allows the description and classification of symmetries in numerous geometric environments.

This richly and compellingly illustrated book addresses the phenomenological, analytical, and mathematical aspects of symmetry on three levels that build on one another and will speak to interested lay people, artists, working mathematicians, and researchers.

 

How to prove square root of 2 is irrational?

A rational number is a number that can be expressed in a fraction with integers as numerators and denominators.

Some examples of rational numbers are 1/3, 0, -1/2, etc. Now, we know that \sqrt{2}\approx 1.41421\cdots.

Is the square root of 2 rational? Or is it irrational (the opposite of rational)? How do we prove it? It turns out we can prove that the square root of two is irrational using a technique called proof by contradiction. (One of the earlier posts on this blog also used proof by contradiction to show that there are infinitely many prime numbers.)

First, we suppose that \displaystyle\sqrt{2}=\frac{p}{q}, where \displaystyle\frac{p}{q} is a fraction in its lowest terms.

Next, we square both sides to get \displaystyle 2=\frac{p^2}{q^2}.

Hence, 2q^2=p^2. We can conclude that p^2 is even since it is a multiple of 2. Thus, p itself is also even. (the square of an odd number is odd).

Thus, we can write p=2k for some integer k. Substituting this back into 2q^2=p^2, we get 2q^2=4k^2, which can be simplified to q^2=2k^2.

Hence, q^2 is also even, and hence q is also even!

But if both p and q are even, then \displaystyle\frac{p}{q} is not in the lowest terms! (we could divide them by two). This contradicts our initial hypothesis!

Thus, the only possible conclusion is that the square root of two is not a rational number to begin with!

irrational


Featured book:

Math Jokes 4 Mathy Folks

Who says math can’t be funny? In Math Jokes 4 Mathy Folks, Patrick Vennebush dispels the myth of the humorless mathematician. His quick wit comes through in this incredible compilation of jokes and stories. Intended for all math types, Math Jokes 4 Mathy Folks provides a comprehensive collection of math humor, containing over 400 jokes.

– Highly rated on Amazon.com

 

Studies and Studying: How do top students study?

Source: http://www.quora.com/Studies-and-Studying/How-do-top-students-study/answer/Qiaochu-Yuan-1

Check out this post by MIT almost perfect-scorer, on how to study. His secret is to study the material in advance, before the lessons even start! This is really a useful strategy, if implemented correctly. Imagine being in Primary 3 and already knowing the Primary 4 syllabus! Primary 3 Math will be a breeze then. This is one of the reasons why China students are so good at Math – they have already studied it back in China, where the Math syllabus is more advanced!

Do try out this strategy if you are really motivated to improve in your studies. The prime time to do this is during the June and December holidays – take some time to read ahead what is going to be learnt during the next semester.

This is an excerpt of the thread:

I graduated from MIT with a GPA of 4.8 (out of 5.0) in mathematics. I had two non-As, both of which were non-math classes.

That doesn’t imply that I have good study methods, but anyway, here’s how I studied at MIT. My main study method as an undergraduate, for math classes, was knowing a sizable chunk of the material in advance.

This isn’t a method that will work for everybody. I did a lot of mathematics outside of the classroom both in high school and at MIT, and I often saw a substantial portion of the material in a given class before I took it. I can’t emphasize enough how much easier this makes a class, and not just for the reasons you might expect: one of the most valuable things you get out of knowing a lot of the material already is just not being intimidated by it. (And you can get this benefit even if you’ve only seen some of the material before and possibly forgotten some of it too.) You’re much more relaxed, and that makes it easier to process the part of the material that you don’t know.

What that translates to in terms of practical advice is this:

  • cultivate a sense of curiosity,
  • don’t restrict your learning to the classroom,
  • only take classes that actually seem really interesting to you, and
  • try to learn something related to those classes the semester before.

None of this is advice for studying for a class you’re taking now, but it’s advice for reducing the extent to which you will need to study for classes you’ll take in the future.

– Qiaochu Yuan

Math News: Math student from Nanyang Technological University detects OAuth, OpenID security vulnerability

Is it safe to log in through well known sites such as Facebook and Google? Think again, for Wang Jing, a PhD student in mathematics at the Nanyang Technological University in Singapore, has detected critical security vulnerabilities in the OAuth, OpenID security protocols. (Source: http://phys.org/news/2014-05-math-student-oauth-openid-vulnerability.html) [Second article in the list below]

Forward this information to your friends via the Tweet button below to warn them of the potential danger!

Unique Math Learning Center Opens in Lake Forest: Local After-School Program to Provide
Chicago Tribune
Mathnasium – The Math Learning Center opened its doors in Lake Forest in March to students looking for math help and math enrichment. The new …
Math student detects OAuth, OpenID security vulnerability
Phys.Org
(Phys.org) —To get right to the point, a doctoral candidate in math has discovered two holes in OAuth and OpenID that could leak data and redirect …
A math lesson for city: Teachers’ contract likely to cost billions
SILive.com
Mayor Bill de Blasio, who visited Staten Island Wednesday evening to speak at an SIEDC cocktail reception at the Hilton Garden Inn, has cleared his …
Math error halts Bank of America’s stock buy back and dividend increase
UPI.com
WILMINGTON, Del., April 28 (UPI) — Bank of America has had to halt its proposed stock buy back and dividend increase because of a math error in its …
Philadelphia girl makes math a game – and excels
Philly.com
Josephine Nyugen, a sixth-grader at St. Cecelia, right, with her father, Joseph, left, plays the math game ‘Into the Vortex’ on the First in Math website.
Math for public works borrowing bill proves tough
St. Cloud Times
Lawmakers from the Minnesota House Capital Investment Committee look around the prison yard near the loading dock inside the Minnesota …
Houghton Mifflin Harcourt takes Go Math! Academy into the home market
Boston Globe
Houghton Mifflin Harcourt, a Boston-based education company, developed the program for students in kindergarten to sixth grade to learn basic math …
Steve Ballmer’s math on Apple innovation doesn’t add up
ZDNet
His math of Apple innovation appears lacking to this longtime Mac user. Let me add a few more of his “tricks” to the list: The Apple II platform. Ballmer …
Math class helps special needs student try to win a wheelchair van
KOB.com
There’s a lot more going on than just addition and subtraction in Mr. Green’s math class. “We just want to help out this family,” Said Cody Green.
Budget math takes a U-turn: Christie blames federal tax law that brought windfall last year
NorthJersey.com
The same federal tax policy that Governor Christie is now blaming for New Jersey’s $807 million budget shortfall helped save his budget last year …

Education News Update

The Straits Times holds its first Education Forum on Sunday
Straits Times
The Straits Times’ first Education Forum on May 4, 2014, held at the Singapore Management University’s Mochtar Riady Auditorium. — ST PHOTO: …
All 300 places at The Straits Times’ first education forum this Sunday taken up
Straits Times
Mr David Hoe, an undergraduate at the National University of Singapore (NUS), is one of the speakers at the inaugural The Straits Times Education …
Many turn up at E Plus International Education fair
The Hindu
The aspirants evinced keen interest in countries like Holland, Singapore, New … Official boards of all the countries presented seminars on education …
Tuition and divorce
The Independent Singapore News
In September 2013, The Independent Singapore reported on Senior Minister of State for Education Ms Indranee Rajah’s observation on the perceived …
NS committee may propose changes to IPPT management
TODAYonline
SINGAPORE — Suggestions to improve the management of the Individual … Veterans’ League, which was founded to promote National Education.
Should India Embrace Socialism, Singapore Style?
Businessinsider India
This is because the Singapore government only borrows to develop a … What offers a ray of hope to Indian educators is that Singapore’s education …
How does one of the top-performing countries in the world think about technology?
The Hechinger Report
SINGAPORE—Forty students in bright yellow shirts hunched over their … Investments in education technology have been a key part of Singapore’s …
Why Indonesian education is in crisis
Jakarta Post
Does anyone seriously believe “education” in Indonesia is on par with the west, or even Asian countries like Japan, Korea or Singapore? Ask the …
Are you getting a little crazy in your classroom?
T.H.E. Journal
We have asked Dr. Zachary Walker, an assistant professor at the National Institute of Education, Singapore, an American who is traveling the world …
GEMS Education eyes expansion in the region
Business Times (subscription)
GEMS Education, the world’s largest operator of private schools, aims to … from kindergarten to pre-university, will open in Singapore later this year.

Riemann Hypothesis Proof

Latest News: Riemann Hypothesis Proved?

Source: http://arxiv.org/abs/1402.5952

Recently, I saw on Arxiv (an online Math journal) that a professor from South-China Normal University, Mingchun Xu, has proved the notoriously difficult Riemann Hypothesis.

Quote: “By using a theorem of Hurwitz for the analytic functions and a theorem due to T.J.Stieltjes and I. Schur, the Riemann Hypothesis has been proved considering the alternating Riemann zeta function. “

His paper can be downloaded here: http://arxiv.org/pdf/1402.5952v2

More verification is needed to check if it is indeed a proof.


What is the Riemann Hypothesis about? Watch this Youtube Video:


Featured book:

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

In 1859, Bernhard Riemann, a little-known thirty-two year old mathematician, made a hypothesis while presenting a paper to the Berlin Academy titled  “On the Number of Prime Numbers Less Than a Given Quantity.”  Today, after 150 years of careful research and exhaustive study, the Riemann Hyphothesis remains unsolved, with a one-million-dollar prize earmarked for the first person to conquer it.

Rated: 4.5 stars on Amazon

SG Education News: Even Saudis are learning Singapore way of Teaching

Saudis learning the Singapore way of teaching
Straits Times
Since last October, the National Institute of Education (NIE) has taken leadership trainers from the kingdom under its wing, training them in curriculum …
All 300 places at The Straits Times’ first education forum this Sunday taken up
Straits Times
Mr David Hoe, an undergraduate at the National University of Singapore (NUS), is one of the speakers at the inaugural The Straits Times Education …
Singapore Plows Ahead of US With Tech in Schools
NBCNews.com
In the late 1990s, the Singapore Ministry of Education unveiled its master plan for technology. The first phase was spent building up infrastructure and …
Govt mulls more recognition for NSmen in housing, health, education
Channel News Asia
SINGAPORE: More recognition could be given to National Servicemen (NSmen) in areas such as housing, healthcare and education. Defence …
Singapore to beef up nuclear technology expertise
Channel News Asia
Singapore is beefing up its nuclear technology expertise with a newly-announced programme. The 10-year Nuclear Safety Research and Education …
Many turn up at E Plus International Education fair
The Hindu
The aspirants evinced keen interest in countries like Holland, Singapore, New … Official boards of all the countries presented seminars on education …
AWARE’s pushback on more benefits for NSmen ignites debate
TODAYonline
SINGAPORE — The Government’s plan to enhance housing, healthcare and education benefits for operationally ready national servicemen has …

Math News April 28

Can Monkeys do advanced Math?

Read the below news for the answer and more!

Math wiz monkeys providing researchers with insights into human brain activity
Fox News
Monkeys trained to solve math problems are providing researchers with new insights into understanding a human learning disability in which children …
Math and Science Pay, But High Schoolers Care Less
Wall Street Journal (blog)
Math and science are the peas and carrots of the jobs market: great for a career future, but resolutely unpopular with the young. Even amid a relatively …
Math wrath in Pincher Creek?
Pincher Creek Echo
Protesters gather during a rally to support a petition calling for math curriculum reform at the Alberta Legislature Building in Edmonton, Alta., …
Monkey Math and Other Number-Crunching Critters
Discovery News
Rhesus monkeys are able to perform math at an advanced level, reports a study this week from Harvard Medical Medical school. The monkeys were …
Math department wins national award for exemplary program
The Williams record
According to Susan Loepp, professor of mathematics, the College’s department is unique in several regards. “Everyone likes math even if they don’t …
From math failure to savant: How a mugging made a numbers whiz
CTV News
Padgett describes the brutal bar attack and his subsequent transformation into a math savant in his new book, Struck by Genius: How a Brain Injury …

Finding E Maths or A Maths Difficult?

Are you finding Elementary Maths (E Maths) or Additional Maths (A Maths) Difficult?

Do not be discouraged if you find E Maths or A Maths difficult. The main reason why you are finding it to be difficult is that it is new. You have not gotten enough exposure to the type of questions asked. It is like learning to ride a bicycle, at the start it is difficult and you may even fall down. But after you have mastered riding the bicycle, you will be able to ride as fast as you wish. You need to get over the initial difficulty of learning in order to master the art of riding the bicycle.

At our Group Tuition at Bishan, we constantly practice actual exam questions, be it on Trigonometry, Differentiation or Integration (A Maths), or Vectors, Matrices and Probability (E Maths). We learn different methods to check and do the questions. You will find out, at last, that once you master the art of solving O Level questions, all the O Level questions are just repackaging the same questions in different forms. Once you know how to do one question, you will know how to do all similar questions. Expanding your repertoire of questions you know will enable you to get that coveted “A”. Constant practice, as opposed to cramming one month before the O Levels, is absolutely necessary to avoid panic and to consolidate our Mathematical memory.

Some Math formulas like the quotient rule, \displaystyle\frac{d}{dx}(\frac{u}{v})=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}, you will automatically memorize it once you have done enough practice.

In the end, you may even find that E Maths or A Maths is easy!


Motivational Story to motivate you

(Source: http://www.indianchild.com/inspiring_stories.htm)

THE OBSTACLE IN OUR PATH
In ancient times, a king had a boulder placed on a roadway. Then he hid himself and watched to see if anyone would remove the huge rock. Some of the king’s wealthiest merchants and courtiers came by and simply walked around it.

Many loudly blamed the king for not keeping the roads clear, but none did anything about getting the big stone out of the way. Then a peasant came along carrying a load of vegetables. On approaching the boulder, the peasant laid down his burden and tried to move the stone to the side of the road. After much pushing and straining, he finally succeeded. As the peasant picked up his load of vegetables, he noticed a purse lying in the road where the boulder had been. The purse contained many gold coins and a note from the king indicating that the gold was for the person who removed the boulder from the roadway. The peasant learned what many others never understand.

Every obstacle presents an opportunity to improve one’s condition.


Chinese Version

石头的故事

从前有一个国王,他故意派人搬来一块大石头放在道路中间。然后这个国王躲在旁边静静地观察,想知道是否会有人过来把这块大石头搬走。有几个富有的商人经过这块石头,只是直接绕过它。之后又来了几个人,他们只是埋怨国王居然没有找人来清理道路,然后仍然是绕过这块巨石走掉了。又过了一会,一个农夫经过这里,他身上背着沉甸甸的蔬菜。走到巨石的前面,把身上背的蔬菜放下,然后试着去把这块巨大的石头移到道路的旁边。农夫竭尽全力去推那块巨石,终于成功得把石头推到道路的一侧。农夫背起蔬菜准备继续赶路,却发现石头原来所在的地方下面有一个袋子。农夫好奇得打开袋子,里面有许多金币还有一个纸条。纸条是国王留下的,原来金币是用来奖励移走石头的人。
这个小故事讲述了一个简单的道理:困境也有可能是机会。遇到困难的时候,有些人会像商人那样,直接选择放弃。有些人会像那几个埋怨者,只会抱怨却不想着付出行动来改变现状。很少人会像故事中的农夫那样选择迎难而上。然而,最终收获最多的往往就是这部分迎难而上的人。
平常生活中和工作中也是这样,遇到问题,首先要不抛弃不放弃,积极寻求解决方法。无休止的抱怨只会白白错失机会,抱怨多了,好运气也会绕道而行。在困境中寻找机会方法,在挫折中吸取经验教训,那么就会很容易走出困境了,说不定还会有意外的收获哦!

How to Find Asymptotes of Graphs

This post is all about finding  Vertical and Horizontal asymptotes of graphs.

Vertical Asymptotes

Usually, vertical asymptotes come about when there is a rational function with a numerator and a denominator, for instance, \displaystyle y=\frac{2}{x-3}. When the denominator is 0, the function is undefined, and hence there is a vertical asymptote there.

Hence, to find the asymptote, let the denominator be 0. E.g. x-3=0, so x=3.

graph1
\displaystyle y=\frac{2}{x-3}

Another way vertical asymptotes can come about is via logarithmic graphs, e.g. y=\ln (x+2).

\ln 0 is undefined, so when x+2=0 or x=-2, there will be a vertical asymptote at x=-2.

graph2
y=\ln (x+2)

 


Horizontal Asymptote

Horizontal asymptotes usually come about when one of the terms approaches zero as x approaches infinity.

To find the Horizontal Asymptote, find the value of y when x approaches infinity (i.e. when x becomes a very big number).

For example, \displaystyle y=\frac{1}{x}+1. When x is a very big number, say x=10000, y will be close to 1 since 1/10000 is almost zero. Hence, the horizontal asymptote is y=1.

graph3
\displaystyle y=\frac{1}{x}+1

Another time where Horizontal Asymptotes appear is for Exponential Graphs. For instance, y=e^{-x}+1. When x is very large, e^{-x} will be very small, and hence y approaches 1. This means that the Horizontal Asymptote will be y=1.

exponential
y=e^{-x}+1

Note: The graphs above were drawn using the software Geogebra. 🙂

Model-Centered Learning: Pathways to Mathematical Understanding Using GeoGebra

Good Friday Math

Is it possible for Good Friday to fall on the 13th?

Source: http://mathforum.org/library/drmath/view/52364.html
From: Susan Melanson
Subject: Good Friday on the Thirteenth

How many times has Good Friday fallen on the 13th? I have looked at
your formulas and don’t seem to find one that fits this question,
since Good Friday doesn’t fall on the same date each year. I’ve looked
at some other sources as well, to no avail.

Thanks for your help on this,
Susan Melanson

Check out the above website for the answer! It turns out that Good Friday will fall on the 13th approximately once in 29 years.

Wishing all Christians a blessed Good Friday.


Also check out this Wikipedia page on how to calculate the date of Easter (http://en.wikipedia.org/wiki/Date_of_Easter)

The calculation of the date of Easter is called Computus. It turns out calculating the date of Easter is quite complicated. Even the great Mathematician Gauss made a mistake (see http://en.wikipedia.org/wiki/Date_of_Easter#Gauss_algorithm).

In 1800, the mathematician Carl Friedrich Gauss presented this algorithm for calculating the date of the Julian or Gregorian Easter[38][39] and made corrections to one of the steps in 1816.[40] In 1800 he incorrectly stated p = floor (k/3). In 1807 he replaced the condition (11M + 11) mod 30 < 19 with the simpler a > 10. In 1811 he limited his algorithm to the 18th and 19th centuries only, and stated that 26 April is always replaced with 19 April and 25 April by 18 April. In 1816 he thanked his student Peter Paul Tittel for pointing out that p was wrong in 1800.[41]

How to calculate Chinese Zodiac

How to use Math to calculate Chinese Zodiac and Impress your Friends

The Shengxiao (Chinese: 生肖, literally “birth likeness”), also known in English as the Chinese zodiac (“zodiac” derives from the similar concept in Western Astrology and means “circle of animals”), is a scheme and systematic plan of future action, that relates each year to an animal and its reputed attributes, according to a 12-year cycle. It remains popular in several East Asian countries, such as China, Vietnam, Korea and Japan. (Wikipedia: Chinese Zodiac)

calculate chinese zodiac


Simple Mental Calculation: Years after 1900

  1. Subtract 1900 from the year you are finding. E.g. 1988-1900=88
  2. Divide your answer by 12 and find the remainder. E.g. 88/12=7R4
  3. Add 1 to your answer. E.g. 4+1=5
  4. The answer represents the Chinese Zodiac of that year! (1-Rat 鼠, 2-Ox 牛, 3-Tiger 虎, 4-Rabbit 兔, 5-Dragon 龙, 6-Snake 蛇, 7-Horse 马, 8-Goat 羊, 9-Monkey 猴, 10-Rooster 鸡, 11-Dog 狗, 12-Pig 猪) Hence, 1988 is the Year of the Dragon, since 5 represents Dragon.

Calculation for Years after 2000

Very similar to the above, with some slight changes.

  1. Subtract 2000 from the year you are finding. E.g. 2015-2000=15
  2. Divide your answer by 12 and find the remainder. E.g. 15/12=1R3
  3. Add 5 to your answer. E.g. 3+5=8 (Note: If your answer is greater than 12, subtract 12 from it)
  4. The answer represents the Chinese Zodiac of that year! (1-Rat, 2-Ox, 3-Tiger, 4-Rabbit, 5-Dragon, 6-Snake, 7-Horse, 8-Goat, 9-Monkey, 10-Rooster, 11-Dog, 12-Pig) Hence, 2015 is the Year of the Goat, since 8 represents Goat.


Secrets of Mental Math: The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks


Chinese Zodiac List

Test your new skills and check the Chinese Zodiac for the following years from 1900 to 2100!

You may also refer to the table at: http://www.prokerala.com/general/calendar/chinese-years.php

Year 1900: Rat
Year 1901: Ox
Year 1902: Tiger
Year 1903: Rabbit
Year 1904: Dragon
Year 1905: Snake
Year 1906: Horse
Year 1907: Goat
Year 1908: Monkey
Year 1909: Rooster
Year 1910: Dog
Year 1911: Pig
Year 1912: Rat
Year 1913: Ox
Year 1914: Tiger
Year 1915: Rabbit
Year 1916: Dragon
Year 1917: Snake
Year 1918: Horse
Year 1919: Goat
Year 1920: Monkey
Year 1921: Rooster
Year 1922: Dog
Year 1923: Pig
Year 1924: Rat
Year 1925: Ox
Year 1926: Tiger
Year 1927: Rabbit
Year 1928: Dragon
Year 1929: Snake
Year 1930: Horse
Year 1931: Goat
Year 1932: Monkey
Year 1933: Rooster
Year 1934: Dog
Year 1935: Pig
Year 1936: Rat
Year 1937: Ox
Year 1938: Tiger
Year 1939: Rabbit
Year 1940: Dragon
Year 1941: Snake
Year 1942: Horse
Year 1943: Goat
Year 1944: Monkey
Year 1945: Rooster
Year 1946: Dog
Year 1947: Pig
Year 1948: Rat
Year 1949: Ox
Year 1950: Tiger
Year 1951: Rabbit
Year 1952: Dragon
Year 1953: Snake
Year 1954: Horse
Year 1955: Goat
Year 1956: Monkey
Year 1957: Rooster
Year 1958: Dog
Year 1959: Pig
Year 1960: Rat
Year 1961: Ox
Year 1962: Tiger
Year 1963: Rabbit
Year 1964: Dragon
Year 1965: Snake
Year 1966: Horse
Year 1967: Goat
Year 1968: Monkey
Year 1969: Rooster
Year 1970: Dog
Year 1971: Pig
Year 1972: Rat
Year 1973: Ox
Year 1974: Tiger
Year 1975: Rabbit
Year 1976: Dragon
Year 1977: Snake
Year 1978: Horse
Year 1979: Goat
Year 1980: Monkey
Year 1981: Rooster
Year 1982: Dog
Year 1983: Pig
Year 1984: Rat
Year 1985: Ox
Year 1986: Tiger
Year 1987: Rabbit
Year 1988: Dragon
Year 1989: Snake
Year 1990: Horse
Year 1991: Goat
Year 1992: Monkey
Year 1993: Rooster
Year 1994: Dog
Year 1995: Pig
Year 1996: Rat
Year 1997: Ox
Year 1998: Tiger
Year 1999: Rabbit
Year 2000: Dragon
Year 2001: Snake
Year 2002: Horse
Year 2003: Goat
Year 2004: Monkey
Year 2005: Rooster
Year 2006: Dog
Year 2007: Pig
Year 2008: Rat
Year 2009: Ox
Year 2010: Tiger
Year 2011: Rabbit
Year 2012: Dragon
Year 2013: Snake
Year 2014: Horse
Year 2015: Goat
Year 2016: Monkey
Year 2017: Rooster
Year 2018: Dog
Year 2019: Pig
Year 2020: Rat
Year 2021: Ox
Year 2022: Tiger
Year 2023: Rabbit
Year 2024: Dragon
Year 2025: Snake
Year 2026: Horse
Year 2027: Goat
Year 2028: Monkey
Year 2029: Rooster
Year 2030: Dog
Year 2031: Pig
Year 2032: Rat
Year 2033: Ox
Year 2034: Tiger
Year 2035: Rabbit
Year 2036: Dragon
Year 2037: Snake
Year 2038: Horse
Year 2039: Goat
Year 2040: Monkey
Year 2041: Rooster
Year 2042: Dog
Year 2043: Pig
Year 2044: Rat
Year 2045: Ox
Year 2046: Tiger
Year 2047: Rabbit
Year 2048: Dragon
Year 2049: Snake
Year 2050: Horse
Year 2051: Goat
Year 2052: Monkey
Year 2053: Rooster
Year 2054: Dog
Year 2055: Pig
Year 2056: Rat
Year 2057: Ox
Year 2058: Tiger
Year 2059: Rabbit
Year 2060: Dragon
Year 2061: Snake
Year 2062: Horse
Year 2063: Goat
Year 2064: Monkey
Year 2065: Rooster
Year 2066: Dog
Year 2067: Pig
Year 2068: Rat
Year 2069: Ox
Year 2070: Tiger
Year 2071: Rabbit
Year 2072: Dragon
Year 2073: Snake
Year 2074: Horse
Year 2075: Goat
Year 2076: Monkey
Year 2077: Rooster
Year 2078: Dog
Year 2079: Pig
Year 2080: Rat
Year 2081: Ox
Year 2082: Tiger
Year 2083: Rabbit
Year 2084: Dragon
Year 2085: Snake
Year 2086: Horse
Year 2087: Goat
Year 2088: Monkey
Year 2089: Rooster
Year 2090: Dog
Year 2091: Pig
Year 2092: Rat
Year 2093: Ox
Year 2094: Tiger
Year 2095: Rabbit
Year 2096: Dragon
Year 2097: Snake
Year 2098: Horse
Year 2099: Goat
Year 2100: Monkey

 

Mathematicians to find MH370 Debris?

Source: http://www.slate.com/blogs/future_tense/2…for_the_black.html

Australian authorities have announced that satellite images taken of a stretch of ocean 1,550 miles southwest of Perth, Australia, are believed to show floating debris that could be part of missing Malaysia Airlines Flight 370. “It is probably the best lead we have right now,” said John Young, a spokesman for the Australian Maritime Safety Authority. Confirmation of the material’s provenance will likely have to wait, however. While a merchant vessel has arrived in the area to help with the search, poor visibility has prevented search aircraft from locating the debris, and the nearest Australian Navy ship is several days’ sail away.

The search for Air France 447 offers a useful template for how investigators can whittle away at the seemingly unsolvable mystery of a midocean airliner disappearance. After the Airbus A330 went missing over the middle of the equatorial Atlantic in 2009, search aircraft took just one day to locate the first pieces of floating wreckage. The recovery of the black box, however, took another painstaking two years, and a full assessment of its implications another year after that.

The first step after determining the debris’ location is to call in the mathematicians. Based on all the data available—the aircraft’s last known position, route of flight, altitude, prevailing winds, sea currents, ocean depth, and so on—a probability is assigned to each variable, and a distribution map of probable locations on the sea floor is generated. Searchers can then deploy their underwater assets to scour the vastness of the deep, working back and forth along grid lines laid out in the areas of maximum probability.

There’s a deep problem inherent in this approach, however, and it’s that the probabilities are themselves only guesses. Searchers are uncertain even as to the extent of their own uncertainty. In the case of Air France 447, the set of base-set assumptions turned out to be wrong, and the first two search seasons scoured thousands of square miles in vain.

What turned the tide for AF447 searchers, in the end, was better math and better undersea technology. A recalculation of the location probabilities using a different mathematical approach led to the redrawing of the search grids much closer to the site of the plane’s disappearance. And a new type of autonomous undersea vehicle—a robot sub, in other words—became available for the first time. Called Remus 6000, these subs were able to navigate on their own along precise grid lines, ascending and diving to match the contours of the undersea terrain. On April 3, 2011, less than a week after the refined search began, one of the three submersibles deployed in the search returned to its mother ship bearing images of a debris field scattered across an abyssal plain. AF447 had been found. A month later another type of unmanned submersible brought the black boxes to the surface.

Math in Nature Video (3 million views!)

Math in Nature Video (3 million views!)

A movie inspired on numbers, geometry and nature, by Cristóbal Vila. One of the most popular Math videos on Youtube.

In this video, you can see how Fibonacci Numbers, the Golden Ratio, and Fractals are often found in nature.

Like the video? Leave your comments below, and share this page with your friends using the Facebook, Twitter sharing buttons below!

H2 Maths Tuition: Foot of Perpendicular (from point to line) (Part I)

Foot of Perpendicular is a hot topic for H2 Prelims and A Levels. It comes out almost every year.

There are two versions of Foot of Perpendicular, from point to line, and from point to plane. However, the two are highly similar, and the following article will teach how to understand and remember them.

H2: Vectors (Foot of perpendicular)

From point (B) to Line ( l)

(Picture)

h2-vectors-tuition

Equation (I):

Where does F lie? F lies on the line  l.

\overrightarrow{\mathit{OF}}=\mathbf{a}+\lambda  \mathbf{m}

Equation (II):

Perpendicular:

\overrightarrow{\mathit{BF}}\cdot \mathbf{m}=0

(\overrightarrow{\mathit{OF}}-\overrightarrow{\mathit{OB}})\cdot  \mathbf{m}=0

Final Step

Substitute Equation (I) into Equation (II) and solve for  \lambda .

Example:

[CJC 2010 P1Q7iii]

Relative to the origin O , the points A , B and C  have position vectors  \left(\begin{matrix}1\\2\\1\end{matrix}\right) , \left(\begin{matrix}2\\1\\3\end{matrix}\right) and \left(\begin{matrix}-1\\2\\3\end{matrix}\right) Find the shortest distance from  C to \mathit{AB} . Hence or otherwise, find the area of triangle \mathit{ABC} .

[Note: There is a 2nd method to this question. (cross product method)]

Solution:

Let the foot of perpendicular from C to AB be F.

Equation (I):

\overrightarrow{\mathit{OF}}=\overrightarrow{\mathit{OA}}+\lambda  \overrightarrow{\mathit{AB}}=\left(\begin{matrix}1+\lambda \\2-\lambda  \\1+2\lambda \end{matrix}\right)

Equation (II):

(\overrightarrow{\mathit{OF}}-\overrightarrow{\mathit{OC}})\cdot  \overrightarrow{\mathit{AB}}=0

\left(\begin{matrix}2+\lambda \\-\lambda \\-2+2\lambda  \end{matrix}\right)\cdot  \left(\begin{matrix}1\\-1\\2\end{matrix}\right)=0

\lambda =\frac{1}{3}

\overrightarrow{\mathit{CF}}=\overrightarrow{\mathit{OF}}-\overrightarrow{\mathit{OC}}=\left(\begin{matrix}2\frac{1}{3}\\-{\frac{1}{3}}\\-1\frac{1}{3}\end{matrix}\right)

\left|{\overrightarrow{{\mathit{CF}}}}\right|=\sqrt{\frac{22}{3}}

Area of  \Delta  \mathit{ABC}=\frac{1}{2}\left|{\overrightarrow{\mathit{AB}}}\right|\left|{\overrightarrow{\mathit{CF}}}\right|=\sqrt{11}

For the next part, please read our article on Foot of Perpendicular (from point to plane).

H2 Maths Tuition

If you are looking for Maths Tuition, contact Mr Wu at:

Email: mathtuition88@gmail.com