[AM_20151127DF2D] Differentiation with Chunking and Elimination

Question

Introduction

     Although this looks like a differential equation question, the student is not required to solve the differential equation.  The requirement is just to derive the equation.  This would be a challenging question for secondary 4 (~ grade 10) students taking Additional Mathematics or their counterparts in Integrated Programme schools.

Strategy

     One way to do this is to differentiate the given equation once and again and just verify the equation by substitution.  The problem is that when we repeatedly apply the Product rule

the terms tend to sprawl.  A way to keep things neat is to try to recognise chunks and also use elimination.

Solution

Remarks

     After differentiating once, we notice that  10xe^2x   is twice of  5xe^2x,  and this allows the simplification in [1].  The second differentiation yields  10e^2x   which, we notice, is twice of  5e^2x.  We can get rid of that term.   Multiplying equation [1] by 2 gives  10e^2x  in equation [3],  which matches nicely with the same term in  [2].  So we can eliminate that term via elimination.  After that, we just need to rearrange things to get the final equation.

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘O’ Level Additional Mathematics, “Integrated Programme Mathematics”

* GCE ‘A’ Levels H2 Mathematics (revision)

* AP Calculus AB / BC (revision)

* University / College calculus (revision)

* other syllabuses that involve differentiation

* any learner interested in calculus

[EM_20151120QGXS] Quadratic Graphs by Completing the Square

Question

Introduction

     This “Elementary” Mathematics question is usually pitched at secondary 3 (» grade 9).  It is quite a standard type of question, but from my experience, many students do not know how to begin.  The wording of the question (especially part (a)) may throw off some people.  The first stage in solving any mathematics question is always to make sure you understand the question, to know what is expected of you.  Once this is done, this question is actually quite straightforward if you know what to do and know where to look.  The coefficient of  x²  is  a = 1, so the question is not that tricky.  Part (a) is simply instructing you to complete the square.

Review

 

Solution

Tip:  You could get the y-intercept by using your fingers to cover the x² and x terms.  There is an even faster way: just look at the constant term!  You get the answer  5  immediately!

Heuristics Used

H02. Use a diagram / model

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H11. Solve part of the problem

Suitable Levels

* GCE ‘O’ Level “Elementary” Mathematics

* any syllabus that includes quadratic functions and parabolas

* anyone who is interested 

[AM_20151119] Anticipation and Bridging as Proof Tactics

Question

Introduction

     In a previous article, I have showed an “evil” tactic that can be used against “evil” questions.  Here is another “evil” trigonometric proof question.

     Many traditionalist school teachers insist on starting either from the LHS or the RHS, and working all the way to the other side.  Personally I do not mind any form of presentation as long as it is logical.  But not many students are able to do that.  People tend to fall into the trap of beginning a proof with the statement that they are supposed to prove in the first place.  This is called circular reasoning (or “begging the question” or petitio principii).  It is definitely a no-no.  So there is some advantage to sticking to the traditionalist mould.  The disadvantage is, of course, that it stifles creativity and this gives a misleading image of mathematics to the learner.

More “Evil” Tactics

     Now, if we do not want to “break the rules”, perhaps we can “bend the rules” a little.  On a piece of rough paper, or in your mind, secretly work from both sides and try to bridge them in the middle.  Let us compare

Think:  How are they similar?  How are they different?

As you can see, the LHS already has a preponderance of  cos 75°.  One of these  cos 75°  must somehow disappear.  The  RHS  has  4 sin 75°  which the LHS does not have.  So if we start from the LHS, we can use our magic wand [SV4] to create  4 sin 75°  out of thin air, remembering to divide by the same thing, so that the original value does not get changed.

Solution

Reflection

     Have you learned anything from this problem?  Let us review the heuristics used to help us solve this problem successfully.

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

Special Variants of Heuristics (Good for Trigonometric Proofs)

SV1   Work from both sides and try to bridge them in the middle [~ H04]

SV2   comparing (similarities/differences) what you have now with what you want [~ H05]

SV3   anticipate what will happen in the end and what you must do now [~ H05]

SV4   Magic Wand or create something out of nothing (无中生有) [~ H09]

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* IB SL & HL Mathematics (revision)

* other syllabuses that involve trigonometry

* just about anyone who is interested, really

[AM_20151117] An “Unorthodox” Technique for Trigonometric Proof

Question

Introduction

     When proving trigonometrical identities, one usually starts from the “more difficult” side and try to work towards the other side.  The above identity looks like a tough nut to crack.  Both sides look equally complicated.  Where do we even begin?

     Here is one way to “cheat”.  Starting from, say, the LHS, we multiply the RHS expression and also multiply by its reciprocal i.e. dividing by the same amount.  It is like using a magic wand to create something out of nothing (无中生有 in Chinese), but doing so does not change the value of the LHS expression.  Now we do not touch the part that is equal to the RHS (highlighted in yellow below), but we try to find a way to cancel away the other stuff, as shown here.

Solution

Remarks

     In the third step, I had replaced  cos²A  with  1 – sin²A  and  sin²B  with  1 – cos²B.  This leads to the required cancellation and we are left with the yellow patch, which was never touched since the first step and it is the RHS.  This completes the proof.

     Just as in martial arts where deadly opponents require deadly strokes to counter them, evil questions require “unorthodox” techniques.  Even though most people would not have thought of it, all the steps presented above are actually legitimate.  This is because at every step, the equality is preserved.  All the “=” are really equal, and it’s legit (either work hard or you might as well quit), although my “can’t touch this” tactic seems a bit clairvoyant.

. 

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H08. Make suppositions

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* other syllabuses that involve proof of trigonometric identities

* anyone who loves mathematical challenges

[AM_20150620DFFPPR] Differentiation of Powers from First Principles

Question

Introduction

     This question was posted to a Facebook forum for Secondary School and Junior College mathematics (the equivalent of grade 7 and above).  Most students, except a few enthusiastic ones or maybe some from “integrated programme” schools, will not bother with differentiation from first principles.  However, mathematical formulas are true because mathematicians took the trouble to critically analyse them and show that they always work, not because they are found in the textbook, nor because the teacher says so.

Solution

Remarks

     The answer to the original problem just falls out by letting  p = 17  and  q = 19.

     Some people suggested using Binomial Expansion for rational powers.  I feel that this is not from first principles, since Binomial Expansion relies on differentiation, which relies on first principles.

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* college level calculus

* AP Calculus

* other syllabuses that involve differentiation from first principles, or anyone who is interested

[AM_20150621TGCARF] Sum of sin and cos the cost of tan?

Question

Introduction

     This is a bonus Maths II (rough equivalent of Additional Mathematics syllabus) question from a test by Hwa Chong Institution (HCI) this year 2015.  Independent schools in Singapore like HCI are free to set their own curricula, but they usually end up covering slightly more than the mainstream curriculum, since their students also take the national examinations.  For their internal tests and exams, they can set bonus questions.  These are harder but optional questions that students can attempt and if they are successful, the bonus marks can be added to their normal marks.  It is also OK not to attempt the bonus questions.  That gives students the choice and opportunity to stretch their minds, but they are not penalised if they are unable to solve the bonus questions.     In this article, I present two approaches to tackling this question.  Let us review some important formulas first.

Some Useful Formulas

Solution 1  (via the Pythagorean Identity)

  

Solution 2  (via R-formula)

Reflections / Extension

     Here is another HCI question on trigonometry that involves sine, cosine and tangent.

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* other syllabuses that involve trigonometry

[AM_20150621TGSCQT] sin and cos Embroiled in some Quadratic

Question

Introduction

     This is a bonus question from another version of a test set by Hwa Chong Institution this year (2015).  It turns out that different students take the test at different dates, and the school took the trouble to set different versions of the test.  They have the manpower resources to do that!

     It is good to know what topics each question involves.  In this example, students’ knowledge of quadratic theory and trigonometry are being tested in a combined fashion.  Let us first review the relevant material.

Reminders

Solution

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* other syllabuses that involve trigonometry

[AM_20150616LGDFCA] Logarithms and Knowing that You are Correct

Question

Introduction

     This is a relatively straightforward question once the student has learned the rules of logarithms.  When I was first learning logarithms it took me quite some time to get used to the idea of “logs”.  Are they fallen trees?  So what are “logs”?  They are just the exponents or indices.  For example  2³ = 2 ´ 2 ´ 2 = 8  and we can write  log₂8 = 3.  Logarithm to base  2  of  8  is 3, because  3  is the index  i.e. to get  8  you need to multiply  2  by itself  3-fold.

     In general,  log_ba = x   Û   a = b^x.  Why?  Because that is exactly what logarithm means!  One way to remember this definition is to imagine: if you transport the log to the other side of the equation, the log drops off and you get the base  b  propping up the  x.  You can also do it the other way round.  If the base  b  of a power moves to the other side, it becomes a  “log”  with base  b.  [active mnemonics]

     What about the “common logarithm” lg?  It is the logarithm with base  b = 10.  In the days before pocket calculators were prevalent, students used books and slide-rules with logarithms of base  10  for multiplying and dividing large numbers.  Base 10 logarithms are still commonly used in today for the Richter Scale (in seismology, to measure earthquakes), for decibels (to compare the loudness of sounds or gain / loss in amplifiers), for pH (measurement of acidity / alkalinity in Chemistry) .... etc.  The aforementioned rule works exactly the same way, with  b = 10.

                      lg a = x   Û   a = 10^x                 (lg means log₁₀)

Note that in many calculators, their “log” button is for  lg  or logarithm of base 10.

Solution

Checking Your Answer

     The person who posted this question on Facebook got  33 333 333.3  as his answer, but did not realise that his answer is the same as the “model” answer, which is given to three significant figures in standard scientific notation.  Many students have the habit of checking their answers against the “model” answer usually given at the back of the book or worksheet, which may sometimes be wrong!  Anyway, in tests and examinations, you do not have the luxury of checking your answers like this.  In real life, if an engineer makes a calculation mistake, buildings may collapse and people die.  It is better to make it a habit to check your answers on your own and to know and be sure that you are correct.  One way to do this is to substitute the value of  x  back into the original equation to see if it works.  Nowadays, many models of calculators have a “store” function indicated by a button labelled with “STO” or an arrow “®” or something like that.  You can store the value into a variable (or memory location) like  X  and then key in something like  “log(3X) ”  and see whether you get  9  or something close.  Be aware that calculators can have rounding errors. 

Notations for “log”

     School students are taught to use “lg” to mean “log₁₀”  and  “ln”  to mean the natural logarithm “log_e”  where the special number  e = 2.7182818284 ...  discovered by the visually impaired but brilliant mathematician Euler.  Many calculators take “log” to mean “lg”  or  “log₁₀”.  For adult working professionals, “log” (without indication of the base) usually depends on what field they are in, or on the topic being discussed.  As mentioned before, base 10 is used for Richter scale, decibels and pH.  Computer scientists tend to use base  2  because of the binary system.  For rate of reaction (chemistry) or radioactive decay (chemistry / physics), the natural logarithm “ln” is often used.  In school, for the purposes of learning, we make the logarithm bases explicit.  Do not simply write “log”.  Write “lg”, “ln” or “log₂” or “log₇” or “log_b” (for whatever  b  is).  Note also that the letter “l” in all these notations is not the letter “i” or “I”, but it is the smaller case “L” (for logarithms).

Heuristics Used

H05. Work backwards

H09. Restate the problem in another way

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘O’ Levels Additional Mathematics

* International Baccalaureate (IB) Mathematics (revision)

* other syllabuses that involve logarithms and exponentials