[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

[H2_20150512SSSTRR] Finding a Recurrence Relation for Terms in a Series

Question

Introduction

     This question pertains to the relationship between the partial sums of a series and its terms.  I am not sure if all the junior colleges teach this explicitly, but students are expected to know or be able to observe this relationship.  Let us follow our nose and focus on the first part first.

Reminders

     For the series  u₁ + u₂ + ¼ + u_n–1 + u_n + ¼  ,  the n^th partial sum

                    S_n = u₁ + u₂ + ¼ + u_n–1 + u_n

                 S_n–1 = u₁ + u₂ + ¼ + u_n–1

Taking the difference, we see that

                    u_n = S_n – S_n–1

Innocuous looking, this is actually a very powerful formula.  It is applicable to all sequences and series (not only for arithmetic and geometric series).  That means this formula can always be used!

     Another thing to note is that sequences  u_n  and partial sums  S_n  (which are themselves another sequence) behave like functions.  [In advanced mathematics, they are in fact defined as functions with domain as the positive integers.]  What this means is that  S_n-1  has the same formula as  S_n  except that  n  is replaced with  (n – 1). 

Solution

Checking

     Actually, the question setter forgot that the formula works for  n > 1. 

     OK, let us check whether the formula really works.  We know that  u₁ = 3.  Let us tabulate and compare the recursive formula with the explicit formula.  You can do this on a piece of rough paper.

n	recursive un = f(un–1)	explicit un = 3´2n–1
1	u1 = 3	3´21–1 =  3
2	u2 = 2´  3   = 6	3´22–1 =  6
3	u3 = 2´  6  = 12	3´23–1 = 12
4	u4 = 2´12 = 24	3´24–1 = 24
5	u5 = 2´24 = 48	3´25–1 = 48

Challenge

     What if the question wanted a recurrence relation for  S_n?

  

Heuristics Used

H04. Look for pattern(s)

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘A’ Levels, H2 Mathematics 

* International Baccalaureate Mathematics 

* other syllabuses that involve sequences and series

[AM_20150302BTRCUK] Binomial with known Ratio of Coefficients but Unknown Power

Question

Introduction

     This Additional Mathematics question is challenging because the index  n  of the binomial power is unknown but you only know the ratio of a pair of coefficients.  Usually the  n  is given and you just plug in the formula and apply the binomial expansion formula.

Reminder

Solution

   

Heuristics Used

H05. Work backwards

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* GCE ‘A’ Levels H2 Mathematics (revision)

* IB Mathematics (revision)

* other syllabuses that involve Binomial Theorem for higher powers

[Pri20150402PTSMAP] A Staircase with Higher Steps

Question

Introduction

     This pertains to the sum of consecutive numbers with constant skips.  I set this question

to illustrate the heuristic of looking for patterns [H04].  It is similar to this question, except that now the numbers jump or skip by  2  instead of just   1.  The more knowledgeable reader will doubtless recognise this to be an arithmetic progression.  The challenge now is how can a primary school pupil do it without having learnt about any more advanced mathematics or algebra, relying purely on pattern recognition.

Solution

     As in the previous solution, imagine the sum as a series of vertical bars.  The numbers all jump by  2  this time.  Because the jump amount  2  is constant, you see a nice staircase pattern (shown in violet).  Each step of the staircase is of height  2  units.  If we make a copy of it and turn it upside-down (shown in green), the two staircases join together nicely to form a rectangle.  Notice that  101+3 = 99+5 = 97+7 = ... etc and they are all equal to  104.  If we know the number of columns, we can work out our desired sum.  How many columns are there?

     The number of columns is the same as the number of terms in  our sum.  OK, but then how many terms are there?  How to calculate this?  Let us look at a few simple cases first [H10. Simplify the problem].

Let us try to observe the pattern.  Note that the size of each skip is always  2.  If there are  2  terms, it is just  3  and  5,  there is one skip of  2.  From  3  to  7,  there are  3  terms, there are two skips of  2  each.  From  3  to  9,  there are  4  terms,  the difference is  6  and there are  3  skips.  From  3  to  11,  there are  5  terms,  the difference is  8  and there are  4  skips.  If you go from  3  to  13,  the net jump is  10  and there are  5  skips  and  6  terms.  We can tabulate the data into a table [H02] below:-

        skip size = 2

Start	End	Total Skip	# skips	# terms
3	5	5 – 3 = 2	2 ¸ 2 = 1	2
3	7	7 – 3 = 4	4 ¸ 2 = 2	3
3	9	9 – 3 = 6	6 ¸ 2 = 3	4
3	11	11 – 3 = 8	8 ¸ 2 = 4	5
3	13	13 – 3 = 10	10 ¸ 2 = 5	6

Do you notice some things?  [H04]

The total skip is the difference between the starting and ending numbers.

The number of skips is the difference divided by the skip size.

The number of terms is always one more than the number of skips.

Since our last term is  103,  the total skip is  101 – 3 = 98.  The number of skips is  98 ¸ 2 = 49.   So there are  50 terms  i.e.  50  columns.

Hence the size of our rectangle is  50 × 104.  But we only want half of this rectangle (shown in violet).   Hence the sum is  ½ × 50 × 104 = 2 600.

Ans:   3 + 5 + 7 + ... + 99 + 101 = 2 600

Summary

     This article illustrates the heuristic [H04 Look for pattern(s)].  Our first pattern we notice is the staircase pattern.  After making a copy and turning that around, we notice that it forms a rectangle, with columns of size  104  each.  Now we look for a pattern that enables us to find the number of columns, which is the number of terms in our sum.  We note that the number of terms is always the same as the number of skips, which is the same as the difference between the start and the end all divided by the skip size.  This enables us to solve the challenge in a way similar to my previous example.

Reflections
     Do you think this method will work for different starting numbers and different ending numbers?  For different skip sizes?  Why not set up your own similar question and try it yourself and see whether it works?

Heuristics Used

H02. Use a diagram / model

H04. Look for pattern(s)

H09. Restate the problem in another way

H10. Simplify the problem

Suitable Levels

* Primary School Mathematics

* GCE ‘O’ Level “Elementary” Mathematics (Number patterns, with algebra)

* GCE ‘A’ Levels H2 Mathematics (sequences and series, with algebra)

* IB Mathematics (sequences and series, with algebra)

* anyone who loves patterns and relishes a challenge

[Pri20150609PCBA] Members of a New Fitness Club

Question

Introduction

     This is a question on percentages.  Percentages are in themselves also units.  One percent (1%) simply means ¹/₁₀₀.  And we can use units (shown circled in the diagrams below) in which each unit is  ¹/₁₀₀  or  1%  of some whole.

     It is useful to think of increases and decreases as multiplying by some percentage.  For example, a decrease by 20%  means multiplying by  100% – 20%  i.e.  80%.  After all, if you work out  100%  of something and subtract  20%  of the same thing,  you will end up with  80%  of that thing.  It is much easier to think of it that way.  Likewise, an increase of  45%  means multiplication by  100% + 45% = 145%.

Solution

     From the information given in the question, we can set up a diagram like this.  I use circles to envelop the percentage units.

We can work out the units in the “after” situation (one year later):-

40 ×  80%  = 40 ×   ⁴/₅  = 32

60 × 145% = 60 × ²⁹/₂₀ = 87

The new total is  119%  or  119 circle units.  The net increase is  19%  or  19 circle units, which we know is equivalent to  228.  Once we got this part, we can work out  1  circle unit  and then  20 circle units, which is the difference between the number of male and female members.  [Remember the check that you are answering the question that was asked.]

Ans: 240

Summary

     We have used a diagram in the form of a ratio-units model [H02].  The ratio unit used in this example happens to be the same as a percentage.  Be careful that other questions may involve different kinds of units with different bases for their percentages.  In other words, in other questions, the “100%” may stand for different things.  In the diagram, we have used the before-after concept [H06].  Increases or decreases in percentages may be re-stated as multiplications by the appropriate percentages, which, in turn, may be thought of as multiplications by fractions [H09].  It is a good idea to be able to inter-convert between fractions and percentages.  By comparison, we found the link between  19 units (or 19%)  and  228 [H11].  Having solved this part of the problem, we are able to answer the original question as asked.

HeuristicsUsed

H02. Use a diagram / model

H06. Use before-after concept

H09. Restate the problem in another way

H11. Solve part of the problem

Suitable Levels

* Primary School Mathematics

* other syllabuses that involve percentages and ratios