[AP_Calculus_IGSB] Integrating an Exponential with Square Root

Problem

Introduction

     Here is an integration problem that has no clues as to what to do with it.  Hmmm ... the integrand (3 to the power of square root something) does not look like it can be simplified.  [H09, H10]  How about a substitution?  [H12]  But what substitution?  Usually, we substitute the “ugliest” part of the integrand.  What constitutes the “ugliest” requires experience and observation.  In this case, the square root expression looks pretty nasty.  But how do we even integrate  3  to the power of something?

How to Integrate the Exponential

Solution

Comment

     In this problem, the original variable of integration is  x.  When doing substitutions, it is usually easier to make  x  the subject, and then replace the  “dx” with its equivalent.  After the substitution [H11], we integrate by parts and then substitute back to express everything in terms of  x.

Heuristics Used

H04. Look for pattern(s)        [look for the “ugliest” part]

H05. Work backwards

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

H12* Think of a related problem

Suitable Levels

* GCE ‘A’ Levels H2 Mathematics (challenge)

* IB Mathematics HL (challenge)

* Advanced Placement (AP) Calculus AB & BC

* University / College calculus

* other syllabuses that involve integration

* any precocious or independent learner who is interested

[AP_Calculus20151201] Integrate something with arcsecant by parts

Question

Introduction

     This question is taken from the Techniques of Integration chapter of Thomas’ Calculus, 12th edition.  It looks pretty nasty in that the arcsecant is just one of those things on the fringes of teachers’ and students’ minds.

Strategy

     We apply the “Integration by Parts” Formula                                                  

with the “d(etail)” heuristic.  The expression  t  is  algebraic whereas the arcsecant expression is of the “inverse” type.  Since “a” comes before  “i”  in “d(etail)”, we choose the algebraic   expression  t  to serve as our  ^dv/_dx.  We realise that we will need the derivative of the arcsecant                                                 

for  sec^-1 x  being a cute angle ... I mean, an acute angle.

Solution

Remarks

     A slight modification of this approach is to first re-express the arcsecant as  arcsec t = arccos(¹/_t).  One needs to work out the derivative of this arccosine expression when doing the integral.

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H10. Simplify the problem

H11. Solve part of the problem

Suitable Levels

* GCE ‘A’ Levels H2 Mathematics (challenge)

* IB Mathematics HL (challenge)

* Advanced Placement (AP) Calculus BC (challenge)

* University / College calculus

* other syllabuses that involve integration and inverse trigonometric functions

* any precocious learner who loves a challenge

[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

[H2_20150514GAS] Grouped Arithmetic Sequences

Question

Introduction

     This question, even though not the most difficult of its type, poses quite a challenge to many students.  Without the curly brackets, the sequence is just an ordinary arithmetic progression (AP) or arithmetic sequence.  Once the curly brackets are in, it messes up our mental schema.  We can no longer use the formulas for AP naïvely.

     Actually, we should never apply any mathematical formula blindly.  Neither we should be stuck with a literal interpretation of the symbols.  We should apply formulas according to their meaning.  For example, in the sum-of-an-AP formula                                       

the  n  represents the number of terms.  But the  n  as used in the question has a different meaning.  It means the set number.  The best students are able to observe this, and hold the difference in meanings in their heads when they apply the formulas.  This requires some mental effort.  It is easy to make a careless mistake if you lose your focus or concentration.  If you are not so confident of doing this mentally, and you want to play it safe, I would suggest that you use another set of symbols (say, capital letters)                                        .

You can also use subscript notations like  A_n  for the first member in set  n.

Observations

     Before trying to do anything.  It is always good to take a step back and make observations, and play with small numbers first.  Once you have observe the patterns, you can plan your strategy to tackle the question.

Solution

Remarks

     As you can see, the actual presentation of the solution is actually quite short.  But there is a lot of thinking behind it.  It is important to make observations, even if some of them seem unnecessary for this question.  This allows you to solve problems even more challenging than this.  For example, what if the bare sequence did not start from  1  and has a common difference more than  1? 

          {5},  {8,  11},  {14, 17, 20},  {23, 26, 29, 32},  ...

What if the bare sequence was a geometric progression?  Like this

          {1},  {2,  4},  {8, 16, 32},  {64, 128, 256, 512},  ...

What if the bare sequence was an arithmetic progression, but the number of terms in the sets follow a geometric progression?  Like this

          {3},  {5,  7},  {9, 11, 13, 15},  {17, 19, 21, 23,  25, 27, 29, 31},  ...

Happy figuring these out!

HeuristicsUsed

H04. Look for pattern(s)

H09. Restate the problem in another way

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘A’ Levels, H2 Mathematics 

* International Baccalaureate Mathematics 

* other syllabuses that involve arithmetic and geometric progressions

[H2_20150512PCD] Quadratic Discriminant for a Parametric Curve

Question

Introduction

     To begin with, do you notice that there are many letters (in italics) in the above?  There seems to be a confusing mix of variables and constants.  There are only three variables:  x,  y,  and  t.  The constants are  a,  s,  and  p.  There is another letter ‘l’,  which is the name/label for a straight line.  It is good to highlight or mentally mark these different things as different.

     Since the topic is on parametric differentiation, it seems that you need to use differentiation to solve this question.  Notice that the equations involved are at worst quadratic?  No square roots, cosines, logarithms, cubes, exponentials ... etc.  Whilst it is not wrong to use differentiation, there is a slick way – using quadratic discriminants.  In fact, this is the first thing you should think of if you see that the equations involved link to a quadratic equation.

Solution

Remarks

     We should always make it a habit to check and justify division by zero.  It is dangerous to divide an equation throughout by a variable or constant if you do not know what it is, or whether it is zero.  Make sure it is not zero before dividing.

     The quadratic discriminant method cannot be used unless you have things that reduce to quadratic equations.  But when it can be used, it is very powerful and it gives a direct answer.  Note that here we are not solving for the variable  t.  We are solving for the constant  s  in the first part, and for the constant  p  in the second part. 

Heuristics Used

H04. Look for pattern(s)

H09. Restate the problem in another way

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘A’ Levels, H2 Mathematics 

* International Baccalaureate Mathematics 

* other syllabuses that involve quadratic discriminants

[H2_SAJC2006PromoQ1] Skipping Terms in an Arithmetic Progression

Question

Introduction

     This difficult-looking question has become pretty standard already.  There are some principles that the schools may or may not teach explicitly, but they expect students to know.  Let us review some of these principles.

Reminders

Refer also to this article.

Solution

Summary

     Remember that when you apply a formula (e.g. like the formula for the sum of an AP), you need to apply it with the appropriate numbers substituted.  Do not get stuck with the letters.  They are not meant to be taken literally, but change according to the situation.  For example, the “d” in the later part is  4  but it is different than the  d = 2  in the earlier part.

Heuristics Used

H04. Look for pattern(s)

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

H12* Think of a related problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘A’ Levels, H2 Mathematics 

* International Baccalaureate Mathematics 

* other syllabuses that involve arithmetic and geometric progressions

[H2_ACJC2000P1Q15b] Adders use Logs to Multiply

Question

Introduction

     Here is a question on arithmetic progressions, and not the first question of its type.  As you know, the schools in Singapore mimic questions from the GCE ‘A’ Levels, as well as from one another.  Before we go into the solution, let us go through some things you need to know.

Prerequisites

Solution

Remarks

     You might observe that the argument of the logarithm,  pq^n–1,  forms a geometric progression.  Indeed, any logarithm of a geometric progression will form an arithmetic progression.  However, this is not something that you should memorise.  Just stick to the basic principles and work it out.  Mathematics is not about memorisation.  It is about observing and understanding links between things.  If you want to memorise, ask: Why do adders like to live among logs?  Answer: That’s the way they multiply.  J

Heuristics Used

H04. Look for pattern(s)

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 ‘A’ Levels, H2 Mathematics 

* International Baccalaureate Mathematics 

* other syllabuses that involve logarithms, arithmetic and geometric progressions