[H2_Expository] Derivative of the Arcsecant function

Introduction

     This article explains how to differentiate, i.e. find the derivative of, the arcsecant function, which is seldom discussed in class.  If you look at the various syllabus outlines, sometimes they do not explicitly mention or imply the arcsecant, neither do they explicitly exclude this.  The important thing is: we should be able to figure it out from our basic knowledge, which is the whole point of mathematics.  The people who talk about and focus on mathematics syllabus content as if it is the only or most important thing are missing the point of mathematics education, and there are a plenty of these idiots around.  Do not follow them.

     The arcsecant, written as arcsec or sec^-1, is also known as the inverse secant function.  So arcsecant means that, given the value of a secant function, you want “the” angle whose secant is that given value.  The problem is there are many possible values.  Look at the graph below.

Defining the Inverse of the secant properly

     On the graph, a horizontal line can pass through (infinitely) many points.  Like any periodic trigonometric function, the secant function is not a one-to-one (a.k.a. “injective” or  “one-one”) function.  As such, it is not invertible.  However, we can restrict the function so that its domain is  [0, p] \ { ^p/₂ }.  This is highlighted in yellow on the graph.

     Remember that sec x = ¹/_{cos x}.  Basically we follow the principal values  [0, p]  of the arccosine function   except  ^p/₂  where the cosine is zero and its reciprocal the secant is undefined.       With this restriction on the domain, we get a one-one secant function, with range (-¥, -1] È [1, +¥).  We can now define the inverse function, and its graph is obtained by reflecting the above graph along the mirror line  y = x.  We get this:-

     Observe that in the yellow regions in both graphs, the gradients at the points are non-negative.  We have chosen the domain of the secant function, which is the same as the range of the arcsecant function, such that the derivatives will be non-negative.

Deriving the Derivative (refer to the“Onion” Method for differentiation)

Remarks

     Do not confuse  arccos y   with  (cos y)^-1.  They are not mean the same thing.

     In case you are wondering, the prefix “arc-” means the angle, which (if you use radians) is literally the same as the arc-length when the radius equals to  1.  In symbols, s = rq  with r = 1 means  s = q.  Although the term can be used with degrees or other units, when doing advanced mathematics like calculus, we would usually be using radians anyway. 

Suitable Levels

* GCE ‘A’ Levels H2 Mathematics

* International Baccalaureate (IB) HL Mathematics

* Advanced Placement (AP) Calculus AB & BC

* University / College calculus

* other syllabuses that involve differentiation

* any learner interested in calculus

[AM_20151127DTCR] The “Onion” Method for Differentiation

Question

Introduction

     The differentiation of the secant function is not taught directly as part of the Additional Mathematics syllabus.  It can be derived from known facts.  I first show the standard application of the (extended) Chain Rule for novices, and then show a more effective way of applying the chain rule, which I called the “Onion Method”.  This looks like peeling onions or unpacking Matryoshka dolls (“Russian dolls”). 

Reminders

Solution 1  (for beginners)

Solution 2  (a more expedient way)

Final Remarks

     When we peel onions, we peel from the outer layer inwards.  Likewise, when we have a composite function, we differentiate from the outer layer first, and then work to the inner layers.  Every time we differentiate a layer, we write down the changed layer and then copy and paste everything within that layer.  With regular practice, this should become second nature.
     For another example of the “onion”, take a look at the derivative of the arcsecant function.

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

* GCE ‘A’ Level H1 Mathematics

* GCE ‘A’ Level H2 Mathematics (revision)

* International Baccalaureate SL & HL Mathematics

* AP Calculus AB & BC

* other syllabuses that calculus

* anyone who loves to learn!

[S2_20151122IXBQ] A Bonus for your Index Fun?

Question

Introduction

     This is a “bonus” question which most likely came from an Integrated Programme (IP) school.  It really tests the wits of students’ knowledge of indices and problem solving tactics.  I present two solutions without the use of logarithms.  The first solution uses reciprocal indices and the matching of the base ^a/_b.  In the second solution, we match up the indices to  xy.

Review of Important Laws of Indices

Solution 1

 

Solution 2

Remarks

     No logs from any forest were harmed in the process of making this blog post.  J

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* challenge for Lower Secondary Mathematics (Secondary 2)

* GCE ‘O’ Level “Elementary” Mathematics (revision)

* other syllabuses that involve algebra and indices

[AM_20150510PLRF27] Looking for a Polynomial's Missing Link

Question

Introduction

     The first part of this question is rather standard.  The second part is quite challenging, especially if you are trying to connect with the earlier part.

Reminder

     To solve polynomial equations of degree  3  or higher (that are tested in school tests and exams), we often need to guess a rational (fraction or integer) root.  By the way, whole numbers are rational numbers because we can always put them into fractions upon  1  as denominator.  So how do we guess the roots?  The following is a very important theorem that guides us as to what numbers to try.

So we consider all the possible factors of the constant term  a₀  for the numerator and

all the possible factors of the coefficient  a_n  of the highest power for the denominator and consider the + and the – of all the possible fractions formed.  Usually, we try those with denominator 1 i.e. the integers first.

Solution

Remarks

     Note that the solution consists of only the part in blue.  Black is used for explanations, which are lengthy because of the dense interplay of ideas and subtleties involved.

     For the second part, if you are not able to see the connection, then use the standard method to solve the equation.  Here we realise that when the  x  is replaced by  ^v/₂,  the coefficients are reduced to the original coefficients.  However, these are in reverse order.  This indicates that one needs to use the reciprocal, so you divide throughout by  v³,  so that the highest power becomes just a constant.  I know you would not have thought of this if you have not seen this kind of question before, but this is the trick to use.

     Do not be discouraged by difficult question.  Have a growth mindset.  Every time you encounter a difficult question, learn how the trick ticks.  Your brain muscles will get stronger.  Try to apply the same trick when you see a similar question next time.

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

* GCE ‘A’ Level H2 Mathematics (revision)

* IB Mathematics (revision)

* other syllabuses that involve polynomials, Remainder and Factor Theorem