[AM_20160125DAHT] Horizontal Tangents via Quadratic Discriminants

Problem

Introduction

     This is a Additional Mathematics textbook problem.  This question is of an intermediate level of difficulty.  The general method is by differentiation.  The equation of the curve happens to be capable of being put into a quadratic equation in  x.  Hence we can also use the theory of quadratic discriminants.  I present both methods of solution.

Method 1 (Using differential calculus)

Method 2 (Using quadratic discriminants)

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H11. Solve part of the problem

H12* Think of a related problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* other syllabuses that involve differentiation or quadratic discriminants

* any independent learner who is interested

[AM_20151223DATP] A Horizontal Tangent and a Faux Asymptote

Problem / Question

 

Solution 1

Solution 2  (not using differentiation)

Remarks

     This problem just happened to be put as an exercise in a textbook under the applications of differentiation.  But who says one must use differentiation?  Once again, there are at least two ways to solve a mathematical problem.  In this instance, it happened that the equation of the curve can be put into a quadratic form that is amenable to analysis by the discriminant.  Mathematics is about mental flexibility and creativity, actually.

     An asymptote is a straight line that the curve goes near to (but does not touch), as  x  gets large or gets very negative.  The book’s use of the phrase  “tends towards the line  l” may be wrong or imprecise.  Technically, the line  l  is not an  asymptote, because if you analyse or plot the graph, the gap between the curve and the line does not really get closer and closer.  However, the ratio of  y  over  x  gets nearer and nearer to  -1  and the gradient of the curve also gets nearer and nearer to  -1.  What really happens is: as  x  increases, eventually the curve becomes almost parallel to the line, but does not go near it.

Heuristics Used

H02. Use a diagram / model

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,  “IP Mathematics”

* revision for  GCE ‘A’ Levels H2 Mathematics

* revision for IB Mathematics HL & SL

* revision for  Advanced Placement (AP) Calculus AB & BC

* other syllabuses that involve differentiation and/or quadratic functions

* any precocious or independent learner who wants to learn

[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!

[IBHL_SOTA201304_1B10c] Quadratic Discriminants

Question

Important Reminders

Solution

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

*  IB Mathematics HL / SL

* other syllabuses that involve quadratics

[IB-HL H&H_Ex13e] pg 374 Q19 Angles with nice tans add up nicely

Question

This is taken from Haese and Harris textbook for IB Mathematics page 374.  It involves an interesting connection between certain angles whose tangents are nice fractions and the 45º degree angle i.e. ^p/₄.

Solution

Here is another look at the problem.

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* revision for GCE ‘A’ Level H2 Mathematics

* revision for IB Mathematics HL / SL

* other syllabuses that teach further trigonometry

[IB-HL H&H_Rev6C Q11] Factors of a Complex Polynomial

Question

Introduction

     This question is taken from the Haese & Harris textbook for IB HL Mathematics and is rather challenging.  Here I present two solutions.  In the first solution, I use substitution to make a variable “disappear”.  [ heuristics: H10. Simplify the problem, H11. Solve part of the problem]  And that allowed me to crack the rest of them problem.  In the second solution, I rephrased the problem in terms of tangents to curves. [ heuristic H09. Restate the problem in another way ] This gives another angle from which to tackle the problem.

Solution 1

Solution 2

Thinking Back

     Both solutions are related in the sense that they hinge on some form of the relation maked as [*].  That led to an equation in  a.  Once a  is found,  k  can be found, and the solutions proceed similarly.  The equation [*] is a manifestation of the fact that for a repeated root, both P(x) and P’(x) share a common factor.