[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

[U_Calculus_20150520DCPL] Different Coordinate Systems, Same Lengths

Question

Introduction

     I got this question from a student who as studying AP Calculus under a school teacher who went beyond the syllabus.  If we try to do this question directly, it will be a tedious mess without any insight.  Is calculus just a mindless torture?  Is there a better way to look at the problem?

Solution

Remarks

     Observe that both the LHS and the RHS are squares of lengths of the vector gradient  Ñw  of  w  in their respective coordinate systems (polar for LHS and rectangular for RHS).  The key insight is that the Jacobian-like matrix  J  represents a rotation, which common sense tells us preserves lengths.  So it is not surprising that the LHS and RHS are equal.  This is one of the “evidences” that the vector gradient is a concept that transcends coordinate systems, and represents something “real and physical”.  Indeed the gradient  Ñw  is the vector that represents the change of  w  per unit distance in its direction of maxium increase.

HeuristicsUsed

H04. Look for pattern(s)

H09. Restate the problem in another way

H10. Simplify the problem

H12* Think of a related problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* University Level Mathematics  (Calculus, Vector Calculus)

* AP Calculus students who wish to stretch themselves / are being stretched

* other syllabuses that involve calculus and coordinate systems

[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.