[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