**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**an***not**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.
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

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