[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

AJC 2009/I/14(a)(ii) Application of Integration: Area

figure 0 – problem statement

Introduction

     In this article, we look at question 14 part (a)(ii) taken from the preliminary examination paper of Anderson Junior College H2 Mathematics Paper 1 in year 2009.  In Singapore schools, you either get challenging questions or very challenging questions.  This part of the question is challenging, and yet is worth only worth 3 marks’ credit.  We can apply the same processes of metacognition (self-monitoring, self-awareness, self-questioning) and heuristics (guidelines, rules-of-thumb, tactics) to solve this problem.  These processes are generally applicable in problem-solving, and more important than the mathematical content itself (which you probably will forget anyway after you graduate from school).  One should learn mathematics not just for the sake of clearing examinations, but to get educated.

     “Education is what remains after one has forgotten what one has learned in
     school. ”                                                                              – Albert Einstein

Stage 1 – Understanding the problem

What topic is this under?

     Integration Applications (Area)

What are you trying to find?

     The area of the shaded region.

Stage 2 – Planning

What methods can you use?  Which is easier?

     We can integrate by cutting the area vertically or horizontally.  [Imagine cutting the area into very thin rectangular strips.]  Horizontal slicing looks easier.

What is the correct formula for that?

     Observing that the area is the area between two curves/lines, the formula is

Which means … ?

     Obviously, y_upper = ⁸/₃  and  y_lower = ¹/₆.  These integration limits correspond with the variable of integration ‘y’.  If the integration is ‘dy’ (with respect to y), then the upper and lower values must be  y  values.  If the integration is ‘dx’ (with respect to x), then the definite integral’s limits must be  x  values.

     x_right is the equation of the (straight) oblique line on the right boundary of the region, with  x  expressed in terms of  y.

     x_left is the equation of the elliptical curve on the left boundary of the region, with  x  expressed in terms of  y.

What heuristics can you use?

     1.  I can split this task into smaller sub-tasks.

     2.  Try to use the result in the previous part, part(a)(i).

What do you need to do?

     I need to find the x_right and x_left and then do the calculation.

Stage 3 – Execution

     Let us find the  x-formulas for the right and left lines.

figure 1 – determining the formulas for right and left parts

Remark: Most of this is straightforward for a JC student.  You are expected to be very familiar with secondary school algebra by now.  Regarding the last two lines: there are two choices for the equation of the curve.  Since we are using the left side of the ellipse (x < 1), we choose the one with the ‘–’ square root instead of the one with the ‘+’.  This is a common trick that schools like to catch students with.  Make sure you don’t stumble on this point.

     With these formulas, we can now work out our solution.

     Applying the formula (line #2) we have line #3 and taking out the brackets leads to line #4.  We do a bit of algebra, and then split the integration into two parts (#line 5).  For lines #6 and #7, the left integral is a straight-forward calculation and the right integral is the answer from part (a)(i).  In line #8, we consolidate our answer by pulling out 25 as common factor.

Stage 4 – Evaluation

Is your answer correct?

     The upper and lower limits match the variable of integration and they make sense.

     Although an exact answer (i.e. non-decimal) is required, we can use the Graphing Calculator to check the calculation numerically.  Here is one way (there are other ways too) to do it:-

We are correct.  The slight differences in the 7^th decimal place is because the Graphing Calculator itself uses an approximation to numerically calculate this definite integral.

Stage 5 – Reflection

What lessons did you learn by solving this question?

     ·  decide whether do to the area by slicing “horizontally” or by slicing “vertically”.  Whichever is easier.

     ·  if doing the integral by slicing “horizontally” always take the right curve
         minus the left one.

     ·  split the task into smaller sub-tasks:-

         1.  determine the upper and lower limits of the definite integral

         2.  if you integrate by slicing “horizontally”, determine the equations of the right
              and left curves, with  x  as the subject.

     ·  for the equation of the elliptical curve with  x  as the subject: choose ‘–Ö’ for the left

         half and ‘+Ö’ for the right half.  [But usually Singapore schools like to set ‘–Ö’ to
         catch unwary students off-guard, and that becomes so predictable: If you do not
         know which to choose, just choose the ‘–’ and you’d probably be right!  LOL! J ]

What if we took the left minus the right?

What will happen if we did the integration by slicing vertically (i.e. with respect to x)?

     Type your comments below.