Friday, November 27, 2015

[AM_20151127DF2D] Differentiation with Chunking and Elimination

Question

Introduction
     Although this looks like a differential equation question, the student is not required to solve the differential equation.  The requirement is just to derive the equation.  This would be a challenging question for secondary 4 (~ grade 10) students taking Additional Mathematics or their counterparts in Integrated Programme schools.

Strategy
     One way to do this is to differentiate the given equation once and again and just verify the equation by substitution.  The problem is that when we repeatedly apply the Product rule
the terms tend to sprawl.  A way to keep things neat is to try to recognise chunks and also use elimination.

Solution

Remarks
     After differentiating once, we notice that  10xe2x   is twice of  5xe2x,  and this allows the simplification in [1].  The second differentiation yields  10e2x   which, we notice, is twice of  5e2x.  We can get rid of that term.   Multiplying equation [1] by 2 gives  10e2x  in equation [3],  which matches nicely with the same term in  [2].  So we can eliminate that term via elimination.  After that, we just need to rearrange things to get the final equation.

H04. Look for pattern(s)
H05. Work backwards
H10. Simplify the problem
H11. Solve part of the problem
H13* Use Equation / write a Mathematical Sentence


Suitable Levels
GCE ‘O’ Level Additional Mathematics, “Integrated Programme Mathematics”
GCE ‘A’ Levels H2 Mathematics (revision)
* AP Calculus AB / BC (revision)
* University / College calculus (revision)
* other syllabuses that involve differentiation
* any learner interested in calculus







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