[AM_20161105ITFF] False Friends in Integration (Calculus)

Question

Introduction

          False friends are words in two languages that look/sound alike, but differ significantly in meaning.  Do you know that there are also false friends in mathematics?  Can you distinguish and explain the difference between the two integrals?

Solution

          The integrand on the left has the variable  x  as the base and the constant  e  as the index.  So we integrate it using the Power Law.

          By contrast, for the integrand on the right, the base  e  is a constant whereas the index is the variable  x.  Integrating  e  to the power of  x  is the eeeeeeeeeeeeeeeasiest.  You just get back the same thing, plus the arbitrary constant of course.

Remark

          Many students make the mistake of trying to apply the Power Law for the exponential.  As a learner of mathematics, one needs to cultivate the habit of being observant and paying attention to detail.  This is part of developing one’s identity and character which is important in life.

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* GCE ‘A’ Levels (revision)

* revision for IB Mathematics HL & SL (revision)

* Advanced Placement (AP) Calculus AB & BC

* University / College Calculus

* other syllabuses that involve integral calculus

* whoever is interested

[AM_20150302BTRCUK] Binomial with known Ratio of Coefficients but Unknown Power

Question

Introduction

     This Additional Mathematics question is challenging because the index  n  of the binomial power is unknown but you only know the ratio of a pair of coefficients.  Usually the  n  is given and you just plug in the formula and apply the binomial expansion formula.

Reminder

Solution

   

Heuristics Used

H05. Work backwards

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* GCE ‘A’ Levels H2 Mathematics (revision)

* IB Mathematics (revision)

* other syllabuses that involve Binomial Theorem for higher powers

[JCH2BXQNSR_20150429] Binomial Expansion for a Quotient

Question

Solution

Comment

     Note in the above working that  3(x + kx²)² = 3(x² + 2kx³ + k²x⁴),  but since we do not need the  x³  and  x⁴  terms, we omit them and just write “3(x² + ... )”.

Heuristics Used

H12* Think of a related problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* revision for GCE ‘A’ Level H2 Mathematics

* AP Calculus

* other syllabuses infinite binomial series