[AM_20150620DFFPPR] Differentiation of Powers from First Principles

Question

Introduction

     This question was posted to a Facebook forum for Secondary School and Junior College mathematics (the equivalent of grade 7 and above).  Most students, except a few enthusiastic ones or maybe some from “integrated programme” schools, will not bother with differentiation from first principles.  However, mathematical formulas are true because mathematicians took the trouble to critically analyse them and show that they always work, not because they are found in the textbook, nor because the teacher says so.

Solution

Remarks

     The answer to the original problem just falls out by letting  p = 17  and  q = 19.

     Some people suggested using Binomial Expansion for rational powers.  I feel that this is not from first principles, since Binomial Expansion relies on differentiation, which relies on first principles.

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* college level calculus

* AP Calculus

* other syllabuses that involve differentiation from first principles, or anyone who is interested

[AP_Calculus_20150516LIM] Large abyss with no Limit?

Question

Introduction

     This looks like a scary limit to calculate.  Indeed, it is quite tricky, as  x  goes towards negative infinity and there is a square root.  However, we can often work out the correct answer informally first.

Informal Solution (intuitive approach)

Formal Solution

Final Remark

     Having the informal solution (which can be worked out on rough paper or even mentally) gives us an idea of the answer.  We can counter-check our formal solution with the informal solution for confirmation.

Suitable Levels

* Advanced Placement (AP) Calculus AB & BC

* University / college level calculus

* other syllabuses that involve limits