The gradient of a smooth curve, , at a point is the gradient of the tangent to the curve at the point . Point is on the curve and is a neighbouring point whose value is displaced a small quantity, .
The idea behind differentiation is that as becomes very small, the gradient of tends towards the gradient of the curve. In the limit as becomes infinitesimally close to zero, the gradient becomes the gradient of the curve.
there is a fair bit of analytic work missing (higher education) to make these ideas sound.
We also write:
Standard results can be proved for different functions.
If , then we need to consider the small angle approximation that is if radians is very small (infinitesimal), then and , and compound trigonometry from which follows,
The differentiation process described above is linear and extends to more complicated functions. That is to say that if, where ,
Students in search of extension work in maths, particularly as far as Sixth Term Entrance Papers (STEP) are concerned, should peruse Carr’s Synopsis of results in Elementary Pure Mathematics.
This book is famous for being the volume used by the autodidact Ramanujan to teach himself mathematics. The book is merely a list of results and student how works through them, proving each one, will develop a strong ability in the subject. This life line into higher mathematics could not be produced in the same form in today’s educational and publishing context but, perhaps, Carr deserves a place as one of the subject’s great educators.
Higher mathematics is about generalisation and abstraction. Because of this, school mathematics students can find university text books unreadable. In such a context how can harder, higher level questions be set using the frame work of sixth form mathematics? The answer to this question is to wind the clock back to the 1880s and see how the subject was constructed then. Carr does this for us; many useful stages in school mathematics are listed here with extensions and generalisation written in generally sixth form recognisable form.
Take question 1 from this year’s (2016) STEP I exam. Pupils will only have to be familiar with the first six results to tackle this quickly.
The question play is all about the divisibility of,
which result 6, above, tells us is
Question game play leads us to factorise large numbers (without use of calculator):
in a difference of two squares (result 1) construction.
Functional notation and transformations is always tricky to teach and understand. GCSE students will meet this in Year 11. In general, transformations applied after the function are more easily understood:
Something very un-intuitive happens when the transformation is applied to the argument of the function:
with things stretching when they look like they should be compressing and other things moving the wrong way.
Mathematics is not always obvious, if it were we wouldn’t need it.
The Google facsimile of the 1621 edition of Diophantus’s Arithmetica lays out, in original Greek and Latin translation, the essentials for success in teaching and learning mathematics. Also included is the translation of the second half by Heath.
When I first studied and learned to love mathematics I was beset by friends and family who stridently claimed that this field of study was closed to them because of some biological, or other, reason. This always seemed to undervalue my own effort in getting off the first page.
Teachers of mathematics cannot be permitted to admit that nature is able to restrict the ability to learn the subject but the oriental wisdom of the student making their way to the master is a precondition for success when things seem difficult; in tales of martial arts the student must somehow prepare themselves to learn.
Mathematics appears rather difficult if one is not familiar with it.
Students need energy, eagerness and enthusiasm.
When such motivation is backed up by good teaching rapid learning results.
In school we struggle because of item 1 and bear our hearts to gain item 2, being so often rebuffed. I suppose that we get paid for 1 and 2 because, when item 3 is achieved the job is it’s own reward.
Edexcel’s January 2013 Mechanics 2 exam paper has massively intricate last part of last question to determine existence of third collision. Examiner’s report says not many candidates supplied an answer, which is not a surprise. This, and much more, is freely available on my exam bucket website: http://jped.co.uk/ExamBucket/exambucket.php.
Carr’s Synopsis – 19th century text for self taught geniuses – marks out this A-star topic for I/GCSE maths hopefuls.
Arithmetic Progressions, much loved by students of AS Core Mathematic, is an early result in my favourite Victorian textbook and has a good visual proof too.