r r
How to Use This Book xi
have been chosen to illustrate the new methods introduced in the section, and
should be sufficient preparation for the questions of the following exercise.
The Order of the Topics:
We have presented the topics in the order we have found
most satisfactory in our own teaching. There are, however, many effective or-
derings of the topics, and the book allows all the flexibility needed in the many
different situations that apply in different schools (apart from the few questions
that provide links between topics).
The time needed for the algebra in Chapter One will depend on students’ expe-
riences in Years 9 and 10. The same applies to other topics in the early chapters
— trigonometry, quadratic functions, coordinate geometry and particularly curve
sketching. The Study Notes at the start of each chapter make further specific
remarks about each topic.
We have left Euclidean geometry and polynomials until Year 12 for two reasons.
First, we believe as much calculus as possible should be developed in Year 11,
ideally including the logarithmic and exponential functions and the trigonometric
functions. These are the fundamental ideas in the course, and it is best if Year 12
is used then to consolidate and extend them (and students subsequently taking
the 4 Unit course particularly need this material early). Secondly, the Years 9
and 10 Advanced Course already develops much of the work on polynomials and
Euclidean geometry in Options recommended for those proceeding to 3 Unit, so
that revisiting them in Year 12 with the extensions and far greater sophistication
required seems an ideal arrangement.
The Structure of the Course:
Recent examination papers have included longer ques-
tions combining ideas from different topics, thus making clear the strong inter-
connections amongst the various topics. Calculus is the backbone of the course,
and the two processes of differentiation and integration, inverses of each other,
dominate most of the topics. We have introduced both processes using geomet-
rical ideas, basing differentiation on tangents and integration on areas, but the
subsequent discussions, applications and exercises give many other ways of un-
derstanding them. For example, questions about rates are prominent from an
early stage.
Besides linear functions, three groups of functions dominate the course:
The Quadratic Functions: These functions are known from earlier years.
They are algebraic representations of the parabola, and arise naturally in
situations where areas are being considered or where a constant acceleration
is being applied. They can be studied without calculus, but calculus provides
an alternative and sometimes quicker approach.
The Exponential and Logarithmic Functions: Calculus is essential for
the study of these functions. We have chosen to introduce the logarithmic
function first, using definite integrals of the reciprocal function y =1/x.This
approach is more satisfying because it makes clear the relationship between
these functions and the rectangular hyperbola y =1/x, and because it gives
a clear picture of the new number e. It is also more rigorous. Later, however,
one can never overemphasise the fundamental property that the exponential
function with base e is its own derivative — this is the reason why these func-
tions are essential for the study of natural growth and decay, and therefore
occur in almost every application of mathematics.
www.cambridge.org© in this web service Cambridge University Press
Cambridge University Press
978-1-107-63332-2 - Cambridge Mathematics: 3 Unit Extension 1: Enhanced: Year 11
Bill Pender, David Sadler, Julia Shea and Derek Ward
Frontmatter
More information
