المحاضرة 6

These lecture notes are written for a two-semester calculus course which meets four hours a week. The course
is designed for engineering students who have a precalculus back ground, and
whose primary interest lies in the applications of calculus. We try to focus
on the topics which are of greatest importance to them and use engineering science
examples to illustrate them. At the same time, we try of stay mathematically
coherent without becoming technical. There is less of an emphasis on by hand calculations.
In this sense, we are trying to adopt several ideas from
calculus reform. Among them is a more visual and less analytic approach.
We typically explore new ideas in examples before we give formal defnitions.
In one more way we depart radically from the traditional approach to
calculus. We introduce differentiability as a local property without using
limits. The philosophy behind this idea is that limits are the a big stumbling
block for most students who see calculus for the first time, and they
take up a substantial part of the first semester. Though mathematically
rigorous, our approach to the derivative makes no use of limits, allowing
the students to get quickly and without unresolved problems to this concept.
It is true that our defnition is more restrictive than the ordinary one,
and fewer functions are differentiable in this manuscript than in a standard
text. But the functions which we do not recognize as being differentiable
are not particularly important for students who will take only one semester
of calculus. In addition, in our opinion the underlying geometric idea of the
derivative is at least as clear in our approach as it is in the one using limits.