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A
Primer on Derivatives Book Description:
.This
36-page Microworld
is an active excursion into one of the most basic concepts
of the Calculus: the Derivative. It develops this seminal
notion with many simulations, illustrations, examples and
exercises. Throughout the book, readers may ask their own
questions and study their own examples.
It
begins by illustrating the graphical representation of derivative
as slope with lively animations: physical notions of speed,
steepness of ascent, and growth of geometric objects. Here,
the author describes in an informal way, the common-sense
ideas behind the constructions.
Next,
it defines the derivative in a formal way in terms of limits,
and illustrates this limiting process for two-sided and left-
and right-hand derivatives. After that, the book goes on to
explore the relationship between continuity and differentiability.
In particular, it studies some interesting counterexamples
to the erroneous hypothesis that continuity implies differentiability.
Moving
into the algebraic content of the subject, the next section
presents a number of algebraic rules for calculating derivatives
of functions. This begins with the usual formulas for specific
classes of functions: polynomial functions, trigonometric,
exponential, logarithmic, etc. But then it backs up and explores
where these rules come from with several exploratory exercises
that offer a way to visualize the rules and to compare the
algebraic procedures with their pictorial and graphical correspondents.
Having
stated and illustrated the basic differentiation rules, the
Chapter on "Arithmetic of Derivatives" puts it all
together in an interesting new way. In that chapter we explore
step by step such rules as "The Sum Rule", "The
Product Rule", "Quotient Rule" by allowing
the student to supply her own examples and then giving a step-by-step
application of the relevant rule to calculate the derivative.
In this way, the student can see the rule applied to problems
that she supplies, and therefore, the student will be more
likely to understand the calculation.
Next,
follows a discussion of the "Chain Rule" within
the context of the same pedagogic strategy outlined above.
This is very rich, because the student can easily supply examples
whose calculation will lead to surprises. Since the calculations
are always explained step by step, and there is an unbounded
set of possibilities...
After
presenting a "general strategy" for differentiation,
the book moves to its Center: the Exercises. The Exercise
section can be used to generate thousands of problems that
will give practice in solving the derivative problems based
on the basic derivative laws listed earlier. When the student
clicks on any of the four buttons (Sum or Difference of functions,
Product of Functions, Quotient of Functions, or Composition
of Functions), a function of the listed type is printed. The
student attempts to solve the problem by hand or with the
Symbolic Calculator which is available from most pages. Then
they may check their answer using the visual and symbolic
"Derivative Checker."
As
in previous examples, students may attempt the randomly generated
problems, or make up their own problems of each type. They
may of course pick problems on finding derivatives from any
source for further practice. The book finishes with explanation
and examples of implicitly defined curves. It then allows
the student to supply example equations of the form f(x,y)
= 0 for which it shows, step by step, how to calculate the
implicit derivative.
You
might like to press F11 to view the Microworld in Full Screen
mode.