Welcome back. You've just completed

the first two modules which provided you with a comprehensive tool kit from precalculus,

and an extensive repertoire of functions and techniques for manipulating them,

and also for visualizing them using graphs in the x-y plane.

This next module is the first of two modules

introducing and developing techniques for differential calculus.

We begin by discussing average rates of change which

become instantaneous as time intervals become vanishingly small,

leading to the notion of a derivative.

The motivating example velocity is used throughout,

which turns out to be the derivative of displacement with respect to time.

If you're used to looking at and processing information from

the speedometer of a moving vehicle like a car or truck,

then you already have practical experience with derivatives.

The derivative is the limit of certain fractions associated with curves.

Namely, the slopes of secants which approach

the slope of the tangent line to the curve at a point of interest.

Tangent lines are very good approximations to the curve,

and you'll learn techniques involving differentials that exploit this fact.

The mathematics uses the beautiful and elegant Leibniz notation due to Gottfried Leibniz,

one of the founders of calculus along with Newton in the 17th century,

and has endured to the present day.

You become experts very quickly with Leibniz notation and see how to use it

to get information easily about the derivative of a function and how to apply it.

Again, we hope that you'll find the material interesting and stimulating.

That you'll find these videos helpful and the practice and

challenges provided by the many exercises beneficial.

I look forward very much for your continued attention and participation.