the design of the course

This is a response to a very good question posed very articulately by a student in this course. The question is:

Question:

I think that I am also having trouble knowing where we are. I feel incredibly behind in assignments and am not sure where to begin with things.

The links are a little confusing they seem more like a labyrinth than a path at this point. Can you possibly leave some bread crumbs behind?

Response:

I'm concerned about the confusion caused by the fact that the course isn't laid out in a linear fashion. The design of this course isn't immediately clear from the homepage. This is due to the fact that during the first 1/3 of the course we are really doing two things at once, reviewing Precalculus and laying a concrete hands-on foundation for understanding the rest of the text. This is, in my experience, necessary, and cannot be done in a linear fashion. We must for awhile do two things at once to allow those whose preparation is less than ideal to lay the necessary founcation for understanding Calculus.

The hands-on foundation is built within the context of a series of three concrete mathematical modeling problems. Within the context of these problems we introduce most of the essential ideas of the Calculus.

We do this simultaneously with the review of Chapter 1, so that when we finally reach Chapter 2 we can begin to move relatively quickly through the material and make up for the time we have spent on review:

The three situations we model include the following:

The flow of water out of a uniform cylinder from a hole near the bottom of the cylinder, where we model water depth vs. clock time, obtaining an excellent model in the form of a quadratic function.
The approach of the temperature of an object (e.g., a hot potato) to that of the room in which it is placed, where we model temperature vs. clock time, obtaining an accurate model in the form of an exponential function.
The way the volume and the diameter of a series of geometrically similar sandpiles change as the sandpiles are built a tablespoon at a time, motivating a study of proportionality and power functions.

Within the context of these problems we review central ideas of Precalculus and introduce the main concepts of Calculus. You should reread this list after each exercise; as these ideas become clearer you will see your progress.

We introduce linear, quadratic, exponential, logarithmic and power functions.
We consider average and instantaneous rates at which these functions change, again within the concrete context of the models.
We derive at the corresponding rate-of-change functions (derivatives) for quadratic and power functions.
We explore the relationship between the rate-of-change functions and the original functions, obtaining a familiarity with the concepts of integration and differentiation.
We see how knowledge of the instantaneous rate of change of a function at a point allows us to approximate the behavior of the function at nearby points.
We find that we can write equations involving the rate of change (i.e., differential equations) of a function and obtain approximate solutions to these equations.
We generalize these concrete and specific ideas about functions, rates, changes and their relationships into symbolic form, and in so doing we will have experienced in a very tangible way the main ideas of the Calculus, including limits, the derivative, the integral, the differential and differential equations.
We will also begin to see the importance of basic algebra and geometry in the problem solving that is so important in this course.

When we later encounter these ideas in the rest of the course, we should therefore have a clear picture of what everything we are doing means, and where it fits in.