The ``chapters'', or ``units'' involved in this course correspond roughly to the chapters in your text, but we won't be following the text exactly. You really should work through the units in the order they are presented, though in each one there is a lot of room to explore in the order that makes the most sense for you.
This chapter introduces the major ideas of calculus: slope of curves and velocity, limits, and finding areas by successive approximation, so you can get an idea of what is to come.
The notion of a limit is the single idea, more than any other, that separates calculus from all other mathematics. It is a difficult idea to master completely, and we will be coming back to this notion in various guises throughout the course.
The first real use of this notion of a limit is to make sense of the ``slope of a curve'', the derivative of a function. Although the idea of the slope of a curve is not difficult, and simple tools like a speedometer can find the derivative of the position of a car very easily, there are many serious applications of this notion that help understand the behavior of functions.
Calculus is of course the study of functions. There is a collection of standard functions that are the standard fare for calculus, which we call elementary functions. These include all of the functions used to model most physical and man-made phenomena. This chapter introduces, with reviews of the basics when you need them, all the functions we will be encountering.
This chapter discusses the primary theoretical tool of differential calculus, the Mean Value Theorem, and the many applications that actually owe their existence (or at least their theoretical justification) to that tool, such as optimization problems and qualitative descriptions of functions (curve-sketching).
The theory of integration, the area under a curve, is as basic a notion in calculus as the slope of a curve. It's introduced late in the course since the main method of calculation of integrals, using the Fundamental Theorem of Calculus, is another application of the Mean Value Theorem. This chapter also includes the ``standard'' techniques for computing integrals, which is a topic that will be covered in greater depth in the second semester.
Any computation that can be broken down into simplified sub-computations, and then added up to solve the more complicated original question can be analyzed in terms of integration. A simple example is the area of a plane region, which can be sliced up into little strips, whose area is just the width times the height. The sum of the areas of the strips gives the area of the original region. We will go over several such applications of integration to various measurements. More importantly, we will discuss how to go about turning a complicated computation into an application of integration.
Factoring, quadratic formula, dealing with terms and factors in equations. Inequalities.
What is an x ? Domain and Range, graphing, composition and inverse.
Differentiability and continuity, integrability and continuity, uniform continuity,
Logistic growth, differential equations.
Integrable functions. Step functions and impulses
Simple electric circuits, basic economic models, examples from physics
Newton versus Leibniz, l'Hospital and Bernoulli. Cauchy and the advent of rigor. What did the Greeks know? How close were they to developing calculus? What about the Arabs, Phoenicians, or Meso-American mathematicians?
Limits and function spaces, optimization and geometry, differential equations.
Copyright (c) by David L. Johnson, last modified
On May 1, 2000..