Description

The first area to explore is that involving various functions and the limits of functions. We start by exploring various functions and their graphs. Afterward, we refresh our knowledge of trigonometric functions and of setting up functions. We then investigate the notion of a limit of a function at a point; theorems on limits; limits that involve infinitity; and continuity.

Sections of this topic include:

1. Functions and Graphs

   Since Calculus is based on the idea of functions, we begin our study by refreshing functions and their graphs. Groups within this section include: evaluating functions, finding the Difference Quotient; finding domains of functions; finding the range of functions; graphing functions; arithmetic of functions; and composition of functions.

2. More on Graphs of Functions

   We explore more concepts of graphing and functions including intercepts, symmetry, transformations and even using a graphing calculator. Groups within this section include: graphing using intercepts, symmetry, asymptotes, shifted graphs, and reflections; graphing functions using transformations; graphing quadratic functions using completing the square; graphing functions using a graphing calculator.

3. Trigonometric Functions

   To completely explore Calculus, we should become familiar with the subject of Trigonometry. To this end, we refresh in concepts such as radians and degrees, the unit circle, identities, solving equations, formulas, graphing, even and odd trigonometric functions, and using a calculator. Groups within this section include: converting degrees to radians; converting radians to degrees; using the unit circle; using identities to find trigonometric values; solving trigonometric equations; using sum/difference and double/half angle formulas; graphing trigonometric functions; multipliations, division and composition of trigonometric functions; even and odd trigonometric functions; and using a calculator.

4. Setting Up Functions

   This brief section explores how to set up piece-wise functions given information. The group within this section is called Creating Functions.

5. Limit of a Function at a Point

   We are now ready to explore the first major concept of Calculus called the Limit. Here, we explore graphs to understand the fundamentals of limits. Groups within this section include: using graphs to find teh limit of a function; using graphs to determine if a limit statement is correct; using a calculator to investigate limits.

6. Theorems on Limits

   With a foundation of limits, we now explore theorems on limits to algebraically find limits of functions. Groups within this section include: using theorems on limits to find a limit of a function; evaluation limits with functions; the Squeeze Theorem.

7. Limits that Involve Infinity

   One extension of the idea of a limit of a function allows to explore limits that involve infinity. Groups within this section include: finding limits that contain infinti or result infinity; graphing functions that satisfy criteria; finding limits of rational functions.

8. Continuity

   An application of limits is that of exploring continuous functions. Groups within this section include: finding points of discontinuity; determining if a function is continuous over an interval; finding values to make a function continuous; continuity of composite functions; determing where composite functions are continuous; and the Intermediate Value Theorem.