MAT123 Fall 2003 Review for Midterm I

Review Chapter 1 and make sure you are on top of the techniques covered in that chapter.

2.1 Understand the "four ways to represent a function" (p.137): Algebraic, Visual, Verbal, Numeric. Be able to convert the expression of a function from one "way" to another. (Verbal <--> Algebraic: Exercises 3 and 7; Algebraic --> Numeric: Exercise 11; Visual <--> Verbal Exercises 61 and 68).

2.2 This section treats Algebraic <--> Visual and Numerical --> Visual to some extent. Understand the concepts domain and range and where the domain and the range of f appear in the graph of f (Example 7 p.161, Exercises 1,2). Know some examples of non-polynomial functions (absolute value Example 3 p.158; step functions like the "greatest integer" function Example 4 p.158). More generally, understand piecewise-defined functions (Example 2 p.157 Exercises 63,67) and their graphs. Understand the "vertical line test" (p.161) which tells which curves can be the graph of a function (Exercises 5,7).

2.3 This section has some important examples of functions that turn up everywhere. Understand "d is proportional to t" (Example 1) and "y is inversely proportional to x" (Example 2) as sentences representing the functions d = kt and y = k/x respectively. Be able to calculate k from data (Exercises 23, 26).

2.4 Understand "average rate of change for the function y = f(x) between x = a and x = b" (Box p.175). Be able to calculate it in examples: from formula for f(x)(Example 1, Exercise 3), from table (Example 3, Exercise 20), from graph (Exercise 15). Understand the concept "f is increasing on an interval I" (Box p.180), and "decreasing". Be able to use a graph to find intervals on which f increases or decreases (Example 5 p.180, Exercises 23-26).

2.5 Understand how the graphs of f(x+a), f(x)+a, f(cx), cf(x) are geometrically related to the graph of f(x), for a positive and negative, and for positive c greater or less than one. (Examples 1,2,3,4,7; Exercises 1,3). Understand how the graphs of f(-x), -f(x) are geometrically related to the graph of f(x) (Example 5, Box p.188)

2.8 Understand how to combine functions arithmetically (f+g, f-g, fg, f/g) and how to figure out the domain of a combination. Examples 1,2; Exercises 3,5. Understand the operation of composition in terms of the "machine" representation (Figure 3). Be able to work out the composition when both functions are given by graphs Exercises 23,24.

2.9 Be able to apply the "horizontal line test" to the graph of f to see if f has an inverse (Figures 3,4,5). Understand how restriction of domain can change a function to one which satisfies the horizontal line test (Figures 4,5). Exercises 62,64. Be able to calculate the inverse of a function given by a formula (Box, p.233; Exercises 31,38). Understand that if g is the inverse of f, then the graph of y=g(x) and the graph of y=f(x) are related by reflection through the diagonal y=x (Figures 8,9,10) Exercises 52,54.

Use the Chapter Review on pages 240-245 for further reviewing.

3.1 Understand the terminology "monomial" and "polynomial." Understand that the large-scale behavior of a polynomial is determined by its leading term (Example 2) and that the end behavior has exactly four possibilities (Box on p.260). Understand that a polynomial of degree n can have AT MOST n-1 bumps (local extrema)(Example 7, Exercises 37-42).

3.2 Long division of polynomials. Understand the "Division Algorithm" (Box, p.273) and be able to calculate quotient and remainder in "polynomial long division" (Example 1, Exercises 3,5).

October 8 2003