MAT125 Fall 2007 Review for Midterm I

1.1 Understand the definition of function (box, p.11) and the importance of the word "exactly." The Vertical Line Test (p.17) shows exactly what "exactly" means in terms of graphs (Exercises 5,8). Be able to ascertain the domain of a function either from the formula (Example 7, Exercises 27-31) or the graph (Exercises 6, 7). Practice sketching graphs (Exercises 36,37) especially for piecewise-defined functions (Exercises 41,44).

1.2 Be familiar with the "essential functions." Linear (Example 1, Exercises 10,11,14,18); be able to interpret slope and y-intercept. Polynomial: know what the degree is (p.29), (Exercises 1c, 2de).

1.3 Understand how translations work (Box, p.38 and Figure 1); remember that f(x-c) has the graph of f(x) shifted c units to the right, and understand why! (Exercise 3bd). Same for stretching and reflecting (Box, p.39 and Figure 2); remember that for example f(2x) has the graph of f(x) compressed by a factor of 2, and understand why. Also understand why -f(x) and f(-x) do very different things to f(x). Exercises 1,2,3.
   Understand that when two functions are combined by +, x, - and / the domain of the combination is the intersection of the two domains: where both functions are defined, plus no zeroes in the denominator! Exercises 13,14,16. Be able to "complete the square" (Example 2, Exercise 12). Remember that in the composition f(g(x)) the function g is applied first. (Example 8, Exercises 35,37).

1.5 Be able to sketch the graph of an exponential function f(x) = ax: f(0)= 1 always. Increases from 0 to infinity if a > 1, constant and equal to 1 if a = 1, decreases from infinity to 0 if 0 < a < 1. Only positive a are considered! (Exercises 7,8,9). Be familiar and completely comfortable with the Laws of Exponents (box, p.57). Know what e is. (Example 4, Exercises 13,14).

1.6 Understand that a function can only have an inverse if it is one-one, i.e. it satisfies the Horizontal Line Test (box, p.64). Examples 1,2 are fundamental. Exercises 5-8. Understand and be able to apply the algorithm (box, p.66) to calculate the inverse of a one-one function. (Example 4, Exercises 21,22).
   In particular the exponential function f(x) = ax for any a not equal to 1 has an inverse, called "logarithm to the base a," and written loga. Understand and be comfortable with the equivalences between logax = y and x = ay, etc. (Boxes on p.67!!). Be comfortable with the Laws of Logarithms (Box, p.68). (Example 6, Exercises 35,36). Know that the natural logarithm ln x = logex, and know the change-of-base formula logax = (ln x)/(ln a) (Example 10 -requires a calculator).

2.1 Be able to sketch secant lines through a point on a curve, and the tangent line to the curve at that point, and understand how to use slopes of secants to estimate the slope of the tangent (Exercises 3, 4 -require calculator). Be able to calculate average velocities from distance-time data and to use them to estimate instantaneous velocity (Exercises 6.7).

2.2 Understand that the limit of a function f at a does not involve the value f(a) (which may not even be defined). (Definition, p.99). (Example 1). Also definitions of left- and right-hand limits (Box, p.103 and below). Be able to estimate limx->af(x) as well as one-sided limits from inspection of the graph of f (Exercises 4,5).

2.3 Understand that limits interchange with the arithmetic operations +, x, -, / except when they lead to division by zero (Example 1, Exercise 2,3,8). In particular since limx->ac = c (here c is a constant function) and limx->ax = a (be sure you understand what this means), the limit at x=a of a polynomial function of x is the value of that function at a (Example 2a ). This does not always work for quotients of polynomials (Examples 2b, 3, Exercises 12, 13, 14).

2.4 Know the definition of "f continuous at a" (Box, p.117) (Exercises 3a,13,14). Also "continuous from the right" and "from the left" (Box 2, p.119) (Exercise 3b). Understand that polynomials are continuous everywhere, and that rational functions are continuous wherever they are defined (Theorem 5, p.120) (Example 5). Be able to apply the Theorem on compositions (Theorem 8 p.125) (Examples 8,9). Understand how to use the Intermediate Value Theorem to calculate roots of equations by repeated approximation (Example 10, Exercise 37).

Use the Chapter Reviews for further reviewing.

Chapter 1

Chapter 2

October 1, 2007