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 (

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) = a ^{x}`:

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) = a ^{x}`
for any

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 `lim _{x->a}f(x)` as well as one-sided
limits from inspection of
the graph of

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
`lim _{x->a}c = c` (here

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

- Concept Check 1 through 12.
- True-False 1 through 11.
- Exercises 1 through 16, 18,19,20.

- Concept Check 1, 2, 3, 7, 8
- True-False 1 through 8.
- Exercises 1 through 16

October 1, 2007