1.1 Understand the definition of function (box, p.12) 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* 10,11,15,18); be able to interpret slope and
`y`-intercept. Polynomial: know what the degree is (p.29),
(*Example* 4, *Exercise* 3). Power functions: understand
that the `1/n` power (`n` a positive integer)
corresponds to the `n`th root; and that if `n` is
even, `f(x) = x ^{1/n}` is only defined for positive

Trigonometric functions: keep this diagram firmly and
permanently in mind to understand sine, cosine and tangent
as functions of `x` in radians. Starting at O (the right-hand
intersection of the unit circle with
the horizontal axis),
go a distance `x`
counterclockwise along the circle. Projection onto the horizontal
(cosines)
axis then gives `cos(x)`; projection onto the vertical (sines) axis
gives `sin(x)`; the line through C (center of the circle)
and `x` intersects the tangents axis at `tan(x)`.
(The tangents axis is tangent to the circle at O.)

Elementary properties of `sin, cos, tan`
can be retrieved from this diagram, e.g. -1 ≤ `sin(x)` ≤ 1,
-1 ≤ `cos(x)` ≤ 1, `tan(x)`
= `sin(x)`/`cos(x)` (similar triangles),
`sin ^{2}(x)` +

Exponential and logarithmic functions,
also "essential," are covered in § 1.5.

Understand that when two functions are combined by +, x, - and / the domain of the combination is the

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.61).
*Examples* 1,2 are fundamental. *Exercises* 5-8.
Understand and be able to apply the algorithm (box, p.64)
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 (*Examples* 1,2,*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.95). (*Example* 1). Also
definitions of left- and right-hand limits (Box, p.100 and below).
Be able to
estimate `lim _{x->a}f(x)` as well as one-sided
limits from inspection of
the graph of

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, 4
- True-False 1 through 8.
- Exercises 1 through 18