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 nth root; and that if n is even, f(x) = x1/n is only defined for positive x; understand negative powers as reciprocals of the positive: x-n = 1/xn; be comfortable with the law of exponents in this context: xaxb = xa+b, xa/xb = xa-b, (xa)b = xab; a and b can be integral, fractional, positive, negative, anything.
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
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), sin2(x) + cos2(x) = 1 (Pythagorean theorem), sin(x + 2π) = sin(x), cos(x + 2π) = cos(x) (2π is length of circle), etc.
Exponential and logarithmic functions,
also "essential," are covered in § 1.5.
1.5 Be able to sketch the graph of an exponential function f(x) = ax: f(0)= 1 always. The function increases from 0 to infinity if a > 1, it's constant and equal to 1 if a = 1, it 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.54). Know what e is (p.57). (Example 4, Exercises 14,15,16).
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,
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.65!!). Be comfortable with the Laws of Logarithms (Box, p.65). (Example 6, Exercises 35,36). Know the natural logarithm ln x = logex, and know the change-of-base formula logax = (ln x)/(ln a) (Box, p.67 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 (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 limx->af(x) as well as one-sided limits from inspection of the graph of f (Exercises 3,4,5,6,12).
Use the Chapter Reviews for further reviewing.