Chapter 1, entitled Preliminaries, is a review of
pre-calculus topics. There is very

little in Chapter 1 that we will actually lecture on in class; students are
required to review

it on their own by reading Sections 1.1, 1.2, 1.5, as well as handout materials
on solving

inequalities and sign /direction/concavity of functions. Section 1.6 is the only
section that

will be taught in class.

There are several homework as signments that cover all of
Chapter 1 for students to work

on during the first two weeks of the semester. Our aim is to sharpen/refresh the
students'

knowledge of pre-calculus topics and to reinforce the message to students that
failing to

review this material adequately will hurt their performance in the course. Be on
the lookout

for students in 10A who are having difficulty with this review and reiterate to
them the

importance of having this material at their fingertips.

Many of the difficulties students have in this chapter are
similar to those in 5A so I will

refer you back to the math 5A tutoring notes (see below).

**A. Sections 1.1,1.2, & 1.5.**

• The math 5A tutoring notes for Chapters 1 and 2 should
guide you with most questions

10A students have here.

• Some students have a hard time recalling their trig. Since they probably used
calcu-

lators in high school, they don't know some of the basic trig values that they
must

memorize for 10A. Remind them that a review of basic trig at this point will
help them

throughout the course.

• The only trig identity that students have to memorize in this course is sin^{2}θ+cos^{2}θ = 1.

They should also know:

- The trig values for sin, cos and tan for the angles 0, 30, 45, 60, 90, 180,
270 and

360 degrees (the so called "π/2, π/3, π/4 and π/6" families).

- How to find the angle when the sine, cosine, etc., of the angle is given.

- How to convert from radians to degrees and back.

- The basic relationships between the six trig functions such as sec
, etc...

- American students learn a word to help memorize the trig ratios: SOHCAHTOA.

It reminds them that the ratios for a right triangle are

- Some students find trig equations hard . To be consistent we always ask for
answers

in the interval [0; 2π]. For an equation like sin
you might want to say

something like: " What are the values of µ in the interval [0; 2π] whose sin is

?" After students have studied Section 1.6, they can also approach these

equations using inverse trig functions.

**B. Polynomial and Rational Inequalities.**

• Look at the Chapter 1 tutoring notes for math 5A (Section 1.7) for the
algorithm we

use to solve polynomial and rational inequalities. We will use the same method
in

math 10A.

• The difficulties students have here are also the same as those listed in the
math 5A

Section 1.7 tutoring notes.

**C. Sign, Direction and Concavity of Functions.**

• We define (without reference to the derivative) what it means for a function to
be:

- positive/negative on an interval

- increasing/decreasing on an interval

- concave up/concave down on an interval

• At this point we mainly use the graphical definitions of the above terms .

• As in math 5A remind students that for checking increasing/decreasing we look
at the

graph from left to right

(not right to left)

**D. Inverse Functions and Logarithms (Section 1.6).**

• Students can identify 1-1 functions using the horizontal line test; we don't
insist on

more formal reasoning. (Students generally feel more comfortable with graphical
ex-

planations than with abstract algebraic expressions. This is one thing you
should bear

in mind when tutoring.) They also need to be able to ¯nd the inverse of a
function

like. Remind them to go through the fol lowing three steps rigorously :

- Rewrite the ex pression into

- Solve for x in the resulting equation, by regarding y as a constant.

- You will get something like
In this expression, replace x with

and y with x.

As you will learn, some students will have trouble with the algebra involved in
the

second step.

For some kinds of inverse function problems, it is helpful to remind the
students that

• The only inverse trig functions we introduce in this course are sin^{-1} x and
tan^{-1} x.

Such questions as finding the value of tan
require drawing an illustrating

triangle. The unit circle method is frequently used, and the graphs of sin x and
tan x

are some times very useful.

• Of course we use the identities
and to solve exponential
and

rational equations however students usually say something like \take ln of both
sides"

when they are solving exponential equations and \raise both sides to the e" when
they

are solving a log equation. (This last phrase is wrong, of course, but when
students

say it they are generally doing the right thing. They will also understand if
you use

the correct phrase, \exponentiate both side".)

• is an important
example. Students frequently say that ; some of

them make this kind of error repeatedly through the semester.

• In an example like
students forget to notice that x = -1/2 is not a valid

solution.

• 4 ln x = -8 is also an important example. Students often claim that an
equation like

this has no solutions , since \natural log can't be negative". What they're doing
is

confusing the input and output of ln x.

• is another basic example. Students tend to either start by taking ln
on

both sides of the equation | of course they could succeed if doing it correctly
| or

simply get lost. It is a good point to tell them to \choose the right base when
taking

logarithm".

• ln(2x - 1) - 2 ln x = 0 is yet another very important example since students
often

exponentiate incorrectly as follows: In can be hard to
convince

some students that this is wrong. To do so, start with a simple equation like
2+3 = 5

and point out that 2^{2} + 2^{3} ≠ 2^{5}.

• Solving exponential and logarithmic inequalities is not covered in the text.
However

these come up, of course, when students are analyzing derivatives involving e^{x}
and ln x

later on in the course, so we teach them now. Usually students find these difficult
at

first. We need to remind them to use the number line (0;∞) when ln x is present.

• Two examples of inequalities:

1. xe^{x} + 2e^{x} > 0 is a relatively simple inequality. Students must remember what

they learned earlier in the section about solving inequalities and remember that

e^{x} > 0 for all x.

2. x - 2x ln x > 0 is harder for students. First they have to notice that since
the

domain of ln x is (0;+∞), their number line analysis takes place only on the

positive real line. They also have to figure out how to find \test points" in the

intervals and
. They're used to choosing integer test points, of

course, which don't work well here without a calculator . Point out that the best

kind of test point is a power of e . For instance, for the interval
, they can

choose the test point -or any point of the form
, where p <1/2. Similarly

for they can choose e, e^{2}, etc., or any power of e higher than
1/2.