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
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
knowledge of pre-calculus topics and to reinforce the message to students that
review this material adequately will hurt their performance in the course. Be on
for students in 10A who are having difficulty with this review and reiterate to
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
lators in high school, they don't know some of the basic trig values that they
memorize for 10A. Remind them that a review of basic trig at this point will
throughout the course.
• The only trig identity that students have to memorize in this course is sin2θ+cos2θ = 1.
They should also know:
- The trig values for sin, cos and tan for the angles 0, 30, 45, 60, 90, 180,
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
- 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
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
use to solve polynomial and rational inequalities. We will use the same method
• The difficulties students have here are also the same as those listed in the
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
- 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
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
more formal reasoning. (Students generally feel more comfortable with graphical
planations than with abstract algebraic expressions. This is one thing you
in mind when tutoring.) They also need to be able to ¯nd the inverse of a
like. Remind them to go through the fol lowing threesteps 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
For some kinds of inverse function problems, it is helpful to remind the
• The only inverse trig functions we introduce in this course are sin-1 x and
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
are some times very useful.
• Of course we use the identities
and to solve exponential
rational equations however students usually say something like \take ln of both
when they are solving exponential equations and \raise both sides to the e" when
are solving a log equation. (This last phrase is wrong, of course, but when
say it they are generally doing the right thing. They will also understand if
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
• 4 ln x = -8 is also an important example. Students often claim that an
this has no solutions , since \natural log can't be negative". What they're doing
confusing the input and output of ln x.
• is another basic example. Students tend to either start by taking ln
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
• ln(2x - 1) - 2 ln x = 0 is yet another very important example since students
exponentiate incorrectly as follows: In can be hard to
some students that this is wrong. To do so, start with a simple equation like
2+3 = 5
and point out that 22 + 23 ≠ 25.
• Solving exponential and logarithmic inequalities is not covered in the text.
these come up, of course, when students are analyzing derivatives involving ex
and ln x
later on in the course, so we teach them now. Usually students find these difficult
first. We need to remind them to use the number line (0;∞) when ln x is present.
• Two examples of inequalities:
1. xex + 2ex > 0 is a relatively simple inequality. Students must remember what
they learned earlier in the section about solving inequalities and remember that
ex > 0 for all x.
2. x - 2x ln x > 0 is harder for students. First they have to notice that since
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
. 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, e2, etc., or any power of e higher than
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