Module 1
Section 1.2
(see Text Sections 1.2 and 1.3)
Introduction
This section reminds you of various ways of rearranging algebraic expressions.
Your objectives for this include both understanding and speed.
Understanding of these operations will be important for you to successfully follow the remainder of the course, and speed will also be important if you later need to be able to follow classroom presentations in subsequent courses.
Study Notes and Discussion
This material should all be familiar to you so you should be able to proceed quickly and feel comfortable with doing so. But don’t skip it. Be sure to read all of the text sections carefully, and work the examples as you go.
One point worth emphasizing
with regard to
the material in Section 1.2 is that the radical sign is used just for
the positive
square root of a positive number. So
only, not
.
Sometimes
is called the principal square
root, but it is often just called “the square
root” - without a modifier.
The latter is the most common
usage in
current North American calculus courses and it is what we shall do in
this
course. It is true that some places people do talk of a positive number
a as having “two square roots”. But even
then the root sign is almost always taken as referring only to the
positive one
as we have indicated above. So, again, if you say
,
it will be considered wrong.
You should get help at once if you have trouble either understanding or applying any of the techniques introduced in Section 1.3. Notice that in Example 6 on p.43 it is the numerator rather than the denominator that is being ‘rationalized’. You may have spent lots of time rationalizing denominators in high school, but there was never any really good reason for doing that rather than rationalizing numerators instead. (Actually what turns out to be useful most often in calculus is rationalizing differences - either in the denominator or in the numerator.)
Practice Exercises
Check your understanding and practice for speed by working through some of the Exercises on pp29-31 and 45-47
Do enough of the odd numbered questions of each type to convince yourself that you can get the right answers quickly. (Note that, as usual, the answers are in the back of the text and complete worked solutions are in the student study guide - but try to avoid looking at answers or solutions until you have made your own best effort).
As a bare minimum you should do ##5,15,25,55,65,85 and 97 from Section 1.2
and ##5,15,25,35,53,63,75 and 85 from Section 1.3.