Module 1
Section 1.4
(see Text sections 2.1, 2.2, and 2.3)
Introduction
The use of Cartesian Coordinates allows us to bring together algebra and geometry in a way that works to our advantage in both. By expressing geometric ideas in terms of coordinates we can use algebraic techniques to prove geometric theorems and calculate properties of geometric objects, and by graphing equations and inequalities we can use our physical and geometric intuition to understand their algebraic properties. Your objectives for this section include both of these aspects.
Study Notes and Discussion
The technique of graphing equations by plotting points is a reasonable starting point if you can’t think of anything else, but it can be misleading if you don’t plot enough points and a waste of time if you can do the graph more quickly using geometric ideas.
For example if you just plotted and joined up the points from the table in Example 2 on page 104, you might put a corner at the vertex which would be wrong. On the other hand if you know the basic parabolic shape of graphs of that type, then you could probably sketch a pretty good graph with no more than three points.
As another example, the book’s
Example 8 on
page 111 shows how the idea of completing the square (which we saw when
solving
quadratic equations) can be used to recognize that the graph of
is a perfect circle
something which might have taken a
while to
see just by plotting points. Where does this circle cross the x-axis?
(see answer #1)
The simpler case of linear equations gets a whole section of the text to itself. This is partly because of the significance of straight lines in geometry and partly because of the convenience of linear models for approximating more general relationships. In fact the main idea of calculus is to study the use of linear approximations to study small parts of more complicated graphs. A recent survey of postsecondary math instructors in B.C. identified an understanding of the concept of slope of a straight line as the single most important prerequisite skill for subjects as diverse as calculus and statistics. So be sure that you have read all of the text’s section 2.3 very carefully.
In Example 4 from that section
the line
through two points is found by first calculating the slope,
,
and then applying the point-slope form
to get the equation of the line.
We can motivate this in just one step by just
saying that if
is on the line through
and
,
then the slope from
to
must be equal to that from
to
.
This leads at once to the two
point form
.
This can be useful for quickly
solving
linear modeling problems like the Example 10 on page 129 of the text
especially when the data points don’t
include
an intercept.
Here’s another example. The
length of the
mercury column in a thermometer is 3cm when the temperature is
and 5cm when it is
.
Assuming a linear relationship, find an equation relating the length x
in cm and the temperature y in
,
and use it to predict the length when the temperature is
,
and determine the temperature when the length is 4.5 cm.(see
answer #2)
Further
In the text’s Example 8 of Section 2.2 (discussed above) the method of completing the square was used to convert a quadratic in both x and y (in which the squares both have the same coefficient) into the standard form for a circle.
The same idea can also be applied
when the coefficients are different, but then instead of ending up with
a
circle equation of the form
(or equivalently
),
we get different denominators for the two terms and so end up with
something
like
.
If the signs are both positive, this looks like a circle equation but “with different radii in the x and y directions”. I am sure you remember that such a shape is called an ELLIPSE, and can be created geometrically by cutting a cylinder or cone at an angle.
If the squares are subtracted
rather than
added the resulting graph (called a HYPERBOLA) can also be obtained by
slicing
a double cone
but in a direction more along its axis
rather
than across it.
You probably also remember from previous courses that these curves can also be defined in terms of the sums or differences of distances from special points called FOCI, and that there is a similar definition for a parabola in terms of a FOCUS point and a line called the DIRECTRIX.
These topics are discussed in the first three sections of Chapter 10 of the text. Since these sections do not depend on any of the intervening material, you could read them now to get a bit more practice with the ideas of graphing and geometry in Cartesian Coordinates, but it is not essential as the foci and directrix definitions are not required for calculus.
However, even if you do not read and learn all the details of foci, etc, you should at least be able to convert an equation to “standard form” by completing the squares and be able to describe and plot the graphs.
Use Example 5 on page 779-780 in Sec10.2 and Example 5 on page 793-794 in Sec10.3 as models to help you do Exercise #11 on page784 and #15 on page797 (but don’t worry about finding the foci if you don’t want to), and then compare your solutions with those given in the Solutions Manual.
Further Practice
Check your understanding, and practice for speed, by working through some of the Exercises on pp101-103, 116-119, and 132-137.
Do enough of the odd numbered questions of each type to convince yourself that you can get the right answers. (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 minimum, start with ##1,5,11,15,21, and 31 from Section 2.1, ##11,31,41,51,61,71 and 73 from Section 2.2 and ##1,11,17,27,35,43,53,59 and 67 from Section 2.3