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W 3/1/06
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HW due:
Finish Monday’s assignment completely, plus the following tasks:
1. Read the blue box on p. 526. Read it several times, since the conclusion
is not necessarily true if the coefficients
are complex numbers.
2. Visit the Plane Graphic
Calculator (PGC). Wait a while for the Java applet to load. When the
control panel for the calculator appears, click the “Details” button and the
“Detach” button so that you have two separate windows. One window allows you
to drag points around on the complex plane, as well as to define the
function(s) that will manipulate those points. The second window, the Details
window, shows you the exact real and imaginary coordinates.
3. Using the PGC, carefully copy and paste the following into the f= and g=
windows:
f= (-b+sqrt(b^2-4a*c))/(2a)
g= (-b-sqrt(b^2-4a*c))/(2a)
4. Answer the following questions:
(a) What three complex points immediately appear after you have performed
step 3?
(b) In the Details window, notice that each point has both an x and a y component. What do these represent?
(c) Set the y components of a, b,
and c to be 0, and press TAB so
that all settings take effect. What do you notice about the new positions of
points a, b, and c?
(d) Play around with the positions of points a, b, and c. Hold down the SHIFT key as you move
them left and right, so that they will stay attached to the real axis.
Describe what happens to f and g as you do this.
(e) What is the purpose of f and g? In other words, what have we
accomplished by defining them as we did in step 3?
(f) Write down a combination of values for a, b, and c that will cause f and g to coincide,
i.e., pile up on top of each other. Remember to hold SHIFT down as you move a, b,
and c left and right.
(g) Write down a combination of values for a, b, and c that will cause f and g to be in two
distinct places. What relationship do f
and g have now? Remember to hold
SHIFT down as you move a, b, and c left and right.
(h) Repeat steps (f) and (g) above without using the SHIFT key. In other
words, find random locations for a,
b, and c. What do you notice now about the positions of f and g? Do they have any special symmetry?
5. Erase f and g and replace them with the following:
f= h^4
g= k^4
Then enter the following (copy and paste are strongly recommended!):
h= |a|^.25(cos (.25*arg a) + i sin (.25*arg a))
k= |a|^.25(cos (.25*arg a+pi/2) + i sin (.25*arg a+pi/2))
m= |a|^.25(cos (.25*arg a+pi) + i sin (.25*arg a+pi))
n= |a|^.25(cos (.25*arg
a+3pi/2) + i sin (.25*arg a+3pi/2))
Move point a around to various
random locations. What do you notice about f and g.
6. It appears that h and k both have the property that when
raised to the 4th power, they equal a.
How do you know this? (Note: I am
not asking for a formal proof. For that, you will have to wait a year. What I
want to know is if you can see any evidence to support the idea that h4 = k4 = a.)
7. Do m and n have the same property that h
and k seem to have? How can you
tell?
8. Write down the values of a, h, k,
m, and n for a random position of a.
Remember that each of these numbers will have both a “real part” and an
“imaginary part.” You should have 5 ordered pairs in your answer. Do not copy
someone else’s answer; use your own.
9. Develop a conjecture for the positions of h, k, m, and n. Describe how far out from the origin they are, as well as
their angles relative to a. We call
h, k, m, and n the “fourth roots” of the complex
number a.
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Th 3/2/06
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HW due:
§10-3 #4-40 mo4. Because I really want you to do yesterday’s on-line computer
assignment if you have not already done it, I will also be scanning that HW a
second time.
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F 3/3/06
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HW due: Read
§10-4 (skimming permitted in the examples); write §10-3 #46-56 even. You may
use your QUADF program throughout. Warning:
#48, 50, 54, and 56 are not as easy as you think. Work the corresponding odd
problems and check answers in the back of the book to figure out what is
going on.
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M 3/6/06
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HW due:
Four-part assignment.
1. Read §10-6. No reading notes are required this time, because this is a
review of old material.
2. Write §10-4 #2-10 even. Show work for synthetic substitution, and check
answers by calculator.
3. Write §10-4 #39. Be sure to read the clues on p. 541.
4. Write §10-6 #1.
Note: We are skipping §10-5. The
test on Chapter 10 will be postponed until the end of the school year,
because I had previously announced that there would be no more tests before
the end of the quarter on March 10. We will proceed into Chapter 11.
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T 3/7/06
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HW due:
§10-6 #1, 6. If you have already done #1, then so much the better. If you cannot
get #6, then you should do #5 first and compare answers with the back of the
book as a way of warming up for #6.
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W 3/8/06
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HW due:
Read §11-1; write §11-1 #13-18 all (required) plus 21 (optional). See the
examples of #13 and #16, which are worked for you below.
13. 1, 2, 4, 7, 11, 16, . . .
In words (optional): Each term is a certain amount larger than the preceding
term, where that “certain amount” increases by 1 each time.
Mathematical description (required): tn
= tn – 1 + n – 1.
Translation back into English (optional): The nth term equals the previous term plus (n – 1).
Calculation of t7 and t8 (required): t7 = 22 and t8 = 29 by inspection.
[These values agree with the formula given.]
16. 1, 2, 6, 24, 120, 720, . . .
In words (optional): Each term is the product of all integers from 1 through
the current term number. For example, the 4th term equals 1 · 2 · 3 · 4, and
the 5th term equals 1 · 2 · 3 · 4 · 5. This is what we call the “factorial”
function. The notation is an exclamation point ( ! ) after the number. For
example, 9! = 362,880. You can find factorials easily on your calculator by
punching in the number followed by MATH PRB 4 ENTER. (Or, if the number is
not too large, you can simply multiply out to find the answer.)
Mathematical description (required): tn
= tn – 1 · n.
Translation back into English (optional): The nth term equals the previous term multiplied by n.
Calculation of t7 and t8 (required): t7 = 5040 and t8 = 40320 by calc. [These
values agree with the formula given.]
Quiz (postponed from yesterday):
Calculator-based solution of a system of quadratic equations. This is a
repeat of material tested on the Chapter 7 test.
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Th 3/9/06
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HW due:
Read §11-2; write §11-2 #2-20 even, 26, 33, 40, 41.
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F 3/10/06
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HW due:
Read §11-3; write §11-3 #2-28 even.
In class: Turn in homework, then work §11-2 #19-31 odd, §11-2 #38-44
even, §10-6 #2, 3, 4, §11-3 #29-35
all. Hint: For §10-6 #2, start by
writing a general equation y = ax3 + bx2 + cx + d, and then plug in all of the known
facts. You will get a system of linear equations having four unknowns (a, b,
c, and d). You can pool your heads to figure out how to solve that.
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M 3/13/06
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HW due:
All problems from last Friday will be collected. It is to your advantage to
get as many of them as possible solved and answered during class with your
substitute teacher. If you waste your time, you will have lots of homework.
If you use your time effectively, you may have little or no homework.
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T 3/14/06
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HW due:
Read §11-4; write §11-4 #1-6 all.
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W 3/15/06
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HW due:
Read §11-5; write §11-5 #7, 8, 9, 10, 19, 20, 23. Hint: For #9, #10, and #23, please start by writing out several
terms so that you can see what you’re dealing with. For example, in #9, you
should write 3 + 2(0) + 3 + 2(1) + 3 + 2(2) + . . . = 3 + 5 + 7 + 9 + . . . before you start
trying to solve the problem.
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Th 3/16/06
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HW due:
Read §11-6; write §11-6 #1, 2, 4, 6, 8, 9, 10.
In class: CFU on arithmetic and geometric
sequences and series. The answer key is
also available now.
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F 3/17/06
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HW due: No
additional written assignment, but see notes below.
Quiz on §§11-1 through 11-6. Time
limit: 14 minutes (21 for extended time). Content will be similar to yesterday’s CFU. Feel free to take it again
for practice, and check your answers with the answer
key.
In response to several student suggestions, I have relented and have decided
to require no additional written HW for today. However, you should spend time
studying for your quiz and patching up any gaps in previous assignments in
Chapter 11. Bring all your Chapter 11 homework for a general homework scan.
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Spring break.
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