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M 10/2/06
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HW due: You didn’t really think there would be additional
HW due over Homecoming weekend, did you? Please patch up your existing
problems.
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T 10/3/06
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HW due: Read §§3-1 and 3-2.
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W 10/4/06
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HW due: §3-2 #15, 16; §3-3 #7, 10, 11, 12; §3-4 #1-21 all.
Note: Problems #1-18 in §3-4 are
somewhat monotonous. You may stop after doing 6 minutes’ worth of those. See
if you can finish all of them in 6 minutes. (My time was 5:20 to finish all
18.) However, do #19-21 as normal problems, showing the limit calculations.
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Th 10/5/06
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HW due: Read §3-6; write §3-5 #8, 10; §3-6 #6, 7. Optional
reading assignment: §3-5.
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F 10/6/06
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No
school.
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M 10/9/06
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No
school.
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T 10/10/06
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HW due: Read §3-7 and 3-9; write §3-7 #2-22 even, 25, §3-8 #4,
§3-9 #3-18 mo3, 23.
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W 10/11/06
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HW due: Review problems on pp. 126-127: #T3, T5-17.
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Th 10/12/06
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Test
on Chapters 2 and 3. Optional
additional review to help you prepare: pp. 123-126.
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F 10/13/06
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HW due: Read §4-2.
Optional Bonus Opportunity (exact
number of points TBD): Answer all the questions on yesterday’s
test correctly. You may work with friends, as long as you are thoroughly
clear on what is going on and can survive oral questioning.
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M 10/16/06
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HW due: §4-2 #10, 23, 32; §4-3 #12, 16. For #16, do the problem
two ways: (a) by using QR and (b) by rewriting t(x) as 51x–17. Contrary to the
instructions given in your textbook, simplification is not expected.
Note: This assignment is shorter
than usual so that you will have time to fill in some gaps from previous
assignments.
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T 10/17/06
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HW due: Use induction to prove the generalized product rule
(same as §4-2 #24, except that you must not only make the conjecture, but
prove it as well). Then write §4-4 #3-12 mo3.
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W 10/18/06
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HW due: We are taking a pause from new material in the
calculus to make sure that everyone understands and can perform mathematical
induction. Do the following:
1. Memorize all 6 of the derivative formulas on p. 142. There is not much
memorization in HappyCal, but these 6 are expected to stick with you for the
entire year.
2. Write §4-4 #16-38 even. These should take you approximately 1 minute each
(12 minutes total).
3. Prove each of the following by using induction to develop a conjecture and
then by using mathematical induction to prove your conjecture. During the
first phase (“true induction”), you simply make a table of values, as we did
in class, and see if you can determine through inspection or analysis what
the pattern is. During the second phase, you will use the techniques of
mathematical induction (basis case, then showing that the kth case implies the (k + 1)st case).
(a) The sum of the first n odd
positive integers is . . .
(b) Let n be an integer, n ³ 3. The number of diagonals for a convex n-gon (polygon with n sides) is . . .
Hint for (b): Think very carefully
about where the new diagonals are added as you convert from a quadrilateral
to a pentagon, or from a pentagon to a hexagon. You can imagine “bumping out”
a wall on an n-gon to form an (n + 1)-gon.
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Th 10/19/06
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HW due: §4-5 #9, 10, 12, 19-24 all, 29.
In class: Review of Test on Chapters 2 and 3.
The quiz originally scheduled for today will occur after chapel today (for
seniors) or in class tomorrow morning (for everybody else).
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F 10/20/06
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No
additional HW due. Seniors are not required to attend today, but they must
take the quiz on Thursday, 10/19, if not attending today.
Quiz on mathematical induction (10
pts.). Here are a couple of recommended practice problems:
1. Prove that the sum of the first n
squares (i.e., 12 + 22 + 32 + . . . + n2) equals n(n
+ 1)(2n + 1)/6.
2. Prove that the sum of the degree measures of a convex n-gon equals 180(n –
2).
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M 10/23/06
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HW due: Read §4-6; write §4-6 #1-12 all. Remember to make a
rough sketch for each. Because this is a very quick assignment, you should
(please) use the leftover time to fill in the gaps in your §4-5 problems from
last Thursday.
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T 10/24/06
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HW due:
1. Without peeking at your notes, use the product rule to prove the quotient
rule.
2. Write pp. 173-174 #R1ab, R2cd, R3cde, R4abcd, R5ab, R6abcd.
3. (Strongly recommended.) Prove that if f
is differentiable, then f even Þ f ¢ odd, and f
odd Þ f ¢ even.
In class: Review.
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W 10/25/06
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No
class (University of Maryland math competition, Trapier Theater).
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Th 10/26/06
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Quest
(70 pts.) through §4-6, including mathematical induction. You will have approximately 5 minutes for the
induction proof, not 20 minutes as some people took last Friday.
Here are the two recommended proofs from the HW due 10/24.
Proof 1 (f differentiable and f even Þ f ¢ odd):
By def. of even, ("x Î Df) f (–x) = f (x).
Let v(x) = –x, so that ("x Î Df) f (v(x))
= f (x).
Differentiate both sides (since derivatives of equals are equal) to get the
following:
("x Î Df ¢) f ¢(v(x)) v¢(x) = f ¢(x)
Since v(x) = –x, v¢(x) = –1.
Therefore,
("x Î Df ¢) –f ¢(v(x)) = f ¢(x)
("x Î Df ¢) –f ¢(–x) = f ¢(x)
Multiply both sides by –1 to get
("x Î Df ¢) f ¢(–x) = –f ¢(x),
which is precisely the condition on function f ¢ needed to show that f ¢ is odd. (Q.E.D.)
Proof 2 (f differentiable and f odd Þ f ¢ even):
By def. of odd, ("x Î Df) f (–x) = –f (x).
Proceed as in Proof 1 above, the only difference being an extra negative sign
on RHS.
In the last line, we instead obtain
("x Î Df ¢) f ¢(–x) = f ¢(x),
which is precisely the condition on function f ¢ needed to show that f ¢ is even. (Q.E.D.)
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F 10/27/06
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Last
day of first quarter. Bring in your HW binder for a spot check. We will
discuss yesterday’s quest. When I hand back your
paper, amnesty points may be available for some of the problems if you have a
completely correct (and highly legible) solution ready to show me.
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M 10/30/06
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HW due: §4-7 #2abcde, 3abcde(f), 7abc; §4-8 #2-22 even.
Problem #3f in the first set is a good problem. However, it is optional since
it can chew up a lot of time if you are unable to find the trick.
Hints for #3f: What does x/3 equal? What about y/5? Why must they have an interesting
and familiar relationship involving their squares? Write down that equation,
first using nothing but functions of t,
and then using x/3 and y/5 substituted where appropriate.
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T 10/31/06
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HW due: Read §5-3; write §5-2 #2-16 even, §5-3 #2, 3, 4.
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