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M 1/1/07
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New Year’s Day (no school).
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T 1/2/07
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Cathedral funeral for former President Gerald R. Ford
(no school).
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W 1/3/07
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HW due: As you study for your midterm exam, prepare three
(3) questions that are suitable for inclusion in a midterm exam, including
complete solution key and partial-credit scoring guide. Make your questions
difficult enough that they help you as you review. Mix up the problems (I
suggest one from Chapter 1, one from Chapter 2, and one from Chapter 3).
Note: It is acceptable to use
textbook problems, provided you furnish a complete solution key (not just the
answer, but a complete solution key showing all steps). However, you will
probably learn more if you mix things up a bit, by changing the numbers
and/or the specific way in which the problem is posed.
In your partial-credit scoring guide, indicate how many points the problem
would be worth if the student answered the question completely correctly, and
then indicate the point deductions that would be made for all the standard
types of mistakes that students make: sign error, forgetting the chain rule,
using wrong derivative formula, forgetting to take opposite reciprocal,
forgetting how the point-slope equation of a line works, etc., etc.
There will be relatively little memorization required for the midterm exam.
However, you are expected to know all 12 of the trig and inverse trig
derivative formulas by heart, as well as the chain rule, the product rule,
the quotient rule, the power rule, and the derivatives of exponential and
logarithmic functions.
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Th 1/4/07
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HW due: Prepare another three (3) questions, following the
same instructions as for yesterday. If you did not do yesterday’s assignment
(even if you received an excuse because of Lessons and Carols), then you must
prepare six (6) questions in order to earn points. Don’t forget the
partial-credit scoring guide!
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F 1/5/07
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HW due:
1. Prepare a substantive question to ask regarding the midterm exam. (Do not
ask a question about the format. I have already stated that the format will
be primarily short answer, fill-ins, and problem solving.) If you wish, you
may come to Room R between 9:05 and 9:15 a.m. to pose your question.
2. Consider the curve y sin x + 2 cos (x2 – p2/4) =
1. Find the equation of the normal line at (p/2, –1). Show your work.
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M 1/8/07
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Midterm
Exam, Steuart 202, 8–10 a.m. The
conflict exam will be offered in the same room from 11 a.m. to 1 p.m. for the
two students who spoke to me about having a conflict. Since nobody else told
me about a conflict before the deadline (Wednesday, Jan. 3), there will be no
other times offered. The conflict exam will be of approximately the same
difficulty but may cover different material.
Both versions of the exam will be targeted for 90-minute completion time
(i.e., 18 minutes of teacher completion time). However, you may use the full
2 hours if you wish.
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W 1/17/07
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Classes resume.
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Th 1/18/07
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HW due: Write out two (2) examples, different from those
given in class, of real-world variable-factor products that might be found
with the calculus. Follow this format for each one:
By integrating the variable factor, ______________ , with respect to
______________ , we could determine the ______________ .
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F 1/19/07
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HW due:
1. Revise your two examples from yesterday so that they are good examples. If
they are truly unsalvageable, then come up with replacements. If you have
forgotten what you wrote, then start afresh.
Note: Usually the function that is
being integrated is a rate (e.g., velocity in miles per hour), but not
always. In our example of force being applied through a distance, the
variable factor (force) is being multiplied by a little bit of distance for
each of the rectangular areas under the curve to produce a product, namely
force times distance. But remember, the units of force times distance are
work units (newton meters or ft. lbs.). Hence work is the result of the integration.
2. Use Bobby’s modified temperature function, y = 45 – 10 cos(px/12), to find the total number of heating degree days for a 24-hour
period. If you left early and missed the explanation of how to do this, then
ask someone who listened or figure it out on your own. Remember, we are using
70 degrees as our target temperature.
3. Verify that Bobby’s function satisfies y(0)
= 35, y(12) = 55, and y(24) = 35. Show your work.
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M 1/22/07
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HW due: Read §4.1; write p. 183 QR #1-12; write p. 184
Exercises #11, 14, 17, 20. If you do not have time to read or understand the
entire section, then read the yellow box on p. 180 and the sentence
immediately underneath it. From that short bit of reading, you will have
enough information to do all the problems. (Most of them are a review of
earlier material.)
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T 1/23/07
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HW due: Read §4.1 properly (with good light, calculator and
pencil by your side, with reading notes). Then write §4.1 #20, 24, 26, 30,
45-48 all, plus 11, 14, 17, and 20 from yesterday’s set if necessary. Two
problems are worked for you below as examples.
11. Find extrema of f (x) = 1/x + ln x on [0.5, 4].
Solution: Because f is
differentiable (and hence continuous) on [0.5, 4], extrema can occur only at
critical points or endpoints.
To find the critical points, find all places where f ¢ is 0 or DNE. DNE is not an issue since f ¢ has a domain that is a superset of [0.5, 4], but set f ¢(x) = 0 to
find the others.
f ¢(x) = –1/x2 + 1/x, set = 0 to get x = 1
f (1) = 1/1 + ln 1 = 1
Therefore, is (1, 1) the only possible extremum? No, we must also check the
endpoints.
f (0.5) = 1/0.5 + ln 0.5 = 2 – ln 2
» 1.307
f (4) = 1/4 + ln 4 » 1.636
Conclusion: Absolute min. at (1, 1) and absolute max. at (4, 1.636). There is
also a local max. at (0.5, 1.307) since f
¢(0.5) = –2 < 0,
showing that f is decreasing there.
23. y = (x2 – 1)1/2 Þ y ¢ = 0.5(x2
– 1)–1/2 (2x) with
critical values at x = ±1, x = 0
Since there are no endpoints to check, we
note that for large absolute values of x,
y gets larger and larger without
bound. Therefore, y has no absolute
max. We must check the 3 critical values:
y(±1) = 0
y(0) = (–1)1/2 = DNE
Conclusion: Absolute min. at both (–1, 0) and (1, 0); no absolute max.
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W 1/24/07
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No additional written HW due, but if you did not
finish the previous two assignments, I expect you to put in at least another
35 minutes of good work, including reading notes if you have not already made
some. Remember, the short reading excerpt on p. 180 tells you most of what
you need to know. Anybody feel like writing a catchy jingle that uses the
phrase, “Critical points and endpoints”? I don’t know, maybe something like
this:
Critical points and endpoints
Don’t be a numbskull
Don’t suboptimize
Check them all
I know you can do better than this! Chocolate for the best legit song (or
rap). Competition will be held at the start of class.
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Th 1/25/07
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HW due: Read §4.2; write §4.2 #QR 1-5 all, Exercises #1-5
all. Note that these problems are all review problems and do not use any of
the MVT (Mean Value Theorem) material in §4.2. We will do more difficult
exercises after we have had a chance to discuss the MVT in class.
Note: “Analytic methods” means that
you must use algebra to determine where the derivative is positive
(indicating an increasing function) or negative (indicating a decreasing
function). Answers based on a sketch or a table of values will not earn full
credit.
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F 1/26/07
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HW due: Write §4.2 #15, 16, 17, 19, 20, 25-29 all (#30 for
1-point bonus).
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M 1/29/07
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HW due: Write §4.2 #31-34 all, 41, 51.
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T 1/30/07
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Quiz (50
pts.) through §4.2. The quiz is
cumulative since the beginning of the course. However, on this quiz, you will
not be expected to memorize the 12 formulas for trig and inverse trig
derivatives. If any of those are used, the formulas will be provided for you.
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W 1/31/07
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HW due: Re-do yesterday’s
quiz completely, even the parts that you think you did correctly. Please
note that I made an error in #1(b) that I did not catch until after class.
Show all work neatly. You may write directly on the test if you wish, or if
you prefer, you may follow the standard HW format. You may compare answers
and solution techniques with classmates, but no copying is allowed.
SHOW ALL REQUIRED WORK. For
example, in #1(f), you must prove not only that the given point is a critical
point but also either (a) that the sign change in the derivative that occurs
there is of the necessary type or (b) that the second derivative test proves
that the point is a local maximum.
For full credit, SHOW SLIGHTLY MORE
WORK THAN YOU NORMALLY MIGHT SHOW ON A QUIZ. For example, in #1(e), do
not simply write down the answer. Show the correct notation, the plugging in,
and the algebraic resolution leading to the answer. Yes, I realize that you
may be able to do all of this in your head. That’s not the point. The point
is that you are communicating in writing, and you are thinking about what you
are doing while you are doing it.
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