- Quiz 9 scores (out of 3):
Number Score
19 0
3 1
5 2
1 3
- Homework 8 scores (out of 20):
Number Score
2 16
1 17
4 18
- No office hour Thursday, Apr 12, 3:30-4:30. (Earlier office
hour on Thursday remains.)
- Do any two of the four [final HW] problems, but please look at all four.
- Exam 2 Announcements:
- As of March 28, morning:
| Section 11 | All Math 115 |
| Average | 62.138 | 61.639 |
| Median | 66 | 63 |
On each individual problem, the Section 11 median was
greater-than-or-equal-to the Math 115 median. Except for problem 1,
our average was about the same or slightly greater than the
average for all Math 115. (In problem 1, our average was about 1
point lower.) Note the average for our section and for all sections
can be distorted by zeros assigned to students who have dropped the
course.
There was a lot of individual variation and I'd be glad to discuss
your exam during office hours. The scores were as follows.
0
40 D+
41 D+
41 D+
43 D+
47 C-
48 C-
55 C+
56 C+
58 C+
58 C+
59 C+
59 C+
65 B-
66 B-
68 B
69 B
70 B
70 B
71 B
73 B
76 B+
76 B+
77 B+
77 B+
78 B+
82 B+
84 A-
95 A+
- No WebHW 4.6 will be due until well after the midterm.
- Version 3 of the writeup about steps for optimization can be
found here. (Compared with version 2,
version 3 includes a bit more about checking endpoints and end
behavior when looking for global maxima.)
- Review Monday, March 26
| 2:00 to 3:00 | East Hall 4096 |
| 3:00 to 4:00 | East Hall 3088 |
- HW 8 scores:
Number Score
1 12
1 14
1 15
4 16
- Exam 2 Announcements
- Coverage. The exam will cover 3.1-3.7,3.9, and 4.1 to
4.5. Topics from Chapters 1 and 2 will not be emphasized, but are
fair game. Note that
4.6 is not going to be covered; it will appear on the
final. In section 3.9, we need to know the following:
- The approximation f~ to f at a, given by f~(x) = f(a) + f '(a)(x-a).
- The error E(x) = f(x) - f~(x).
- How to evaluate f~(x) and E(x).
- How to estimate the max of |E| on an interval, from a
graph
- Whether f~ is an over- or under- estimate of f, using
concavity of f.
We do not need to know about bounding |E| using the second
derivative of f.
- Exam Location: 170 Dennison, same as last time
- Exam Date: Tuesday, March 27
- Exam Time: 6 to 7:30 pm
- Bring to Exam: Photo ID, index card with notes, calculator set to radians,
pencils, and a watch. (If you tell time with a cell phone, make
sure that it does not ring and make sure that the proctors can easily
verify that you are not receiving any kind of communication.)
- See Math
115 homepage for further instructions and previous exams.
Warning: Don't expect this year's exams to match previous
years'; the exact syllabus varies from year to year.
- An early seating is available for those with a conflict or those
requiring extra time. Advance confirmation is required.
Please report to the Mathlab, B860 East Hall, at 4:15 with photo
id. You will need to remain in the Mathlab through at least 6:00
pm, even if you finish early.
- Formulas. On the exam, we will supply you with the
formula for the volume of a cone (if that arises), but you're on
your own as to the formula for the area of a rectangle. If you're
hazy on some formula, please write it on your index card.
Review
Review Monday, March 26
| 2:00 to 3:00 | East Hall 4096 |
| 3:00 to 4:00 | East Hall 3088 |
Please go over one of the past
midterm 2 exams, in HW team groups, according to the assignment
below. Please note the following:
- Any difficulties you had working the problems and any novel
solutions or good choices one can make among multiple ways to solve
a problem
- How the problem ties in with the class; where to read more in
the text about the concepts tested.
- What concepts are covered in the test as a whole.
- Whether the syllabus differs from this year's. (The changes may be
greater as we go deeper back into history.)
You might also note any useful non-Calculus meta-information, such as
- Was the test under the direction of Karen Rhea?
- Was the test unusually easy or difficult? (Check the scale, if
available, and compare to the scale of other exams. Fall of 2005
was especially difficult.)
Please come up with 1 or 2 problems you think will be instructive to
do during a review session. Please clear the problem with me in
advance, to assure broad coverage of topics.
| Team | Members | Past exam date |
|---|
| 1 | Dawson, Merritt, Woerdeman, Yates | Fall, 2006 |
2 | DeGraaf, Erben, Frysinger, Thompson, Yankee | Winter, 2006 |
3 | Frorenza, Konen, Petoskey | Fall, 2005 |
4 | De, Faasse, Fernandez | Winter, 2005 |
5 | DeWood, Foley, Martin, Storie, Trudeau | Fall, 2004 |
6 | Anderson, Campbell, Franchini, Jain | Winter, 2004 |
7 | Evans, Kim, Patel, Stowers, Lamb | Fall, 2003 |
- Please read Section 4.5. Try 4.5, Example 1 and 4.5.24.
Example 3
(ladder around the corner) is probably too hard. (Previously this
note said that Example 2 is too hard, but Example 2 (walk through the
park) is probably fair
game.)
- Quiz 7 scores:
Number Score
4 0
6 1
4 2
12 3
2 4
1 5
- Homework 7 scores:
Number Score
1 14
2 16
2 17
2 18
- Extra office hours Wednesday, March 21, 7pm to 11:59pm, in 3063
East Hall. You
can take the gateway in my office, if you have not passed. My
office number is 734-763-3005 (you may need to call me to get in to
the building after hours). You can also come to these office hours
to talk about other things, like the midterm. (But call first; I
may leave early once everyone has passed the gateway.)
- The Thursday office hour from 2-3 is canceled on March 22. The
3:30 to 4:30 office hour is on.
- Gateway Announcements
- Final results: 28/29 pass.
- As of March 20:
| All Math 115 | Section 11, official | Section 11, my count |
| Attempted gateway | 91.0% | 93.1% | 28/29 |
| Passed gateway | 81.3% | 82.8% | 27/29 |
Use "official" numbers to compare our section with all Math 115.
Use "my count" to judge progress against our goal of 29/29
students. The official numbers and my count may be updated at
different times throughout the day.
- The gateway
is open for practice and proctored exams.
- If you do not pass the Gateway by 11:59 pm, Wednesday, March 21, then
you lose a full letter grade.
- Please practice the gateway. Do not attempt a proctored
Gateway exam until you can reliably pass a practice exam.
- Everyone who attempts to pass the Gateway in earnest succeeds.
Problems arise because you only get two chances per day to take a
proctored Gateway and the labs get crowded.
- If 26 of the the 29 currently registered students pass the
gateway by classtime Monday, March 19, I'll bring a reward
to share. (This note originally said "Monday, March 21," which is
not a date in 2007.)
- Quiz 6 was difficult. See writeup
for a discussion of problem
1 and the text (Section 4.3, Example 4) for a discussion of problem
2. Scores:
Number Score
2 0
1 1
5 2
6 3
7 4
6 5
2 6
1 9
- WebHW Chap4Sect1Prob7 may contain an error. Apparently
there are different functions that may turn up for different
students (so your version may be ok) and the error I discussed in
class may be fixed by the time you see the solutions.
- There was an arithmetical error in the discussion about the
Bell-shaped curve in class on Friday, March 9. This writeup corrects the error and finishes the
examples.
- If you do pass the gateway by Friday, March 9, then you
get an automatic 100% on your third lowest quiz score. (The two
lowest scores are dropped.)
- Quiz 5 grades:
Number Score
1 1
7 2
12 3
7 4
- In section 3.9, we need to know the following:
- The approximation f~ to f at a, given by f~(x) = f(a) + f '(a)(x-a).
- The error E(x) = f(x) - f~(x).
- How to evaluate f~(x) and E(x).
- How to estimate the max of |E| on an interval, from a
graph
- Whether f~ is an over- or under- estimate of f, using
concavity of f.
We do not need to know about bounding |E| using the second
derivative of f.
- Homework 6 scores:
Number Score
1 7
2 12
2 13
2 14
1 15
- Please complete a Mid-semester survey, found here.
The site closes on Tuesday, March 6. If our class has a 75%
completion rate, the last team homework will be reduced from FOUR to
TWO problems.
- Please skip problem 3.7.34 on the team HW. Please let me know
if you have already done substantial work on this
problem (before about 4:30 pm Feb 20).
- See handout for comments on HW 5. The distribution is as
follows:
Number Score
1 10
1 14
2 15
1 16
1 18
1 19
- The gateway
is open for practice now and opens for proctored exams on Monday
[Feb 19].
Please try a practice. We'll tak about it more on Monday.
- Quiz 4 was graded out of 3 points, no partial credit. Its
main purpose was for you and for me to know how you are doing.
Please take a look at the handout and the alternate solutions.
Number Score
5 0
8 1
10 2
8 3
- A copy of the Feb 14 quiz is available here.
- We will reshuffle homework teams Friday, Feb 9. Please email
me if you need help joining a team. The next team
HW is due the following Friday, Feb 16.
- WebHW deadlines will be a little shorter than previously. This
is so that I can see how you're doing in time to review material in
class before too much time has elapsed.
- First Midterm Exam Announcements
- Do not write on your copy of the exam.
- Sample solutions and the scale for Exam I are available from
the Math 115 course web
page.
- In almost all cases, Exam I grades will be finalized Weds, Feb 14.
- Plesae let me know if you would like to go over you exam or
discuss how you are doing in the class.
- Please let me know which style of problems you like and did
not like. Based on your responses, I can suggest problems for the
second midterm and for the final.
- The section 11 average is 69.5 and the section 11 median is
70. By comparison, the Math 115 average is about 71 and the median
is 72. I don't think the difference between our section and the
overall course is significant.
The distribution of scores is as follows:
Number Score
1 44, D
1 45, D
1 47, D
2 58, C-
1 61, C
1 63, C
2 64, C+
2 65, C+
2 67, C+
1 68, C+
1 69, B-
2 70, B-
1 72, B-
1 74, B-
3 75, B
1 77, B
2 78, B
2 80, B+
1 81, B+
2 82, B+
1 83, B+
1 87, A-
All of the above numbers are current as of around 8 am, Feb 7.
Some minor adjustment of scores may occur after that time.
- Extra office hours: Friday, Feb 2 and Monday, Feb 5, 3-4 pm,
3063 East Hall.
- Exam Location: Dennison 170
- Exam Date: Tuesday, Februrary 6
- Exam Time: 6 to 7:30 pm
- Bring to Exam: Photo ID, index card with formula, calculator set to radians,
pencils, and a watch. (If you tell time with a cell phone, make
sure that it does not ring and make sure that the proctors can easily
verify that you are not receiving any kind of communication.)
- See Math
115 homepage for further instructions and previous exams.
Warning: Don't expect this year's exams to match previous
years'; the exact syllabus varies from year to year.
- An early seating is available for those with a conflict or those
requiring extra time. Advance confirmation is required.
Please report to the Mathlab, B860 East Hall, at 4:15 with photo
id. You will need to remain in the Mathlab through at least 6:00
pm, even if you finish early.
- Extra office hours: Friday, Feb 2 and Monday, Feb 5, 3-4 pm,
3063 East Hall.
- In HW 4, I deducted a point for improper use of the word
"approximately" or for improper/missing units.
For a sinusoid function, the function increases or decreases
fastest at the horizontal position that is halfway between the
horizontal positions of the minimum and maximum of the graph, but
note that this reasoning only applies to sinusoidal
functions. For now, you can look on a graph or table to tell
where a function has steepest slope and where it reaches a minimum
or maximum. Generally speaking, steepest increase and steepest
decrease of smooth functions are associated with an inflection point
and with zero second derivative, and maximum and minimum are
associated with zero first derivative. But restrictions apply. We
will treat this in more depth in section 4.1.
The distribution of scores is as follows:
Number Score
1 13
1 16
3 17
2 18
1 20
- On quiz 3, part 2a, I wanted you to say that f '(t) is negative
because the coffee is cooling.
As for units, the units for
functions (e.g., cm-given-seconds) is an invention of section 11
only, which I intend for you to use as an aid to calculate other
things. It will not be the correct answer on any well-formed
question on a coursewide exam and (for that reason) I will not make
it the correct final answer for any quizzes in section 11,
either. (If the midterm asks for something like the units of
velocity, you should interpret "velocity" as the "velocity at a
point", with units cm/sec or miles/hour or whatever; do not
interpret "velocity" as the "velocity function," which, in our
section only, might have units like (cm/sec)-given-sec or
(cm/sec)-given-cm. Added later: (cm/sec)-given-cm is probably
rare. Usually the velocity function would be something like
(cm/sec)-given-sec.)
If you write that f '(20) is the slope of the graph of f at 20
minutes or the slope of the line tangent to the graph of f, I gave
you credit, since this is technically correct. Please understand,
however, that the graders may consider the term "tangent" and the
like to be too technical, akin to saying "f '(20) represents the
derivative of f at time 20," and deduct points. Please use one of
the approved stories from Section 2.4 instead.
In some other cases, I awarded full credit but wrote the word
"borderline" to indicate that I guess there is a significant chance
another grader would deduct points. Please check.
The distribution of scores is as follows:
Number Score
1 3
7 4
4 5
7 6
5 7
6 8
- See handout for homework 3 for some specific items.
Here is another approach to Problem 1.7.22.b, using the verbal
definition of concavity.
Recall that, for the problem in question, the function is decreasing
for all x>5 and approaches a limit of 9. This means that f(x) is at
least 9 for all x>5. We want to show that the function can not be
concave down for all x>6.
Suppose the function were concave down for all x>6. This means it
is decreasing and concave down is decreasing at an increasing
rate, i.e., the slope becomes more and more negative. In
particular, this means that
0 < f(6) - f(7) < f(7) - f(8) < f(8) - f(9) < ...
We want to show that f(t) is less than 9 for some t. In fact, we can
assume t is an integer. Write
f(t) = f(6) - [f(6) - f(7)] - [f(7) - f(8)] - ... - [f(t - 1) - f(t)].
That is, f(t) is what you get if you start with a value of f(6) and
subtract off the appropriate value to get to f(7), then subtract the
appropriate value to get to f(8), etc. By concavity (above), all of
the jumps [f(7)-f(8)], [f(8)-f(9)], are greater than [f(6)-f(7)], so
we get
f(t) < f(6) - j*[f(6) - f(7)],
where j is the number of jumps, i.e., j = t - 6. If j > [f(6) -
9]/[f(6) - f(7)], then
f(t) < f(6) - j*[f(6) - f(7)] < 9,
violating one of the requirements. It follows that f must be concave
up somewhere.
-
The number of teams receiving each score is as follows.
Number Score
1 14
6 16
1 17
-
For the quiz of Jan 24,
the number of students receiving each score is as follows.
Number Score
1 3
3 4
4 5
8 6
10 7
4 8
1 9
1 10
Both of these questions are quite typical of what can arise on the
midterm. Make sure you know how to do these.
This quiz was hard, but there was no consistent misinterpretation of
anything. So I think the wording is appropriate.
In question 1, about the rational function, there was more than one
possible answer. The denominator needs to be a cubic with only one
root, at x = 0, and with leading term x^3 if the numerator has leading
term 3x^3. One possible denominator is x^3, but another is x(x^2 +
1). The question should have said "find a formula for a
function of this form" instead of "find a formula for this
function," but everyone seemed to interpret the question correctly.
- I will be out Wednesday night Jan 24 to Friday night. Office
hours Wednesday are as usual. Office hours Thurs are canceled
(please email for alternative appointments). There will be a
substitute teacher on Friday.
- Team Homework 2 went well except for the following items. (See
handout for more information):
- We need four conceptual
quantities in order to fix the four parameters A, B, C, D in A +
Bsin(C(t -D)). Everyone knows about period, amplitude, and
midline, but some of you had trouble with a fourth. A fourth
would be the "time of a maximum," "time of a minimum," or "time
of an upward zero crossing." Note that "time" here is a
horizontal position. Knowing the maximum value
gives no additional information beyond the midline and amplitude.
(An alternative approach is to find the horizontal and
vertical coordinates of a maximum and the next
minimum. Other approaches are possible.)
- Don't use the
expression "the cosine graph starts at the maximum." Instead,
say "the cosine template cos(t) has a maximum at t = 0." The
issue is that the term "start" is ambiguoous, because it could
refer to t=0 or to the minimum t-value in the domain.
- In the
last blank about interest, we want the accumulation rate in
dollars/second of the interest part of the debt. Exclude the
principle and focus on the current rate, since this
represents the consequence of government inaction.
This particular problem about interest rates is somewhat
specialized, but the idea to take the accumulation of debt in
dollars, averaged over a short period, is an extremely
important and central concept. It is the key to Chapter 2 and,
hence, to the entire class.
The number of teams receiving each score is as follows.
Number Score
1 16
2 17
3 18
2 19
- Sections 1.7 and 1.8:
| Section 1.7 |
Read pages 44 and 45 up to the Intermediate Value Theorem, but
skip IVT. Include the paragraph "Which Functions are Continuous?"
Make sure you understand Figures 1.69, 1.70, and 1.71. |
| Section 1.8 |
Read Examples 1 and 2. Skip page 50. Read from Theorem 1.2
to the end. |
You should know about the continuity and limits of polynomials,
rational functions, and piecewise-combinations (like the postal
function in Figure 1.71 and problem 1.7.20). For this course, you do
not need to know about the IVT or epsilon-delta definition. You
should probably look at Figure 1.82, but that's about as hard as it
should get.
On the other hand, some of you may want to go a little deeper here,
e.g., because it is relevant to other or future classes. If so,
please see me in office hours.
- Team Homework 1 went well, overall. Some of you may be surprised
by the lowness of the scores, but please note
that a score of 4/5, or 80%, means "basically correct." This does
not mean a grade of C+/B-. I am trying to be deliberately hard
on these homeworks to avoid unpleasant surprises on the midterms and
to get you to look at my comments and the handout. It is still way
too early to project what grade you will get in the class, but please
come to office hours if you want to talk about the kinds of things
that you find easy or difficult.
The number of teams receiving each score is as follows. (One team
has not yet submitted):
Number Score
1 14
1 16
6 18
- In class Jan 12 I said that the exponential function is concave up and
its inverse is concave down. That is a correct statement, but it is
not true in general that the inverse of a concave up function is
concave down. For example, if f(x)=1/x, then f is concave up for
positive x. But f is its own inverse, so f-inverse is also concave
up.
- Some notes on various coffee problems is available here. This includes a remark comparing the
coffee to the yams in the Team Homework.
- Quiz scores:
Problem 1(a) didn't count. The policy is: You should read the
text before coming to class. Do not wait for me to give an
overview before reading the text. Of course, you are welcome to
read the text again after I give an overview.
Each problem was worth 4pts and if you filled out the Student
Data Sheet on or before Jan 10, you got an extra 4 points.
The number of students receiving each score is as follows:
Number Score
1 0
1 4
1 6
2 10
1 12
11 14
15 16