Math 242C
Midterm Exam 2 Review Sheet
Practice exams
Here are some exams I have given in the past years covering roughly
the same material. There are some differences in both the syllabus and in
the notation and terminology, since some of the earlier exams
were based on a different text (by Stewart) or a different edition of the
Edwards/Penney text we are using now.
In terms of the difficulty level and nature of the problems, these
exams are pretty representative of what you can expect in our exam.
General Information
- Review session: I will hold an exam
review session Monday, October 23, 5 pm - 6 pm, in Room 245
Altgeld.
- Exam date and time:
The exam will be given during the regular class time,
Wednesday, 10/25/2006, 4 - 5 pm.
- Exam location: The location will be the same
as for the first midterm, 112 Greg Hall.
Greg Hall is located on the east side of Wright Street, near the
library.
Note: Seats will again be assigned, but in a more
streamlined (and faster) manner. In particular, if the exam room is
unoccupied during the 3 pm - 4 pm slot, we'll start passing out seat
assignments at around 3:45.
- Exam rules: Same as for the first midterm.
Academic honesty is expected, and cheating will not be tolerated.
Follow the Aggie
Code of Honor.
No earphones, ipods, blackberries, laptops, walkmans, and
similar devices. Student IDs may be checked.
- Calculator policy: As with all exams and quizzes in this class,
calculators are not allowed.
The problems will be such that a calculator
is not needed and would not be of any help.
- Books/notes policy: This is a closed books/notes exam.
Books, classnotes, formula sheets, etc., are not allowed, and must be
securely stowed away without easy reach (e.g., inside a closed backpack).
Exam content:
The exam will cover Sections 12.2 and 12.4 - 12.10,
except for for those parts
that have not been covered in class and explicitly marked
in the lecture summaries
as material that can be skipped. See below for a detailed syllabus.
The majority of the exam problems will be comparable in length and
difficulty to an average homework problem,
and fall into one of the types listed below under
"Typical Tasks."
Some problems will be "quickies", consisting
of a one line calculation involving a single formula, or simply the
recitation of a formula, possibly in multiple choice or true/false
format.
The practice exams above will give you a good idea of what to
expect.
Note: Quizzes serve a rather different purpose than exams
and homework
(mainly, as quick checks of basic formulas and concepts), and
most of the exam problems will be longer than a typical quiz problem
(an exception being the quickie type problems mentioned above,
which are rather like quiz problems).
Grading
Partial Credit:
"Quickie" type questions, and true/false or multiple choice questions,
are all-or-nothing
problems; you either know or don't know the relevant formula, and
giving partial credit for these problems would not make much sense.
For other problems, partial credit will be given if you
make significant progress towards the solution.
The problems will be largely independent of each other, so if you can't do one
problem, it will have no effect on the other problems.
Curve: I do not use a fixed grading scale (such as 90 points = A,
80 points = B, etc.), but rather set a curve after each exam and quiz,
depending
on the performance of the class. The curve used will be announced on the
course web page, and your computer grade reports will show both your raw
and curved scores.
Grading: The exam will be graded by the end of the week
and returned in the discussion sections on Tuesday following the exam.
Scores should be online by Saturday evening. Check the course webpage
periodically; I will make an announcement there when the scores are
online, and provide a link to access the online scores.
As a reminder, each of the three hour exams
counts 15% towards your course grade; the quizzes count 25%
(one drop score);
and the Final Exam counts 30%. See the Course Information Sheet for details on
the grading policy.
Missed Exam policy. If you miss the exam, but have a valid
excuse (e.g., illness), I will count the exam as excused. The absence
must be documented by a letter from the Dean's Office (300 Student Services
Building, 610 East John St.). If you know ahead of time of the
absence, let me know beforehand (email ajh@uiuc.edu), and see someone
in the Dean's office to explain the situation and arrange for a Dean's
letter. If the absence is due to unforeseen circumstances (such as
illness), get in touch with me and with the Dean as soon as possible
after the exam. A missed exam without a valid excuse counts as 0
points.
Tips on preparing for the exam
Concepts and Formulas
As a first step in preparing for the exam you should review the
concepts and formulas that we have discussed in class.
The following is a list of things that you should be familiar
with. Go through that list item by item; if you are
a bit fuzzy about a concept or unsure about a formula,
look it up in the appropriate
section and, if possible, do one or two of the examples in the book.
An excellent idea is to use the list below
to prepare a "cheat sheet" containing
all formulas you need to know for the exam.
Just looking up formulas in the book or in your lecture notes
and writing those formulas down on a sheet of paper
helps you committing those formulas to memory. Of course, you
shouldn't bring these formulas to the exam.
Also, make sure you review any quiz problems you may have gotten wrong.
The quiz problems are intended to probe your knowledge of basic concepts
and formulas; if you missed a quiz problem, that's an indication of
a serious gap in your preparation and you should make sure that you
close that gap. The penalty for a missed quiz problem is very mild, but
making the same error in an exam problem may cost you 10 - 20 percent of
the exam credit if you aren't able to get started on a problem because
you don't know the appropriate formula.
Section 12.2: Functions of several variables
- Function of two or more variables
- Graph of a function of two variables
- Level curves (for functions of two variables)
- Level surfaces (for functions of three variables)
Section 12.4: Partial derivatives
- Definition of partial derivatives, and notations
- Geometric interpretation
- Interpretation as rates of change
- Application: Computation of tangent planes
to surfaces of the form z = f(x,y)
- Higher order partial derivatives
Section 12.6: Differentials and linear approximation
- Differential of a function of several variables
- Linear approximation to a function of several variables
- Application to error estimates
Skip: The latter part of this section, from the bottom of p.894
through the end. This part contains the definition of a
gradient (which is covered again, and much more thoroughly, in 12.8),
as well as some theoretical material (definition of differentiability).
Section 12.7: The multivariable chain rule
- Dependency diagram
- Multivariable chain rule
- Implicit partial differentiation
Skip: Matrix form of chain rule and proof of chain rule
(p. 903 bottom through end of section).
Section 12.8: Directional derivatives and gradients
- Gradient of a function of several variables
- Geometric properties of the gradient
- Relation between the gradient and level curves/surfaces
- Directional derivative
- Rate-of-change interpretation of directional derivative
- Partial derivatives as directional derivatives
- Application: Computation of tangent planes
to surfaces of the form F(x,y,z)=k
Skip: Example 7 (Intersection of two surfaces)
Section 12.5: Maxima/minima of functions of several variables
- Local/global maxima/minima
- Critical points, first derivative test for functions of several
variables
Section 12.10: Critical points of functions of two variables
- Discriminant
- Second derivative test for functions of two variables
- Local maxima, minima, and saddle points for functions of two
variables
Section 12.9: Lagrange multipliers and optimization with constraints
- Method of Lagrange multipliers
- Application to optimization problems with constraints
Skip: Case of two (or more) constraints (p. 924 - end)
Typical computational tasks
- Computing partial derivatives of first and second order (12.4)
- Computing derivatives (partial and ordinary)
by implicit differentiation (12.7)
- Computing tangent plane to surfaces of the form z = f(x,y) (12.4)
- Using differentials to compute approximate values of
a given function f(x,y) near a given point (a,b) (12.4)
- Using differentials to estimate the effect of small changes in the
variables to the value of a function (12.6)
- Computing derivatives via the multi-variable chain rule (12.7)
- Computing gradients and directional derivatives (12.8)
- Computing tangent planes to surfaces of the form F(x,y,z)=k (12.8)
- Finding local/global maxima/minima of
a function of several variables using the first derivative test
(12.5)
- Classifying critical points using the second derivative test (12.10)
- Finding maxima/minima with constraints by Lagrange multiplier
method (12.9)
- Application to optimization problems (12.5, 12.9)
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