Math 241 F1H
Midterm Exam 2 Review Sheet
Practice exams
Below are links to exams I have given in the past years covering the
exam material. Chapter 12 is covered in Exams 2 below (note that most of
this chapter was on the syllabus for the first midterm, and so won't be
on this exam's syllabus); Chapter 13 is covered in Exams 3 below (note
that most of these exams did not cover Section 13.9, which is on our exam
syllabus).
General Information
- Special Open House: In addition to the regular Wednesday Open House
(5 - 6 pm, 241 Altgeld), I will hold a special Open House a few days
before the exam, Sunday, 10/21, 5 pm, in 241 Altgeld.)
- Exam date, time, and location:
The exam will be given during the regular class time,
Wednesday, 10/24/2007, 2 - 2:50 pm, in the usual room, 142 Henry.
- Exam rules: No calculators,
closed books/notes, no formula sheets, and,
of course, no cheating. The problems in the exam will be such that a
calculator is not needed and would not be of any help.
- Missed Exam policy. I don't give make-up exams, but
if you miss the exam with a valid
excuse (e.g., illness), the exam will be counted as "excused". (See the
Course Information Sheet for an
explanation of an "excused" exam score.)
The absence
must be documented by a letter from the Dean's Office at
300 Student Services Building, 610 East John St., phone 333-0050;
other documentation (e.g., note from McKinley) is not sufficient.
Exam content
The exam will cover Sections 12.5, 12.9, 12.10 (i.e., the sections
from Chapter 12 that didn't make it into the first midterm), Section
11.8 (cylindrical and spherical coordinates), and all of
Chapter 13 with the exception of 13.8 (surface area).
The majority of the exam problems will be comparable to an average
homework problem and fall into one of the types listed below under
"Typical Tasks" (or possibly a combination of these). For multiple
integral problems I may only ask for the set-up, i.e., an expression as
an iterated integral, with explicit limits on each integration sign.
Detailed Exam syllabus
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)
Section 13.1:
Double integrals
- Computation of double integrals over rectangular regions
Skip: There won't be any exam problems on Riemann sums
(p. 942/943), or the relation between double integrals and
cross sections (p. 946 middle through end of section).
Section 13.2:
Double integrals over general regions
- Computation of double integrals over general regions R
- Vertically simple region (constant limits on x, variable limits
on y)
- Horizontally simple region (constant limits on y, variable limits
on x)
- Reversing order of integration
Section 13.3:
Area and volume by double integrals
- Computation of volume via double integrals
- Computation of area via double integrals
Section 13.4:
Double integrals in polar coordinates
- Area element dA in polar coordinates
- Computing double integrals in polar coordinates
- Converting double integrals from rectangular coordinates
to polar coordinates
Section 13.5:
Applications of double integrals
- Mass, mass density, mass element dm
- Centroid (center of mass)
- Moments of inertia
Ix,
Iy,
I0 (polar moment of inertia)
Skip: First and second theorem of Pappus
(p. 973 - 975),
formulas for kinetic energy and radius of gyration ((9), (10), (11)).
Section 11.8 Cylindrical and spherical coordinates
- Definition, geometric interpretation
- Conversion formulas between cylindrical, spherical, and
rectangular coordinates
- Conversion of equations in rectangular coordinates to cylindrical
and spherical coordinates
- Surfaces and regions in cylindrical and spherical coordinates:
cylinders, spheres, cones, planes, half-planes
Skip:
Last part of 11.8 (p.844 through end), on longitudes and latitudes.
Sections 13.6/13.7:
Triple integrals
- Triple integrals in rectangular coordinates
- Triple integrals in cylindrical/spherical coordinates
- Volume by triple integrals
- Mass, mass density
- Center of mass (centroid)
- Moments of inertia Ix, Iy, Iz
Section 13.9:
Change of variables in multiple integrals
- Jacobian of a transformation
- Change of variables formula for double and triple integrals
(formulas (7) and (11))
Typical computational tasks
- 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)
- Setting up a double integral as iterated integrals (13.2)
- Evaluation of
iterated integrals by reversing order of integration (13.2)
- Areas by double integrals (13.3)
- Volumes by double integrals (13.3)
- Double integrals in polar coordinates (13.4)
- Volumes by triple integrals (13.6, 13.7)
- Mass, center of mass (centroid), and moments of inertia,
of a lamina (13.5)
- Mass, center of mass, and moments of inertia,
of a solid object (13.6, 13.7)
- Conversion between rectangular, cylindrical, and spherical
coordinates (11.8)
- Triple integrals in cylindrical and spherical coordinates (13.7)
- Computing the Jacobian determinant of a transformation (13.9)
- Changing variables in multiple integrals (13.9)
Miscellaneous hints and comments
- Computing multiple integrals:
The most difficult part in many of these problems is
setting up a double or triple integral
as an iterated integral with appropriate limits on each of
the individual integrals. To this end, you need to express the domain
R of integration in terms of inequalities on the variables x, y
(and z in the case of 3-dimensional integrals). Don't
try to derive these inequalities in the abstract,
but sketch the domain R
and ``read off'' the appropriate inequalities from the picture.
For integrals over rotationally symmetric objects (cones, etc.),
a z-r plot (depicting a vertical cross section of the object)
is often helpful. Last, but not least,
remember the number one rule for setting up multiple integrals:
The limits in the outer integral must be constant, and those
in the middle integral can only involve the outer variable.
- Choosing appropriate coordinates:
Sometimes you don't have a choice. A problem may specifically ask you
to do an integral using cylindrical coordinates
or spherical coordinates
(or do it both ways!). At other times, a problem may include a hint as
to what coordinate system to use. However, if you are faced with integral
and have to decide on a coordinate system, here are some general
guidelines: First, the shape of the region of
integration often makes the choice obvious.
If the integral is over a spherical region,
try to use spherical coordinates. When the integral is over
a rectangular or triangular region, use rectangular coordinates.
For cone shaped regions, either spherical or cylindrical coordinates
should be used; which one depends on the particular case.
Second, look for some tell-tale signs in the function to be
integrated: a term x2 + y2 suggests using polar
(in 2 dimensions) or cylindrical (in 3 dimensions) coordinates;
a term x2+y2+z2 suggests using
spherical coordinates.
- Integration techniques:
You should know how to integrate by substitution and by parts.
(Hint: When doing an integration by substitution,
make sure that you also substitute the limits of
integration.) Some trig integrals (such as the integral of (cos
x)3 (sin x)
can be done
with a simple trig substitution (u = cos x in this case), and you should be
able to do such integrals.
However, you are not expected to be able to compute
more esoteric integrals such as the integral of (sin x)3,
or the integral of Squareroot[1-x2]; if you run into one of
these integrals, it is an indication that you
did something wrong and should try a different approach
(for example, changing to different coordinates, or reversing the order
of integration).
You should know how to handle integrals over
sin2x and
cos2x. (Hint: in each case there is a trig identity that
converts the expression into a simple sine or cosine (without the
square).)
- Trig functions: When evaluating an integral
in polar, cylindrical, or spherical coordinates, you may need to
kow some special values of sine or cosine. Aside from the values at
multiples of pi/2 (which are always 0, 1, or -1),
you should know the values of sine and cosine
at pi/6, pi/4, pi/3 (corresponding to angles of 30,
45 and 60 degrees): at these three angles the sine
increases from 1/2 to Squareroot[2]/2 (or
1/Squareroot[2]) to Squareroot[3]/2, while the cosine goes through the same
three values in reverse order. From the values at pi/6,
pi/4 and pi/3, you can derive values at angles 2 pi/3, 3 pi/4,
etc., using standard identities for trig functions.
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