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note: more information, with possible changes,
will be added as the exam dates near
rules: no calculators, notes, electronic devices (including
headphones) etc. are allowed. such items must be put away (e.g., in
a backpack, pocket, ...) during the exam period. scratch paper will be
provided for your convenience, but no work done on scratch paper will
be graded.
exam 1: fri sep 20 (2 hours)
topics: chapters 12-14 (excluding curvature, the chain rule and
lagrange multipliers).
in particular, here is a (not necessarily comprehensive) list of things
you should be able to do:
- write down equations for lines and planes in 3 dimensions
- determine the angle between two vectors, lines or planes; or
the angle between a line and a plane
- find a (possibly unit) normal vector to a plane through a given point,
or find a plane normal to a given vector which contains a given point
- understand parametric/vector equations for curves in 3 dimensions
- find a (possibly unit) tangent vector (or tangent line) to a curve at a given point
- find the arc length of a plane or space curve
- find/sketch domain, range, graphs for simple functions of 2
variables. also be able to find level curves
and cross-sections (traces).
- determine partial derivatives, gradients
of functions of 2 or
3 variables, and use this to write down equations for the tangent plane
of a graph z=f(x,y) at a point. you may need to use a tangent plane to do
linear approximation to approximate the value of a 2-variable function
at a point.
- compute directional derivatives of functions of 2 or 3 variables,
be able to determine the direction of fastest increase or decrease
of a function of 2 or 3 variables (e.g., convert the gradient to convert to a
unit vector)
- find local/absolute minima/maxima,
as well as saddle points, of functions of 2 variables. (think:
critical points and second derivatives test.) you may have to deal with
regions with boundary points.
here is a list of suggested practice problems.
try them before class thursday (sep 19), or at least determine which
ones you aren't sure how to do, and bring your questions to class
that day. my recommendation is that you try these on your own,
check your
answers, get help with what you don't understand, and then make sure
you can do these problems on your own before the exam. note: being able to
do problems on your own does not just mean without someone's help--it also
means without using other materials such as notes, the book, devices/internet, etc.
- ch 12 review
concept check: 1-6, 8,9, 11-18
true-false: 1-10, 15-20
exercises: 1, 6, 15, 17-19, 28-34, 37
- ch 13 review:
concept check: 1-3, 5, 8a
true-false: 1-4, 11, 14
exercises: 1, 3, 5, 8, 9, 17, 19
- ch 14 review:
concept check: 1-4, 5bc, 6, 7a, 8, 13-17
true-false: 4, 7, 9,
exercises: 1-5, 13, 19, 20, 25, 27, 33, 43-45, 47, 51, 52, 55, 63
in terms of balance, expect that roughly half of the exam will be on
chapters 12 and 13 and half will be on chapter 14.
in terms of types of questions, in addition to numerous specific problems
(similar to quiz problems and the above exercises), be prepared for
some conceptual questions. these may take the form of true-false,
explaining concepts/definitions or why something is or isn't true,
or providing examples of certain conecepts/properties (with a sketch,
with words or with an equation).
final exam: th oct 17 (2pm-4pm)
topics: chapters 12-16, excluding curvature (part of 13.3), 14.5, 14.8, 15.4 and 15.9. that said, most of the exam (perhaps 2/3 to 3/4) will focus on the
material not covered on the midterm exam, i.e., chapters 15 and 16.
in particular, in addition to the topics listed above for the midterm, you
should be comfortable with the following topics:
- compute double and triple integrals
in cartesian (rectangular) coordinates. this includes setting up, and
possibly changing the order of, iterated integrals.
- compute areas of regions in the plane and
volumes of regions in 3-space.
- be able to go back and forth between cartesian coordinates
and polar coordinates, or cylindrical and spherical
coordinates, including translating integrals between these
different coordinate systems.
- vector fields: be able to draw them (in R^2), determine if they are
conservative (in R^2 or R^3), compute curl and div (in
R^3), and know basic facts about div and curl
- line integrals: be able to compute the various kinds
of line integrals (ds, dx, dy, dz and vector field line integrals),
both directly as well as using the fundamental theorem and
green's theorem
- surface area: be able to compute surface area in the plane
with double integrals as in ch 15, with line integrals ds, or using
parametric surfaces as in sec 16.6.
- surface integrals (scalar fields): be able to parameterize
surfaces and compute surface integrals of functions
- surface integrals (vector fields): be able to compute
to surface integrals of vector fields/flux, both directly and with the
divergence theorem, and also use stokes'
theorem to compute a line integral in R^3 as a surface integral.
you should expect the format and length of the final exam to be similar to that
of the midterm (several true-false/conceptual/short answer questions, and
several problems which are more involved)
practice problems: you may review your midterm exam and the practice
problems for the midterm to review the material from chapters 12-14. however,
you should primarily focus on the material from chapters 15 and 16.
here are some suggested pratice problems for the latter:
- ch 15 review
concept check: 2, 3, 6, 9
true-false: 1-3, 7, 9
exercises: 10, 15, 19, 21, 28, 31, 37, 48, 53
- ch 16 review
concept check: 1, 2a, 5, 7, 8, 11bc, 13bc, 14-16
true-false: 1-8, 11-13
exercises: 1a, 2, 5, 10, 11, 17, 25, 27, 33, 34, 37
- individual section exercises
15.8: 23, 27
16.6: 21, 25, 41
caution: while there may be several ways to do a given problem on
the exam, one way may be much easier that others. similarly, there are
multiple ways to do the above pratice problems and going through all of the
practice problems does not necessarily mean you have covered all of the
topics listed above. so i recommend reviewing the above list of topics and
thinking about different ways you can approach the practice problems.
in addition, you may want to look in the book for other examples/practice
problem to work out for topics you feel uncertain about. and as always,
try to make sure by the end of your preparations that you can do these
problems on your own in reasonable amount of time.
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