Review Information for the Second Midterm Exam Fall '04
The next midterm will be on Tuesday, December 7th, from 8:00 to 10:00
pm in Science Center Halls A and B (students in Andy's sections please
go to Hall A, and all others go to Hall B).
As was the case on the first midterm, there are
to be no calculators or notes allowed during the midterm.
There will be a coursewide review held on Friday, December 3rd from 2 to
3:30 in Science Center Hall C (if you weren't able to make it please see
it online at Coursewide
Review for Midterm 2).
Midterm 2 solutions
On this second midterm the emphasis will be on the material covered
since the last midterm. Below please find a list to help you work
out what to study for this midterm – the list covers both the topics that
will be on the test, as well as the ones in the textbook that you are not
responsible for studying (for instance we skipped sections 12.5 and 12.9,
so you don't need to go through either of those sections).On
the midterm you should be prepared to answer questions from any of these
topics. Note that this midterm covers material just up through
section 12.8 on Triple Integrals in Cylindrical and Spherical coordinates
- it will not cover anything after that (i.e. nothing out of Chapter 13
on Vector Calculus).
As before, to get ready for a math midterm be sure to spend as much
time as possible practicing doing math – go back to your homework sets
and try redoing problems pulled out at random.Read
through the textbook and try to do the examples in the sections without
looking at the answers (get a copy of the Student Solutions Manual that
gives worked out solutions for all the odd numbered problems).. Try
going through the review problems at the end of the chapters for more practice
problems.Spend time going over the
practice exams that are being posted on this website.Also
stop by and talk to your TF or CA about the test – they should be able
to quiz you a bit to help you get ready as well.
Probably the easiest way to lull yourself into feeling confident about
your test prep when you perhaps should be doing more to get ready, is to
just reread the textbook – yes you will probably convince yourself that
you are familiar with the topics covered, but if you don’t spend time actually
doing practice problems then you really are not as ready as you could be,
no matter how much time you've put into rereading the textbook.
Finally, if you did well on the first midterm, then great, but make
sure you take this next one seriously as well – do the same prep as you
did for the last test and things should go well.If
it didn’t go too well on the last midterm, then make sure you think hard
about changing your approach to getting ready – talk to your TF or CA about
how you’re preparing, and they should be able to give you some coaching
advice to help you with your studying.Also,
remember, if you’re able to get yourself in good shape by the final exam,
then the final grade can be used to wipe out a lower midterm score!
Topics for second midterm (you should be able to handle concepts
covered in the first part of the semester as well, as the current topics
necessarily depend on understanding the previous ones, but the emphasis
in terms of the questions will be on the following)
Sections 11.5 through 12.8 (skipping 12.5 and 12.9)
-
The Chain Rule
-
Its use and computation in general multivariable function settings
-
Applications of chain rule for calculating derivatives along parametrized
curves
-
Skip Implicit Differentiation (a useful trick, but it won’t be explicitly
covered on our midterm)
-
Gradient Vectors
-
Definition of gradient vector Ñf(x,
y) = <fx, fy> for two variable functions, Ñf(x,
y, z) = <fx, fy, fz> for three,
etc.
-
Note that the graph of a two variable function lives in 3 space, but its
gradient vectors are two-dimensional vectors (i.e. live in xy-plane)
-
Understanding of significance of gradient vectors – point in direction
of maximal increase of function, magnitude equals rate of change of function
in that direction.
-
For two variable functions the gradient vector at any particular point
(when it is nonzero) is perpendicular to the level curve of the function
through that point.
-
For three variable functions the gradient vector at any particular point
(when it is nonzero) is perpendicular to the level surface of the function
through that point, and so gradient vectors are normal to tangent planes
to level surfaces – giving another way to find tangent planes.
-
Directional Derivatives
-
Computation through dot product of appropriate unit vector (note that it
must be a unit vector!!) with gradient vector
-
Understanding – this gives the rate of change of the function in the unit
vector’s direction – so the directional derivative is a scalar quantity,
not a vector
-
Maximum and Minimum Values
-
Definition of critical points as points where gradient vector is zero,
or if one of the partial derivatives does not exist
-
Max/min values of functions occur only at critical points (or on boundaries
if those are given)
-
Knowledge, understanding of second derivative test for finding local max/mins
and saddlepoints
-
Finding absolute max/mins for functions on bounded regions (in addition
to using the second derivative test on critical points on the interior
of the bounded region, one needs to also determine max and min values of
the function on the boundary – check example 7 on page 817)
-
Lagrange Multipliers
-
Knowledge of Lagrange Multiplier Method and examples of its uses in the
one constraint situation
-
Skip two constraint Lagrange Multiplier problems (page 826)
-
Double Integrals
-
Understanding of basic concept behind double integral definition – analogous
to that of single variable calculus with Riemann Sums
-
Computation of double integrals over various regions through use of iterated
integrals
-
Ability to find limits of integration for various regions (should be able
to switch order of integration through recalculating limits - see example
5 on page 859)
-
Understanding of double integral as giving volume over a region, under
a function of two variables (when the function is positive, in any case. When
a function is negative, the double integral will produce a negative result,
just as in the single variable case)
-
Calculating double integrals with polar coordinates:
-
Sub in x = rcosq,
y =rsinq
into whatever function f(x,y) is in the integrand
-
Replace dA (which was equal to dxdy in rectangular/cartesian coordinates)
with rdrdq
-
Describe the region in terms of the polar coordinates r and q
-
Skip section 12.5 (center of mass, moments of inertia, probability and
expected values – again, useful to know about, but won’t be tested on our
midterm)
-
Triple Integrals
-
Again, a basic understanding of the concept behind the Riemann Sum definition
-
Fubini’s Theorem for computation of triple integrals over various volume
regions (so for triple integrals now the region of integration is a volume
in space, not an area in the xy-plane)
-
Ability to find limits of integration for various regions (again, should
be able to switch order of integration through recalculating limits - see
problem 30 on page 892 for a good example)
-
Calculating triple integrals with cylindrical coordinates (basically same
maneuvers as with polar coordinate conversion, with the additional z coordinate):
-
Sub in x = rcosq,
y =rsinq,
z=z (!) into whatever function f(x,y,z) is in the integrand
-
Replace dV (which was equal to dxdydz in rectangular/cartesian coordinates)
with rdrdqdz
-
Describe the region in terms of the cylindrical coordinates r, q,
and z.
-
Calculating triple integrals with spherical coordinates
-
Sub in x = rcosqsinf,
y =rsinqsinf,
z=rcosf,
into whatever function f(x,y,z) is in the integrand
-
Replace dV (which was equal to dxdydz in rectangular/cartesian coordinates)
with r2sinfdrdqdf
-
Describe the region in terms of the cylindrical coordinates r, q,
and f.
-
Surface Area computations
-
Be able to use the formula for surface area for a parametrized surface
r(u,v):
-
A(S) = òò|
ru
x rv | dudv
-
Note that there will be times when you need to come up with the parametrization
(such as in problem 22 in section 12.6), and other times when you will
be given the parametrization (as in problem 8 in section 12.6) – be prepared
to be able to deal with either one!
-
Partial Differential Equations - don't sweat this topic - there might be
one small question on this material, but I will tell you exactly what to
expect at the coursewide review (it will be on the order of "Check that
this function is a solution to a particular PDE with the following initial
conditions..." or something around that level of difficulty)
-
Note there are several topics in the textbook that will not be covered
on this midterm:
-
No implicit differentiation (in section 11.5)
-
Skip the two constraint Lagrange Multiplier problems (in section 11.8)
-
Skip the integration applications in section 12.5
-
Skip the several applications of triple integrals in section 12.8 (bottom
of page 888 through page 890)
Old Exams for practise:
Review Problems for Chapters 11 and 12 from
our textbook
-
The review problems at the end of chapters 11 and 12 also give a wealth
of practice problems to work on. These review problems are good for
going through the topics that we've covered, but you should note that they
are also pretty straightforward applications. To get ready for the
midterm, you should use the other practice midterms from the section above
this one as a better gauge of the level of difficulty.
-
To review for the test, the following lists of problems give the relevant
problems to go over from chapters 11 and 12:
-
Chapter 11 review problems #35 through 65 inclusive (pages 834-835), but
skip #42, 57 and 58
-
Chapter 12 review problems #3 - 34, 38(a), 39, 41, 42, 47 and 48 (on pages
911-913)