Review Information for the Second Midterm Exam Spring '02

The next midterm will be on Tuesday, April 14th, from 7:30 to 9:30 pm in Science Center Hall D.

There will be a coursewide review held on Sunday, April 14th from 6 to 8pm, Science Center E - everyone is welcome to come.  Remember also to take advantage of the Math Question Center which meets from Sunday to Thursday from 8 to 10 pm in Loker.

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 will be covered on 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 (for instance we skipped section 12.9, and nothing will be required out of section 12.5 either).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, and nothing out of the new Rosner textbook for the Biochem section).

For the last midterm, it was unfortunately the case that the previous semester’s midterms were not as helpful in terms of providing practice problems for you, as the semesters covered material in slightly different order, and different topics were emphasized. This time around, the practice exams should be better matched for us.Also, we are in the process of putting together more practice problems for you to use to get ready for the midterm, and will post those over the next several days as well.

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. Try looking at 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.

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!

Enough talk about getting ready – time for a list!

Topics for second midterm (of course, 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)

• 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 covered on our midterm)
• Gradient Vectors
• Definition of gradient vector Ñf(x, y) =  <f_x, f_y> for two variable functions, Ñf(x, y, z) =  <f_x, f_y, f_z> for three, etc.
• Note that the graph of a two variable function lives in 3 space, but its gradient vector is a two-dimensional vector (i.e. lives in xy-plane)
• Understanding of significance of gradient vector – points 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 write tangent planes simply.
• 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
• Normal vectors for planes, use for finding angle between planes
• 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 (basically in addition to using second derivative test on critical points on the interior of the bounded region, on 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 definitions – analogy to single variable calculus of Riemann Sums
• Computation of double integrals over various regions through use of iterated integrals (when f is continuous over the region, i.e. Fubini’s Theorem)
• 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
• Redescribe 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 homework 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
• Redescribe 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
• Redescribe 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 | r_u x r_v | 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!)
• 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)

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Old Exams for practise:

• Here are lots(!) of past semester midterms that you can use for practice.  You should be able to do pretty much all of the questions on any of these midterms at this point.  Unlike the past semester midterms used for practice for our midterm number one, these past semester midterms appear better in line with the material we've covered so far this semester (i.e. these are pretty good practice).  However, please be aware that these are still from different semesters, so even if you can do all the problems on these midterms, you still need to go over the homework, class notes and textbook that we've used this semester to get fully ready for our midterm.
• We will begin posting solutions towards the end of the week as well – but obviously you should be making sure you do the problems on your own first before checking solutions!
• Midterm 2 Spring 1997
• Midterm 2 Spring 1997 solutions
• Midterm 2 Spring 1998
• Midterm 2 Spring 1998 solutions
• Midterm 2 Fall 1999 (with solutions)
• Midterm 2 Spring 2000
• Midterm 2 Spring 2000 solutions
• Midterm 2 Fall 2000
• Midterm 2 Fall 2000 solutions
• Midterm 2 Spring 2001
• Midterm 2 Spring 2001 solutions

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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. We will post solutions to these review problems at the end of the week as well, after you’ve had a chance to do as many on your own as possible.  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 #57 and 58
                       Solutions
• Chapter 12 review problems #3 - 34, 38(a), 39, 41, 42, 47 and 48 (on pages 911-913)
                       Solutions