Review Information for the Second Midterm Exam Fall 2003

The next midterm will be on Wednesday, November 19th, from 7:30 to 9:30 pm.  MWF sections go to Science Center Hall C.  Tu/Thurs sections and Teru Yoshida's MWF section go to Emerson room 105.

There will be a coursewide review held on Monday, November 17th from 4 to 6:30pm, Science Center Hall B - everyone is welcome to come.  We will have the session videotaped in case you can't make it (it will be on reserve in Cabot Science Library in the Science Center)

Suggested solutions for the second midterm are here (pdf).

Solutions for unfinished problems from the review are here (MS Word) (html).

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.  On the midterm you should be prepared to answer questions from any of these topics.  Note that this midterm covers material just up through the end of chapter 11 - nothing from chapter 12.

Towards the end of this page, please find midterms from previous semesters to use for practice problems.  Note that you the old midterms do not line up perfectly with the schedule of topics covered this semester, so you should check to see if any questions aren't useful to do as practice (read the notes posted with the old midterms carefully - they'll let you know which questions to skip off the old midterms).

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 (you should be able to handle concepts covered in the first part of the semester as well, as the current topics depend on understanding the previous ones, but the emphasis in terms of the questions will be on the following)

Sections 10.5 through 11.8

• Surfaces
• Parametric equations for surfaces - understanding of grid curves
• Knowledge of basic examples of surfaces such as cylinders, spheres, paraboloids, etc.
• Finding parametrizations, such as parametric equations for surfaces of rotation (page 740, section 10.5)
• Note, don't overlook the power of the simplest parametrization <x, y, f(x,y)> if you're dealing with the surface of the graph of a function f(x, y)
• Multivariable functions
• Depiction through use of graphs, level curves
• Knowledge of basic examples: linear functions, cones, paraboloids, parabolic cylinder (look back to page 687)
• Partial derivatives:
• Computation (use of analytic definition, in (4) page 768, when necessary)
• Geometric significance (rate of change of function in x, y directions, respectively)
• Skip implicit differentiation (page 771 section 11.3)
• Notation, computation of higher partial derivatives (page 772) and knowledge of Clairaut's Theorem (on equivalence of mixed partials)
• Skip partial differential equations section and the Cobb-Douglas Production Function (in 11.3)
• Tangent Planes - finding equations for them using partial derivatives
• Use of tangent plane for approximating differentiable functions
• Knowledge of what it means to be a differentiable function: one whose linear approximation given by the tangent plane is a good approximation (compare to the z = 0 tangent plane approximation for the function given in figure 4, page 782) (you do not need to know the full blown definition given in (7), bottom of page 782)
• Skip differentials (in section 11.4, page 784 through 786)
• Skip tangent planes to parametric surfaces (in section 11.4)
• 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 grad[f(x, y)] =  <f_x, f_y> for two variable functions, and similarly for higher dimensional analogs (e.g. grad[f(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 two-dimensional 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 (D_u f(x,y) = grad f dot u)
• Understanding – the directional derivative 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 the Lagrange Multiplier method and examples of its uses in the one constraint situation
• Skip two constraint Lagrange Multiplier problems (page 826) - this was covered in some sections, but won't be required as a topic for the midterm
• Note there are several topics in the textbook that will not be covered on this midterm:
• No implicit differentiation (in section 11.3)
• No partial differential equations, Cobb-Douglas Production Function (in 11.3)
• Skip the two constraint Lagrange Multiplier problems (in section 11.8)
• Also, to save you a bit of worry, the following topics that we did cover will not be tested on the midterm:
• Section 11.2 on Limits and Continuity
• Finding tangent planes for parametric surfaces (in section 11.4)
• Two constraint Lagrange Multiplier method (page 826)

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Old Exams for practise:

• Here are lots(!) of past semester midterms that you can use for practice.  Note that the schedule for topics and midterm timing varies semester to semester, so there are a number of problems that cover topics we haven't covered yet.  Be careful to read the notes as to which problems you can skip off each test.  Also, please be aware that it's still very important to go over the homework, class notes and textbook that we've used this semester to get fully ready for our midterm.
• Midterm 2 Fall 1999 (with solutions) (skip problems 6 and 7)
• Midterm 2 Spring 2000 (skip problem 6)
• Midterm 2 Spring 2000 solutions
• Midterm 2 Fall 2000 (skip problems 3 and 5)
• Midterm 2 Fall 2000 solutions
• Midterm 2 Spring 2001 (skip problems 5, 6 and 7)
• Midterm 2 Spring 2001 solutions
• Midterm 2 Spring 2002 (skip problems 2, 3 and 5)
• Midterm 2 Spring 2002 solutions
• Midterm 2 Fall 2002 (skip the two integral problem T/F on problem 1, and skip problems 3 and 8)
• Midterm 2 Fall 2002 solutions

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Review Problems for Chapters 10 and 11 from our textbook

• The review problems at the end of chapter 11 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 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.
• Chapter 10 review problems (page 743) - you can do all the problems for general review (skipping 7, 12-15 and 24 on curvature) , but for 10.5 specifically, there is just problem 21 (and 22 in Mathematica if you'd like)
• For Chapter 11 (review problems on page 833) you can go through almost all the problems (skipping #9 and 10 on limits, #32 and 34 on differentials, #42 on implicit differentiation and #66 on kinetic energy, and the computer problems, #30, 57 and 58)
• Answers to Chapter 10 Review questions
• Answers to Chapter 11 Review questions