The first 21a midterm will be on Thursday, November 4th, from 7:30 to 9:30 pm in Science Center Hall A (Andy's MWF 10 and 11 sections), Sever room 113 (Tues/Thurs sections) or in Science Center Hall D (all other students).

Midterm 1 Solutions

If you missed the coursewide review on Monday, you can view it here: Monday Review Video (or for the low bandwidth version go to at Monday Review Video (56k)). Remember also to take advantage of the Math Question Center which meets from Sunday to Thursday from 8 to 10 pm in Loker.

Please find below a pretty exhaustive list of what we have covered up to this point.  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 11.4 on tangent planes - it will not cover anything after that (i.e. it will not include sections from sections 11.5 , 11.6 and 11.7 of chapter 11, even though we are going over them right now in class).  Also be sure to read through the list of topics that will not be included in this first midterm (these are located at the bottom of the list).

Topics for first midterm:

• Basic definitions related to multidimensional spaces
• Coordinate systems: rectangular, cylindrical, spherical - their use, conversion between systems
• Distance formulas (between points, ability to compute distance between points and lines, points and planes, using scalar projections)
• Standard basis vectors, i, j and k and their use
• Basic vector definitions, operations:
• Notation:  <x, y, z> =  x i + y j + z k
• Addition/subtraction, finding magnitude of vectors, finding unit vectors
• Dot products and their geometric significance (i.e. a · b = |a| |b| cosine(angle between a and b), finding magnitude of a vector in terms of dot products)
• Scalar and vector projections - computation, understanding, use in creating distance formulas
• Cross products - computation and geometric significance (no scalar triple products on midterm)
• Equations of lines and planes -
• For lines: vector and parametric equations  (no symmetric equations on page 677 section 9.5)
• For planes: vector equations, scalar equations, linear equations
• Normal vectors for planes, use for finding angle between planes
• Vector Functions
• Use in finding parametric equations for space curves
• Ability to find and identify basic space curves and their parametrizations - circles, ellipses, helixes, parametrization of intersections (such as in example 5 on page 707)
• Differentiation of vector functions:
• Computation, knowledge/use of differentiation rules (Theorem 3 page 714)
• You don't need to study Integration of Vector Functions (page 715)
• Finding tangent vectors for space curves, equations for tangent lines
• Computing Arc Length for space curves
• No parametrizing space curves with respect to arc length (bottom of page 718, section 10.3)
• No Curvature computations (in section 10.3)
• No Normal and Binormal vectors (in section 10.3)
• Surfaces
• Recognition of equations and traces for basic quadric surfaces: ellipsoids, elliptic paraboloids, cones and hyperbolic paraboloids (all on page 691)
• Use of parametric equations to identify surfaces, how to parametrize surfaces of rotation (both in section 10.5)
• Multivariable functions
• Depiction through use of graphs, level curves
• Knowledge of basic examples: linear functions, cones, paraboloids, parabolic cylinder (page 687)
• Partial derivatives:
• Computation (use of limit definition, page 768 (box 4), when necessary, such as for 11.3 problem  #78)
• Geometric significance (i.e. rate of change of function in x, y directions)
• 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: f_xy = f_yx if both are continuous, and so f_xyx = f_xxy if both are continuous, etc.)
• Skip partial differential equations section and the Cobb-Douglas Production Function (in 11.3) for now.
• Tangent Planes - finding equations for, using partial derivatives
• Use of tangent plane for approximating differentiable functions
• Knowledge of the meaning of 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)

• Note there are a number of topics in the textbook that were not necessarily covered in class, and which will not be covered on the exam:
• No scalar triple products (in section 9.4)
• No symmetric equations (in section 9.5)
• No integrals of vector functions (in section 10.2)
• Nothing  in section 10.3 after page 718 (starting with parametrizing space curves by arc length at bottom of 718)
• Nothing in section 10.4 (skipped completely)
• Nothing on partial differential equations from section 11.3
• Nothing on differentials from section 11.4
• Also, to save you a bit of worry, we covered the following but they will not be tested on the midterm:
• Section 11.2 on Limits and Continuity

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Old Exams for practise:

• Note that although the class is similar from year to year, there are differences in terms of the timing of  what was covered during the semester.  For instance on the spring 2001 mid-term, you don't need to know how to do #4 or #5 (at least not parts c and d for #5) at this point since we haven't covered those topics yet.  In addition to using the following practice tests, you should really go through your old homework problems - they will give you more of a sense of what you need to know for this semester's midterms.
• Although the solutions are posted already, do be sure to try to do these midterms first on your own!
• Midterm practice exam l  (Solutions)
• Midterm practice exam 2  (Solutions)  (note answer to 5a should be x - 5z = 0)
• Midterm 1 Spring 2000  (Solutions)
• Midterm 1 Spring 2001  (skip #4 and #5)  (Solutions)
• Midterm 1 Fall 2001  (Solutions)
• Midterm 1 Spring 2002 (with solutions)
• Midterm 1 Fall 2003 (Solutions)

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Answers to Review Problems for Chapters 9, 10 and 11 from our textbook:

• Since what we've covered is a bit different from what other semesters have covered in the past, another good way to get ready for the midterm is to do the review problems at the end of each chapter that we've covered.  Posted below are the answers to these review problems.  By doing these review problems, you'll be able to get practise for our midterm that's more specifically geared to what we've covered so far this semester.
• Note that certain problems (such as #10 in review for chapter 9) are on topics we have specifically excluded from the midterm (see list of such topics above), so there are some problems that you shouldn't expect to be able to do - you should be able to figure out which ones these are by checking the list of topics covered/not covered.
• Since we've only covered up through 11.4 in chapter 11, don't be concerned if you can't answer the review questions in the Chapter 11 Review that cover material from sections 11.5, 11.6 and 11.7 at this point!
• Answers to Chapter 9 Review questions
• Answers to Chapter 10 Review questions
• Answers to Chapter 11 Review questions