"Anything's fair game" is a mouthful for this course. So what should you study? An easy answer is "what's on the final." As a matter of fact, that isn't as facetious as it sounds. You can pick up some pretty good clues as to what's expected of you by looking at old finals. Towards the end of the semester it is even possible to obtain a copy of several old unit finals from the University Book Store for a small price, compliments of the math department. You should hustle over and buy copies as soon as they become available.

In the interests of helping you sort out priorities, I did a little research on some old unit finals of which I happen to have copies. Since these are a matter of public record, I can provide information for you about their contents. These are the Fall '92, '93, '97, '01, '02, '03, '04, '05, '06, '07 and '08, along with the Spring '01, '03, '04, '05, '07, '08 and '09 exams. Below is a an outline of the sections of the text (formerly Smith and Minton, Weir and Thomas since 2007) that we cover in Math 208, together with a classification of all the problems that I found on these unit exams according to which section they belong. I listed both the frequency of questions on this topic in parentheses and the topic. Some problems could fit into two sections. In such cases I chose the predominant section unless it was very evenly split, in which case I split the question. This should give you a pretty good idea of what's important. Note that since we have switched texts and changed the syllabus, the problems from Chapter 11 are no longer relevant to Math 208 since this chapter is covered in Math 107. Also note that I'm counting the number of problems, not the total points per question, though these numbers correlate reasonably well. ¹

- (4) Do dot, component and/or angle and length calculations on given vectors.
- (1) Express a vector as a sum of a vector parallel and one perpendicular to another given vector.
- (2) Find projection and orth of one vector along another.
- (2.5) Find angle between two vectors.
- (0.5) Find value of a parameter in a vector making it orthogonal to another vector.

- (2) Find area of a parallelogram.
- (1) Find area of triangle given three vertices.

- (1.5) Find parametric equations of line
- (1) Determine if planes are parallel or perpendicular.
- (2) Find equation of plane through given point and parallel to two given vectors.
- (0.5) Find plane perpendicular to a line and through a given point.
- (2.5) Find equation of a plane given three points on it.
- (1.5) Find a normal vector to a plane.
- (2) Find vector parallel to intersection of two planes.
- (1) Find equation of a linear function given a table of data about it.
- (1) Find equation of a plane through a given point and perpendicular to two other given planes.

- (1) Identify a quadric surface by its equation.

- (3) Find velocity, acceleration, speed of an object on given curve, given position vector.
- (4.5) Find position vector, given acceleration and initial data.
- (1.5) Find tangent and normal vector to a curve of motion, and tangential component of acceleration.
- (1) Find points in a motion where a particle has vertical velocity.

- (1) Sketch level curves of a surface g(x,y,z)=0.
- (1) Draw a contour diagram for a function z=f(x,y), including certain contours.
- (2) Use a contour graph to estimate directional derivatives and extrema of z=f(x,y) along a curve.

- (1) Determine points of continuity of a function.
- (1) Either calculate a limit of an expression in x and y as x and y tend to zero or show that the limit doesn't exist.
- (1) Show a two variable limit does not exist by drawing level curves.

- (2) Calculate (and possibly interpret) partials.

- (9.5) Use (multivariate) chain rule given intermediate variables in terms of independent.
- (2) Word problem to find a rate of change of a function of two variables given data by way of chain rule.
- (5.5) Use chain rule to find a partial, given values of intermediate variables and various partials at a point.

- (3.5) Use gradient to find a normal vector and/or equation of tangent plane on surface g(x,y,z)=0.
- (11.5) Calculate gradients, directional derivatives and direction of maximum (minimum) change.
- (1) Find gradient at a point, given directional derivative data.

- (5.5) Use differentials (or linear approximation) to approximate a nearby value of a function.
- (0.5) Compute the differential of a function.
- (6) Use gradient to find a tangent plane to graph of function z=f(x,y).
- (0.5) Find parametric equations for the line normal to graph of function z=f(x,y) at a given point.

- (14.5) Find and classify critical points of function f(x,y).
- (2) Find and classify critical points of function f(x,y), including the possibility of (absolute) maximum and minimum values.
- (0.5) Determine if a function has a maximum value on a given (closed, bounded) domain.
- (1) Find absolute extrema of a function on a disk.

- (4.5) Use Lagrange multipliers to find minimum distance from a curve g(x,y)=0 to a point.
- (1) Use Lagrange multipliers to find dimensions of a box maximizing volume under constraints.
- (10) Use Lagrange multipliers to find maximum (or minimum) value of a given function subject to a constraint.

- (2) Approximate a double integral over a rectangle by a Riemann sum.

- (1) Evaluate a double integral (unstated: needs reversal of order).
- (15) Evaluate a double integral by reversing the order of integration (maybe sketch region of integration).

- (3) Sketch and calculate a plane area.

- (1) Evaluate a double integral of specified region using polar coordinates.
- (1) Set up a double integral of specified region using polar coordinates.
- (3) Convert an iterated integral in rectangular coordinates to one in polar coordinates.
- (2) Evaluate an iterated integral in rectangular coordinates by converting it to polar coordinates (or vice-versa).
- (1) Set up an integral in polar coordinates for a moment of inertia.

- (6) Set up a triple integral over a region Q bounded by surfaces as an iterated integral.
- (1) Set up a triple integral for moment of inertia of a solid about an axis as iterated integral in specified order of differentials.
- (1) Evaluate an iterated triple integral.
- (2) Set up iterated triple integrals for mass of a solid and express the first coordinate of the center of mass of the solid in terms of triple iterated integrals.
- (1) Set up iterated triple integrals for mass of a solid in a specified order in rectangular coordinates.
- (2) Set up and evaluate an iterated triple integrals for mass of a solid.

- (2) Set up integrals for center of mass of lamina bounded by curves.
- (2) Set up and compute integral for moment of inertia of a solid.

- (4) Convert an iterated integral in rectangular coordinates to one in cylindrical coordinates.
- (1) Set up a triple integral over a region Q bounded by surfaces as an iterated integral in cylindrical coordinates.
- (1) Set up and solve a triple integral over a region Q bounded by surfaces as an iterated integral in cylindrical coordinates.
- (5) Use an iterated integral in spherical coordinates to find mass of a solid bounded by surfaces and with given density function.
- (2) Use spherical coordinates to compute an integral over a given solid.
- (3) Convert an iterated integral in rectangular coordinates to one in spherical coordinates (and possibly evaluate).
- (3) Sketch a solid and express a triple integral over it as an iterated integral in spherical coordinates.

- (1) Match a vector field and its graph.

- (12.5) Evaluate a line integral along a specified path given parametrically (possibly first finding a parametric representation for the curve).
- (1) Interpret line integral
**F**·d**r**(e.g., sign), given the graph of the vector field**F**. - (0.5) Compute a line integral
**F**·d**r**around a closed curve.

- (4) Find a potential function for a three dimensional vector field
**F**(x,y,z) (assume that one exists.) - (2) Find a potential function for a two dimensional vector field
**F**(x,y) (assume that one exists.) - (8) Verify a two dimensional vector field is conservative, then find a potential function (and maybe use this function to evaluate a line integral.)
- (2) Compute work done by a given force moving on a given path.
- (2) Compute work done by a conservative force moving on a path between two points..
- (1) Compute work done by a non-conservative force moving on a straight line path between two points.
- (2) Compute line integral between two points along unknown path.

- (11.5) Use Green's Theorem to evaluate a line integral along a closed curve.In some cases, Green's Theorem is not stated explicitly, but line integral can't be evaluated directly.

- (1) Compute a curl.

- (2) Compute surface area of portion of a surface cut off by other surfaces.
- (1) Compute surface integral over portion of a surface cut off by other surfaces.
- (1) Express a surface integral as a multiple integral.
- (2) Express a surface integral as an iterated integral in polar coordinates.
- (4) Compute flux of given function across given surface.
- (1) Set up flux of given function across given surface.
- (1) Find flux integral across a closed surface.
- (1) Compute integral of a function over portion of a surface cut off by other surfaces.
- (2) Express a flux integral over a given surface as an iterated integral not involving vectors.

- (9.5) Use Stokes' Theorem to compute a line integral.
- (1) Use Stokes' Theorem to compute a flux integral.
- (1) Use Stokes' Theorem to convert a line integral around a closed curve to an iterated integral in polar coordinates.

- (14) Use Divergence Theorem to evaluate a surface integral over a closed surface.
- (1) Use Divergence Theorem and spherical coordinates to evaluate outward flux of a function on boundary of a solid.

So there you have it: 275 problems in 18 unit finals for an average of a little over 15 questions per exam. Just for the record, the 18 exams average about 7.5 pages per exam, excluding cover sheets.

Be aware that these frequencies do not tell you the whole story since point values of the problems are not accounted for. You should be a bit careful in interpreting these problem frequencies as well. Consider, for example, the relative paucity of problems from Chapters 10-11. Does this mean that vectors are not important? Hardly! You'll do vector arithmetic, cross products and dot products right and left when you do some of the problems in Chapters 12-14. These concepts are used in later chapters, and that's how they make their importance felt. Another example: there is only one question that explicitly asks for a curl calculation. Not important? Well, in every Stokes' Theorem problem it is necessary to compute a curl. So, again, a basic calculation makes its appearance indirectly.

¹ The usual disclaimer: as always, past performance is no guarantee of future results.