Final Exam Information for Fall 2009 Math 208
As you know, the final exam is a unit exam, meaning that all students of Math 208 take it at the same time, and comprehensive, meaning that anything in the common syllabus is fair game. You should also know that the preparation of the exam is a collective effort by all instructors of 208, so you shouldn't expect it to look like two hour exams from your own section folded together. As a matter of fact, anything's fair game, and I cannot offer any clues or hints as to what is specifically on the final exam you'll be taking. However, I can do my best to help you prepare for the final. It is your responsibility to determine the topics on which you need to get help from me.
"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.
Chapter 10: Vectors and the Geometry of Space (27 problems)
10.1: Three-dimensional Coordinate Systems
10.3: Dot Product
- (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.
10.4: Cross Product
- (2) Find area of a parallelogram.
- (1) Find area of triangle given three vertices.
10.5: Lines and planes in space
- (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.
10.6: Cylinders and Quadratic Surfaces
- (1) Identify a quadric surface by its equation.
Chapter 11: Vector-Valued Functions (10 problems)
11.1: Vector-valued Functions
11.2: Calculus of Vector-valued Functions
11.3: Motion in Space
- (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.
Chapter 12: Partial Derivatives (83 problems)
12.1: Functions of Several Variables
- (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.
12.2: Limits and Continuity in Higher Dimensions
- (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.
12.3: Partial Derivatives
- (2) Calculate (and possibly interpret) partials.
12.4: The Chain Rule
- (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.
12.5: Directional Derivatives and Gradient Vectors
- (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.
12.6: Tangent Planes and Differentials
- (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.
12.7: Extreme Values and Saddle Points
- (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.
12.8: Lagrange multipliers
- (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.
Chapter 13: Multiple integrals (65 problems)
13.1: Double and Iterated Integrals over Rectangles
- (2) Approximate a double integral over a rectangle by a Riemann sum.
13.2: Double Integrals over General Regions
- (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).
13.3: Area by Double Integration
- (3) Sketch and calculate a plane area.
13.4: Double Integrals in Polar Form
- (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.
13.5: Triple integrals in Rectangular Coordinates
- (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.
13.6: Moments and Center of Mass
- (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.
13.7: Triple Integrals in Cylindrical and Spherical Coordinates
- (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.
Chapter 14: Vector Calculus (90 problems)
14.1: Vector fields
- (1) Match a vector field and its graph.
14.2: Line integrals
- (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·dr (e.g., sign), given
the graph of the vector field F.
- (0.5) Compute a line integral F·dr around a closed
14.3: Independence of path and conservative vector fields
- (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.
14.4: Green's Theorem
- (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.
14.5: Curl and divergence
14.6: Surface integrals
- (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.
14.7: Stokes' Theorem
- (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.
14.8: Divergence Theorem and Unification
- (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.