STUDY GUIDE for FINAL EXAM

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Go to Study Guide for Exam 2

The final exam is cumulative and will cover material from sections 11.1-11.6, 12.1-12.5, 13.1-10, 14.1-14.4, 14.6-14.8 and 15.1-15.2. Anything not covered in class or given for homework will not be on the exam. A list of notably excluded topics is included at the bottom of the page.

The test material will be from the following categories:

  1. 3D Cartesian Coordinate System: Spheres, Distance
  2. Vectors: Addition/Subtraction, Length, Standard Basis Vectors, General Properties, Unit Vectors, Writing vector given magnitude and direction
  3. Dot Product and Cross Product: Definitions, Geometric Interpretation, Properties, Orthogonality, Projections
  4. Lines and Planes: Equations, Normal Vector, Parallel/Intersecting/Skew Lines, Parallel/Angle of Intersecting Planes, Intersection of two planes, Distance between point, Lines and Planes
  5. Quadric Surfaces: 6 Basic Forms, Traces
  6. Vector Functions: Space Curves and Tangent Lines, Derivatives, Properties, Integrals, Limits
  7. Arc Length: for space curves
  8. Curvature, Unit Tangent, Principle Unit Normal: formulas for curvature given
  9. Position/Velocity/Acceleration in Space: Newton's 2nd Law (F=ma)
  10. Functions of Several Variables: domain(incl sketching), range, level curves/contour maps
  11. Limits: definition of continuity, finding limit, showing limit doesn't exist
  12. Partial Derivatives: definitions, geometric interpretation, calculating first or higher partial derivatives, tangent planes
  13. Chain Rule: Variations, including writing out multilayer chain rules for general functions
  14. Differentials: Comparison to delta z
  15. Directional Derivatives: Geometric interpretation, calculation
  16. Gradients: Calculation, interpretation and significance
  17. Extreme Values: relative, absolute, saddle points, 2nd Partial Derivatives Test, Lagrange Multipliers
  18. Integrals Over Rectangular Regions: Set up and evaluate any double or triple integral
  19. Integrals Over General Regions: Set up any integral, evaluate any double integral, switch the order of integration
  20. Area and Volume: Find the area using a double integral, volume using a triple integral
  21. Mass and Center of Mass: Find the total mass/charge/population and center of mass/charge/population, given the density (double integrals)
  22. Polar Coordinates: Convert integral from rectangular coordinates, identify that polar coordinates is the best method, set-up integral and evaluate
  23. Cylindrical and Spherical Coordinates: Convert integral from rectangular coordinates, identify best coordinate system choice, set-up integral
  24. Transformations: Find/sketch the image of a given set with a given \transformation
  25. Change of Variables: Find the Jacobian, evaluate double integral using given transformation
  26. Vector Fields: Find a gradient vector field, sketch a vector field, find a potential function of 2D vector field
  27. Curl and Divergence
  28. Line Integrals

The problems will be of a similar type to the homework and examples from lecture. There will be a variety of types of problems asked, some categories of which are discussed below.
Calculation. These include predominantly straightforward problems, such as calculating a dot product, an integral, a partial derivative, or a Jacobian.
Interpretation (and Calculation). Many of the same tools used above are employed, but these problems require notable set-up procedures, and usually a deeper understanding of concepts. Examples include many triple integrals, the geometric interpretation of directional derivatives, variations on the Chain Rule, potential functions, and some problems involving lines and/or planes.
Applications. There is a range of topics in this classification. They include arc length, position/velocity/accleration, local and absolute extrema applications, Newton's 2nd Law, and mass/center of mass.
Graphing. Sketch a curve/plane/region, its domain, or a contour map (i.e., level curves). Another possibility is to match up a function or equation with the appropriate graph, vector field, and/or contour map.
Proof.The only real proof that could be asked would be a vector property from p765, p781 or p791(excluding Property 6). There are some possibilities for pseudo-proofs, such as verifying that a property or theorem is true (e.g., Clairaut's Theorem for mixed second partial derivatives).

You can pretty much guarantee at least one problem will come from each category above. About half the exam will be on material taken from Chapters 14 and 15.

I will provide the paper and any additional scratch sheets you may need. Integration tables and (if necessary) some trigonometric identities will also be provided. You may bring a scientific or graphing calculator (similar to TI-83/84) are permitted, but ones that can do symbolic derivatives/integrals (e.g., TI-89) are not permitted.

Notable Excluded Topics (items with an * were previously included)



Last uploaded July 12, 2011