Exam #2 Review

Scheduling Information:

 Time: Friday, March 24. 7:30 AM - 18:20 AM
Location: UC McConomy
Review Session: TBA.
 

Reading:

 Functions of Several Variables (14.1)
Limits and Continuity (14.2)
Partial Derivatives (14.3)
Tangent Planes and Linear Approximations (14.4)
The Chain Rule (14.5)
Directional Derivatives and the Gradient Vector (14.6)
Maximum and Minimum Values (14.7)
Lagrange Multipliers (14.8)

 

Review Questions:
  1. What is a function of 2 variables? The domain of such a function? Image? Codomain? Range (according to Stewart)?
  2. What is the graph of a function of 2 variables?
  3. What are the level curves of a function of 2 variables? How are they related to traces of a surface?
  4. What is a function of 3 variables? Why do we not think about graphs of a function of 3 variables? What are the level surfaces of a function of 3 variables?
  5. What is the limit of a function of two variables? How is the limit defined?
  6. Why is the subject of limits more complicated for functions of 2 variables then functions of a single variable?
  7. How can you show that a limit does *not* exist?
  8. What is the definition of the limit of f(x,y) as (x,y)->(a,b)?
  9. What does it mean for a function of two variables to be continuous at (a,b)?
  10. Where are polynomials continuous? Where are rational functions sure to be continuous? Where might a rational function be discontinuous?
  11. What is a partial derivative of a function of two (or more) variables?
  12. In practice, how do you go about computing a partial derivative?
  13. What does the value of a partial derivative tell you about the graph of the function?
  14. What is a higher order partial derivative of a function? What makes a second order partial derivative "mixed?"
  15. What is interesting about second order mixed partial derivatives? (at least in most circumstances - what circumstances are those?)
  16. What is the tangent plane to the graph of a function of two variables?
  17. What is the linearization (or linear approximation) of a function of two variables? How is it related to the tangent plane?
  18. If z=f(x,y), what is the "increment" \Delta z?
  19. What does it mean for a function of two variables to be differentiable?
  20. What is the differential of a function of two variables? How is it related to the tangent plane?
  21. If z=f(x,y) and x and y are themselves functions of another variable t, what does the chain rule tell you about dz/dt?
  22. What is the chain rule for functions of several variables?
  23. How can you use a "tree diagram" to help remember the chain rule?
  24. What does the chain rule have to do with implicit differentiation?
  25. What is a directional derivative?
  26. What does a directional derivative have to do with partial derivatives?
  27. What is the gradient vector of a function?
  28. What is important about the direction of the gradient vector? About the magnitude of the gradient vector?
  29. How can you find the equation of the tangent plane to a level surface F(x,y,z)=k?
  30. What is a local maximum of a function? An absolute maximum? A local minimum? Absolute minimum?
  31. What do maxima and minima have to do with the partial derivatives of a function?
  32. What is the second derivative test for a function of two variables?
  33. What is a critical point, what is a critical value?
  34. What is a local minimum (maximum)? What is a local minimum (maximum) value?
  35. How do you find the critical points of a function? How do you determine whether a critical point is a local maximum or minimum or neither?
  36. What is a boundary point of a set in R^2? What is meant by a closed set in R^2? What is meant by a bounded set in R^2?
  37. How do you find the absolute maximum and absolute minimum values of a function on a closed, bounded set in R^2?
  38. In what situation would you use the Lagrange Multiplier Method? How does it work?
  39. How can you find the maximum (or minimum) value of a function subject to a single constraint equation? What if there are two constraints?
 

Exercises:
Section 14.1 #9, 11, 15, 17, 19, 25, 29, 33, 37, 41, 47, 49, 61, 63, 65.
Section 14.2 #5, 9, 13, 19, 21.
Section 14.3 #5, 9, 13, 19, 21, 29, 31, 37, 45.
Section 14.4 #1, 5, 7, 11, 13, 15, 17, 19, 25, 27, 31, 33, 35, 43.
Section 14.5 #1, 3, 5, 21, 23, 25, 39, 45, 47, 49, 53.
Section 14.6 #7, 9, 11, 15, 37, 39, 41, 43, 51.
Section 14.7 #1, 3, 5, 7, 9, 15, 19, 31, 33, 37, 41, 43, 47.
Section 14.8 #1, 3, 5, 11, 15, 17, 19, 21, 23, 25, 29.