General:  The Final will cover sections 10.1-12.6. I will provide a crib sheet that you can use on the final. This will be posted on the course web page by Friday, Oct. 6, but I will also pass this out in class on Monday, Oct. 9.

The following is a breakdown of the material that has been covered since the Midterm. BUT YOU ARE ALSO RESPONSIBLE FOR THE MATERIAL ON THE REVIEW FOR THE MIDTERM. (If you have lost your midterm review, it is available on the course web page).

Bare Essentials (You must be able to do the following to pass the test)

• Find the arc length of a space curve.
• Find level curves of a function of 2 variables and sketch a contour map.
• Calculate limits of a function of several variables.
• Calculate partial derivatives of functions of several variables.
• Calculate mixed partial derivatives.
• Find the total differential of a function of several variables.
• Use the chain rule to calculate the derivatives of functions with respect to a single independent variable.
• Use the chain rule to calculate the partial derivatives of a function with respect to several independent variables.
• Find the gradient of a function or several variables.

"B/C" Material (To get a "C" or a "B" on the exam you should be able to do the following)

• Calculate the curvature of a smooth space curve. (formulas will be provided on the crib sheet). You should try out each of the formulas, so that you get a feel of which is the easiest to use. This can save you huge headaches on the exam.
• Match a surface with its level curves: (Sec 12.1: 45-48)
• Determine whether a function of several variables is continuous at a point.
• Determine the slope in the $x$ or $y$ directions of a surface in three dimensions.
• Determine the rates of change of a function with respect to each of its variables.
• Find the domain and range of a function of several variables.
• Determine whether a function is differentiable at a given point.
• Calculate a derivative or partial derivative using implicit differentiation. (12.5: 21, 26)
• Calculate the directional derivative of a function in a given direction.
• Find the maximum value (over all possible directions) of the directional derivative of a function at a given point. (12.6: 27, 32)
• Use the gradient to find a directional derivative.
• Use the gradient to find a direction normal to the level curves of a function. (12.6: 45, 51)
• Find the direction of maximum or minimum increase of a function. (12.6: 54)
• Simple Theory: (typically tested using True/False questions)
  • Definitions of open and closed sets.
  • Definition of continuity.
  • Definition of partial derivatives.
  • Continuity of sums/differences/products/quotients of continuous functions.
  • Continuity of compositions of functions.
  • Equality of Mixed partial derivatives.
  • Differentiability implies continuity.
• Applications:
  • 12.1: 73
  • 12.4: 25
  • 12.6: 54

"A" Material (In addition to all of the above, you should be able to do the following to get an A on the exam).

• Use the curvature and speed to determine the tangential and normal components of acceleration. (Formula will be on the crib sheet).
• Find the arc length function of a space curve.
• Rewrite a smooth curve as a function of its arc length.
• Find level surfaces of a function of 3 variables.
• Verify a limit by the definition.
• Give an example of a function for which the partial derivatives exist, but which is not differentiable.
• Applications:
  • 11.5: 75, 85
  • 12.1: 77
  • 12.3: 79, 80
  • 12.4: 32, 35
  • 12.5: 42