Reading Assignments for Multivariable Calculus
    Spring 2008, Math 236

    March, 2008



    Be sure to check back often, because assignments may change!
    (Last modified: Wednesday, April 2, 2008, 9:54 AM )


    I'll use Maple syntax for mathematical notation on this page.
    All section and page numbers refer to sections from Smith & Minton's Multivariable Calculus, 3rd edition.


    Due Monday 3/3 at 9am

    Review Sections 12.1-12.3

    No Reading Questions today!

    Reminders:


    Due Wednesday 3/5 at 9am

    12.4: Tangent Planes and Linear Approximations

    E-mail Subject Line: Math 236 Your Name 3/5

    Reading questions:

    1. If f(x,y) is a well-behaved function and has a local maximum at (a,b), what can you say about the linear approximation to f(x,y) at (a,b)?
    2. Let f(x,y)=xy^2. Find a vector normal to the plane tangent to z=f(x,y) at the point (2,3,18).
    3. Let L(x,y) be the linear approximation of f(x,y) at (a,b). What graphical properties of the surface z=f(x,y) would make L(x,y) particularly accurate? particularly inaccurate?

    Reminder:


    Due Friday 3/7 at 9am

    Section 12.5: The Chain Rule

    E-mail Subject Line: Math 236 Your Name 3/7

    Reading questions:

      Suppose that w=f(x,y,z) and that x,y, and z are all functions of r,s, and t.
    1. How many partial derivatives do you need to calculate to determine ∂w/∂t? (Using a tree diagram may prove useful.)
    2. What is the expression for ∂w/∂t?
    Reminder:


    Monday 3/10 through Friday 3/14

    Enjoy your spring break!


    Due Monday 3/17 at 9am

    Section 12.5: The Chain Rule (continued)

    No Reading Questions today!

    Reminder:



    Due Wednesday 3/19 at 9am

    Section 12.6: The Gradient and Directional Derivatives

    E-mail Subject Line: Math 236 Your Name 3/19

    Reading questions:

    1. In order to find Duf(a,b), what must be true about our choice of u?
    2. What type of quantity is the gradient of f? If we have z=f(x,y) is a surface in 3 dimensions, then what dimension does the gradient live in?
    3. For f(x,y)=x^2y-4y^3,
      (a) Compute Duf(2,1) for u=<-3/5,4/5>.
      (b) Compute the gradient of f(x,y)


    Due Friday 3/21 at 9am

    Section 12.6: The Gradient and Directional Derivatives (continued)

    E-mail Subject Line: Math 236 Your Name 3/21

    Reading questions:

    1. Toward the top of page 988, the book says:
      ..notice that a zero directional derivative at a point indicates that u is tangent to a level curve.
      Why is this true?
    2. What information does the gradient give you about f(x,y)?

    Reminders:


    Due Monday 3/24 at 9am

    Section 12.6: The Gradient and Directional Derivatives (continued)

    No Reading Questions Today


    Due Wednesday 3/26 at 9am

    Section 12.7: Extrema of Functions of Several Variables

    E-mail Subject Line: Math 236 Your Name 3/26

    Reading questions:

    1. If the partials fx and fy exist everywhere, at what points can f have a local max or a local min?
    2. Suppose fx(x0, y0)=fy(x0, y0)=0, and at that same point both fxx and fyy are positive (so that in both the x and y direction, f is concave up). Must f have a local min at the point (x0, y0)?
    3. Suppose that f is a "nice" function, and that the following are true:
      • fx(2,-1)=0=fy(2,-1)
      • fxx(2,-1)=-1
      • fyy(2,-1)=-3
      • fxy(2,-1)=2.5
      Will the point (2,-1) be a local max, local min, neither, or do you not have enough information to tell? In any case, justify your answer.

    Reminders:


    Due Friday 3/28 at 9am

    Section 12.7: Extrema of Functions of Several Variables (continued)

    E-mail Subject Line: Math 236 Your Name 3/28

    Reading questions:

    1. Describe the idea behind the method of steepest ascent in your own words.


    Due Monday 3/31 at 9am

    Section 13.1: Double Integrals

    E-mail Subject Line: Math 236 Your Name 3/31

    Reading questions:

    1. If f(x,y) is a positive function of two variables, what does R f(x,y)dA measure?
    2. Explain Fubini's Theorem in your own words. What is its importance?

    Reminder:



    Here ends the reading for March
    Go to the reading assignments for April and May!


    Janice Sklensky
    Wheaton College
    Department of Mathematics and Computer Science
    Science Center, Room 109
    Norton, Massachusetts 02766-0930
    TEL (508) 286-3973
    FAX (508) 285-8278
    jsklensk@wheatonma.edu


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