Reading Assignments for Multivariable Calculus
    Spring 2010, Math 236

    March, 2010

    Be sure to check back often, because assignments may change!
    (Last modified: Monday, March 29, 2010, 9:52 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.

    Monday 3/1 at 9am

    Section 11.6: Parametric Surfaces

    E-mail Subject Line: Math 236 Your Name 3/1

    Reading questions:

    1. Following a process similar to that in Example 1 (not using Maple to graph), identify the surface defined by the parametric equations
      x=2cos(u)sin(v), y=2sin(u)sin(v), z=6cos(v).
      (You might want to review the generic equations for quadric surfaces given in Section 10.6).
    2. In Example 6.3, what might make you think to use cosh(u) and sinh(u) in the parametric equations for x and y? That is, what property that's useful in this situation do they have?


    Due Wednesday 3/3 at 9am

    Exam 1 (in-class portion)- No new reading!

    Due Friday 3/5 at 9am

    Section 12.1: Functions of Several Variables

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

    Reading questions:

    1. Below are the graphs of 4 surfaces. 2 of them are the graphs of the functions
      f1(x,y)=[(x2+y2)-2]3 and f2(x,y)=y2sin(x).
      Match the functions with their graphs.


    Due Monday 3/8 at 9am

    Section 12.1: Functions of Several Variables (continued)

    E-mail Subject Line: Math 236 Your Name 3/8

    Reading questions:

    1. Give the equations of two level curves for the function f(x,y)=x2/4+y2/9. What will these level curves look like?
    2. Below is a contourplot for the function sin^2(x)+cos^2(y). Roughly where does it have local extrema? How can you tell?
      (I used the Maple command
      contourplot((sin(x))^2+(cos(y))^2,x=-3..3, y=-1..2, contours=15, coloring=[blue,red]);
      to create this graph. The "coloring=" option allows you to specify which color will be associated with the lowest contours (blue) and which will be associated with the highest (red). )


    Wednesday 3/10 at 9am

    Section 12.2: Limits and Continuity

    E-mail Subject Line: Math 236 Your Name 3/10

    Reading questions:

    1. Consider f(x,y)=3x2 /(x2+y2). We want to see what we can say about lim(x,y) -> (0,0) f(x,y).
      (a) Find the limit along the path x=0. That is, find lim(0,y)->(0,0) f(0,y).
      (b) Find the limit along the path y=0. That is, find lim(x,0) -> (0,0) f(x,0).
      (c) What -- if anything -- can you conclude from your results?
    2. What is the point of Example 2.5?
    3. Why do you suppose we're studying limits now?


    Due Friday 3/12 at 9am

    Section 12.3: Partial Derivatives

    E-mail Subject Line: Math 236 Your Name 3/12

    Reading questions:

    1. For a function z=f(x,y), what information does fy(1,2) give you?
    2. For f(x,y)=yexy, find fx(x,y).
    3. How many second-order partial derivatives does a function f(x,y,z) have?


    Monday 3/15 through Friday 3/19

    Enjoy your spring break!

    Due Monday 3/22 at 9am

    Review Sections 12.1-12.3

    No Reading Questions today!


    Due Wednesday 3/24 at 9am

    12.4: Tangent Planes and Linear Approximations

    E-mail Subject Line: Math 236 Your Name 3/24

    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 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?
    3. 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).

    Due Friday 3/26 at 9am

    Section 12.5: The Chain Rule

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

    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?


    Due Monday 3/29 at 9am

    Section 12.6: The Gradient and Directional Derivatives

    E-mail Subject Line: Math 236 Your Name 3/29

    Reading questions:

    1. In order to find Duf(a,b), what must be true about our choice of u?

    Due Wednesday 3/31 at 9am

    Section 12.6: The Gradient and Directional Derivatives (continued)

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

    Reading questions:

    1. 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?
    2. 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)
    3. 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?
    4. What information does the gradient give you about f(x,y)?


    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 101A
    Norton, Massachusetts 02766-0930
    TEL (508) 286-3973
    FAX (508) 285-8278

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