Reading Assignments for Multivariable Calculus
    Spring 2008, Math 236

    January and February, 2008



    Be sure to check back often, because assignments may change!
    (Last modified: Monday, February 11, 2008, 9:59 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 Friday 1/25 at 9am

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    Section 10.1: Vectors in the Plane
    Section 10.2: Vectors in Space
    Section 10.3: The Dot Product
    Section 10.4: The Cross Product

    E-mail Subject Line: Math 236 Your Name 1/25

    Reading questions:

    1. Find a unit vector in the direction of <-2,3,-6>.
    2. Let a=<2,-1,3>, b=<4,10,-1>, and c=<3,-1,2> be vectors, and let x.y represent the dot product of any vectors x and y.
      (a) Find a.b.
      (b) Does (a.b).c make sense, and if so, what is it in this case?
      (c) Are b and c orthogonal?
    3. Briefly explain what, geometrically, projba and compba are.
    4. How is a x b related to a and b geometrically?
    5. If a and b are vectors in V2 (that is, in the plane), is a x b defined?

    Reminders:

    (I start out with extensive reminders at the beginning of the semester -- often things I'm afraid I'll forget to mention. I pare them down significantly as the semester goeos by.)

    Please Note:


    Due Monday 1/28 at 9am

    Problem Set Guidelines
    Section 10.4: The Cross Product
    Section 10.5: Lines and Planes in Space

    E-mail Subject Line: Math 236 Your Name 1/28

    Reading questions:

    1. If a, b, and c are arbitrary vectors, should we expect that
      (a x b) x c=a x (b x c)?
    2. How can we tell whether two vectors in V3 are parallel? How can we tell whether two vectors are orthogonal?
    3. What information about a line L do you need to determine an equation for the line?
    4. Find parametric equations for the line through the point (1,2,3) and parallel to the vector <3,0,-1>.

    Reminder:


    Due Wednesday 1/30 at 9am

    Section 9.1: Plane Curves and Parametric Equations Section 10.5: Lines and Planes in Space

    E-mail Subject Line: Math 236 Your Name 1/30

    Reading questions:

    1. Find parametric equations for the line through the points (1,2,3) and (7,-1,0).
    2. What information about a plane P do you need to determine an equation for the plane?
    3. Find an equation for the plane containing the point (1,2,3) with normal vector <3,0,-1>.

    Reminder:


    Due Friday 2/1 at 9am

    Section 11.1: Vector-Valued Functions

    E-mail Subject Line: Math 236 Your Name 2/1

    Reading questions:

    1. The vector-valued function r(t)=cos(t)i+sin(t)j lies in the plane.
      (a) Using Example 1.2 as a guide, what will the graph of this function look like?
      (b) Is it possible to rewrite r(t) as a function y=f(x)?
    2. What are some advantages to using vector-valued functions?

    Reminders:


    Due Monday 2/4 at 9am

    Section 11.2: The Calculus of Vector-Valued Functions

    E-mail Subject Line: Math 236 Your Name 2/4

    Reading questions:

    1. If r(t)=tcos(t)i+exp(t^2)j+ln(t)k, what is r'(t)?
      Remember exp(t^2) is Maple notation for et2.
    2. If r(t) is a vector-valued function, what geometric/graphical information does r'(a) give you?
    3. Let r(t)=cos(t)i+sin(t)j and s(t)=sin(5t)i+cos(5t)j.
      (a) What do the graphs of r(t) and s(t) look like?
      (b) If the graphs of two vector-valued functions r(t) and s(t) are the same, must r'(0)=s'(0)? (Is this a new result, or was it also true for functions f(x) and g(x)?)

    Reminders:


    Due Wednesday 2/6 at 9am

    Problem Set Guidelines
    Section 11.3: Motion in Space

    E-mail Subject Line: Math 236 Your Name 2/6

    Reading questions:

    1. If r(t) is a vector-valued function representing the motion of an object at time t, what physical information does r'(t) give you? How about ||r'(t)||?
    2. Find the velocity and acceleration vectors, if the position of an object moving in space is given by
      r(t)=(5/sqrt(t))i+ln(t3)j-tan(3t)k.

    Reminders:


    Due Friday 2/8 at 9am

    Section 11.4: Curvature

    E-mail Subject Line: Math 236 Your Name 2/8

    Reading questions:

    1. Explain the idea of curvature in your own words.
    2. If the helix in Example 4.5 were changed to r(t)=< 2sin(t), 2cos(t), 4t2>, will the curvature still be constant? Don't actually do the calculation, but give an intuitive justification - think about how changing the z-coordinate to 4t2 from 4t will affect the graph.

    Reminders:


    Due Monday 2/11 at 9am

    Section 11.5: Tangent and Normal Vectors

    E-mail Subject Line: Math 236 Your Name 2/11

    Reading questions:

    1. Suppose you are skiing down a hill along a path that curves left. Describe the direction of the unit tangent and principle unit normal vectors to the curve that describes your motion.

    Reminders:


    Due Wednesday 2/13 at 9am

    Section 10.6: Surfaces in Space

    E-mail Subject Line: Math 236 Your Name 2/13

    Reading questions:
    Consider the surface x=4y2+4z2.

    1. Which coordinate plane does the equation z=0 define?
    2. What does the trace of this surface in the xz-plane look like?
    3. What do the traces of this surface in the planes x=k look like?
    4. What is this quadric surface called?

    Reminders:


    Due Friday 2/15 at 9am

    Section 10.6: Surfaces in Space

    No Reading Questions Today


    Due Monday 2/18 at 9am

    Section 11.6: Parametric Surfaces

    E-mail Subject Line: Math 236 Your Name 2/18

    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 2/20 at 9am

    Section 12.1: Functions of Several Variables

    E-mail Subject Line: Math 236 Your Name 2/20

    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.
      (a)(b)
      (c)(d)

    Reminders:


    Due Friday 2/22 at 9am

    Section 12.1: Functions of Several Variables (continued)

    E-mail Subject Line: Math 236 Your Name 2/22

    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). )

    Reminders:


    Monday 2/25 at 9am

    Section 12.2: Limits and Continuity

    E-mail Subject Line: Math 236 Your Name 2/25

    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?

    Reminders:


    Due Wednesday 2/27 at 9am

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


    Due Friday 2/29 at 9am

    Section 12.3: Partial Derivatives

    E-mail Subject Line: Math 236 Your Name 2/29

    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?


    Here ends the reading for January and February
    Go to the reading assignments for March!


    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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