Reading Assignments for Multivariable Calculus
Spring 2007, Math 236
March, 2007
Be sure to check back often, because assignments may change!
(Last modified:
Tuesday, March 20, 2007,
3:44 PM )
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 3/2 at 9am
Section 12.3: Partial Derivatives
E-mail Subject Line: Math 236 Your Name 3/2
Reading questions:
- For a function z=f(x,y), what information does f_{y}(1,2) give you?
- For f(x,y)=ye^{xy}, find f_{x}(x,y).
- How many second-order partial derivatives does a function f(x,y,z) have?
Reminders:
- The take-home portion of Exam 2 is of course due Friday.
Due Monday 3/5 at 9am
Review Sections 12.1-12.3
No Reading Questions today!
Due Wednesday 3/7 at 9am
12.4: Tangent Planes and Linear Approximations
- To read: All. Don't obsess too much over the details of the section on Increments and Differentials, but do read it, as it leads to the definition of what we mean by being differentiable at the point (a,b).
E-mail Subject Line: Math 236 Your Name 3/7
Reading questions:
- 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)?
- 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).
- 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?
Due Friday 3/9 at 9am
Section 12.5: The Chain Rule
- To read: All (this way, you don't have to read for the Monday after spring break!)
E-mail Subject Line: Math 236 Your Name 3/9
Reading questions:
Suppose that w=f(x,y,z) and that x,y, and z are all functions of r,s, and t.
- How many partial derivatives do you need to calculate to determine dw/dt? (Using a tree diagram may prove useful.)
- What is the expression for dw/dt?
Monday 3/12 through Friday 3/16
Enjoy your spring break!
Due Monday 3/19 at 9am
Section 12.5: The Chain Rule (continued)
- To read: Re-read the section, to refresh your memory after break!
No Reading Questions today!
Reminder:
- Only a few of you let me know for sure what option you'd chosen for the project; please let me know by Tuesday, because by Friday you need to have either read a chapter or gotten a sketch and found a few functions.
Due Wednesday 3/21 at 9am
Section 12.6: The Gradient and Directional Derivatives
- To read: Read through Example 6.3 (page 987)
E-mail Subject Line: Math 236 Your Name 3/21
Reading questions:
- In order to find D_{u}f(a,b), what must be true about our choice of u?
- 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?
- For f(x,y)=x^2y-4y^3,
(a) Compute D_{u}f(2,1) for u=<-3/5,4/5>.
(b) Compute the gradient of f(x,y)
Reminder:
Due Friday 3/23 at 9am
Section 12.6: The Gradient and Directional Derivatives (continued)
- To read: Finish the section
E-mail Subject Line: Math 236 Your Name 3/23
Reading questions:
- 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?
- What information does the gradient give you about f(x,y)?
Reminders:
- Either give me a rough sketch of your parametric design, along with at least two functions you've come up (you can always change your plans later), OR tell me the name of the book you've chosen, and have read a chapter.
Due Monday 3/26 at 9am
Section 12.6: The Gradient and Directional Derivatives (continued)
- To read: Re-read the whole section -- there are a lot of subtle ideas in this section.
No Reading Questions Today
Due Wednesday 3/28 at 9am
Section 12.7: Extrema of Functions of Several Variables
- To read: Through example 7.3 (bottom of page 1001)
E-mail Subject Line: Math 236 Your Name 3/28
Reading questions:
- If the partials f_{x} and f_{y} exist everywhere, at what points can f have a local max or a local min?
- Suppose f_{x}(x_{0}, y_{0})=f_{y}(x_{0}, y_{0})=0, and at that same point both f_{xx} and f_{yy} are positive (so that in both the x and y direction, f is concave up). Must f have a local min at the point (x_{0}, y_{0})?
- Suppose that f is a "nice" function, and that the following are true:
- f_{x}(2,-1)=0=f_{y}(2,-1)
- f_{xx}(2,-1)=-1
- f_{yy}(2,-1)=-3
- f_{xy}(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.
Due Friday 3/30 at 9am
Section 12.7: Extrema of Functions of Several Variables (continued)
- To read: Finish the section
E-mail Subject Line: Math 236 Your Name 3/30
Reading questions:
- Describe the idea behind the method of steepest ascent in your own words.
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
Back to: Multivariable Calculus | My Homepage | Math and CS