Spring 2007, Math 236

March, 2007

Be sure to check back often, because assignments may change!

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

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?

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

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

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?

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

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 dw/dt? (Using a tree diagram may prove useful.)
2. What is the expression for dw/dt?

Monday 3/12 through Friday 3/16

Due Monday 3/19 at 9am

Section 12.5: The Chain Rule (continued)

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

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

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)

Reminder:

• The 7th annual Norman W. Johnson talk will be Thursday at 5pm, on fractals. This talk is supposed to be aimed at a general audience for cross-campus appeal, so bring a friend who ordinarily might not choose to come to a math talk!

• Just in case you didn't see the reminder for Monday, here it is again:

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 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

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?

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.

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

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.

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

1. 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

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