Please spend at least 1 if not 2 hours a night reading the material/looking at the proofs/making sure you can do the algebra. Below is a tentative reading list and homework assignments. It is subject to slight changes depending on the amount of material covered each week. I strongly encourage you to skim the reading before class, so you are familiar with the definitions, concepts, and the statements of the material we'll cover that day.

• Week 1: Friday, February 5
  • Read: Chapter 1, Sections 1.1 and 1.2; also skim my lecture notes for those sections of Chapter 1.
  • Slides for the first lecture (2/5/10) are available online here.
  • HW: Due on Monday, February 8: Section 1.1: #4, #7
  • Extra credit (which of the four formulas models A beating B) is due on 2/10/10.
  • Optional: additional comments on Friday's lecture are available online. These are comments on advanced material / further applications of what we discussed.

   

• Week 2: Monday, February 8 - Friday, February 12
  • Read: Chapter 1, Sections 1.1, 1.2, 1.3, 1.4 and 1.5; also skim my lecture notes for those sections of Chapter 1. You should read 1.1 and 1.2 for Monday's class.
  • Homework:
    • Due Wednesday, February 10: Section 1.1: #13, #16, #22
    • HW: Due Friday, February 12: Section 1.2: #1, #7, #19
    • HW: Due Monday, February 15: Section 1.3: #2c, #4, #6, #15a
  • Suggested Problems (not to be turned in): Section 1.1: #9, #19, #28, #30 (applications in chemistry); Section 1.2: #20, #27 (physics). Also find a relation between the sum of the lengths of the two diagonals of a parallelogram and the sum of the lengths of the sides; prove your relation. Section 1.3: #10, #17a, #21, #35 (physics).
  • Extra credit problems (due 2/12/10): You do not need to do all of them; however, very little partial credit will be awarded (extra credit is basically right or wrong). Hand these in on a separate sheet of paper.

    • (1) Let N be a large integer. How should we divide N into positive integers a_i such that the product of the a_i is as large as possible. Redo the problem when N and the a_i need not be integers.

    • (2) What is wrong with the following argument (from Mathematical Fallacies, Flaws, and Flimflam - by Edward Barbeau): There is no point on the parabola 16y = x² closest to (0,5). This is because the distance-squared from (0,5) to a point (x,y) on the parabola is x² + (y-5)². As 16y = x² the distance-squared is f(y) = 16y + (y-5)². As f'(y) = 2y+6, there is only one critical point, at y = -3; however, there is no x such that (x,-3) is on the parabola. Thus there is no shortest distance!

    • (3) Without using any computer, calculator or computing by brute force, determine which is larger: e^π or π^e. (In other words, find out which is larger without actually determining the values of e^π or π^e). If you're interested in formulas for π, see also my paper A probabilistic proof of Wallis' formula for π, which appeared in the American Mathematical Monthly (there are a lot of good articles in this magazine, many of which are accessible to freshmen).

     

• Week 3: Monday, February 15 - Friday, February 19
  • Read: Chapter 1: Sections 1.4, 1.5; Chapter 2: 2.1, 2.2, 2.3; also skim my lecture notes for those sections of Chapter 2.
  • Mathematica files: click here for the Mathematica file on contour plots and level sets.
  • Homework:
    • Due Wednesday, February 17: Section 2.1: #1 (just find the level sets, no need to graph the function), #24; also if you are doing quiz 2, it is due on Wednesday.
    • Due Monday, February 22:  Section 2.2: #4. Additional Problem #1: Let f(x) = x² + 8x + 16 and g(x) = x²+2x-8. Compute the limits as x goes to 0, 3 and ∞ of f(x)+g(x), f(x)g(x) and f(x)/g(x). Additional Problem #2: Compute the derivative of cos(sin(3x² + 2x ln x)). Note that if you can do this derivative correctly, you should be fine for the course.
    • Due Wednesday, February 24: Section 2.2: #8ab, #17, #21. Note that for these problems you may assume that the exponential, sine and cosine functions are continuous.
  • Suggested Problems (not to be turned in): Section 2.1: #3, #30; Section 2.3: #3 ,#4cde, #10, #15, #18, #19.
  • Extra credit problem (due 2/19/10): Prove that the product of the slopes of two perpendicular lines in the plane that are not parallel to the coordinate axes is -1. What is the generalization of this to lines in three-dimensional space? What is the analogue of the product of the slopes of the line equaling -1?

   

• Week 4: Monday, February 22 - Friday, February 26
  • Summary sheet for the week (reading, quiz, some of the HW, key points to cover during the week)
  • Read: Chapter 2: Sections 2.2, 2.3, 2.4, 2.5.
  • Homework:
    • Due Wednesday, February 24: Section 2.2: #5, #8ab, #17. Note that for these problems you may assume that the exponential, sine and cosine functions are continuous. Section 2.3: #1ad (hint: if you know δf/δx by symmetry you know δf/δy).
    • Due Friday, February 26: Section 2.3: #2b, #4ab, #5, #7c  (instead of giving the matrix of partial derivatives, just give the partial derivatives with respect to x and y of the two coordinate functions f₁(x,y,z) = x + e^z + y and f₂(x,y,z) = yx²).
    • Due Monday, March 1: no written HW
  • Suggested Problems (not to be turned in): Section 2.2: #8c, #9, #10, #25, #26, #27. Section 2.3: #3, #4,cde, #10, #15, #18, #19. Section 2.4: #3, #6, #7, #13, #17.
  • Extra credit problem: Due Monday, March 1: Section 2.2: #21 (you are not allowed to look at the solution in the back of the book).

   

• Week 5: Monday, March 1 - Friday, March 5. Note: First Midterm: March 10 (Wed) in class
  • Summary sheet for the week (reading, quiz, some of the HW, key points to cover during the week).
  • Read: Chapter 2: Sections 2.4, 2.5, 2.6. Read also my handout on the chain rule (my apologies that it's in MSWord -- it was written a long time ago; I was young(er)).
  • Supplemental notes for Monday's lecture on curves in space (Section 2.4) is available here.
  • Homework:
    • Due Wednesday, March 3: Section 2.3: 12a, #13a. Section 2.4: #1, #15. Solution hint (i.e., write-up of some similar problems) is here.
    • Due Friday, March 5: Section 2.5: #2g, #4 (by verify it means use the chain rule as well as substitute for u(x,y) and v(x,y) and then take the derivative considering it as a function of x and y), #7, #12 (there are two ways to do this problem). Solution hint (i.e., write-up of some similar problems) is here.
    • Due Monday, March 8: Section 2.5: #4 (now do it using the chain rule), #5a (do not do #10), #13a or #13b (but not both), #20.
  • Suggested Problems (not to be turned in): Section 2.4:  #3, #6, #7, #13, #17. Section 2.5: #1, #4, #8, #17, #18 (important in physics), #25iv, #26, #28.
  • Extra credit problems: (1) Let f: R⁴ --> R³ and g: R² --> R⁴ be two continuous functions, and let U be an open subset of R². Must f(g(U)) be an open subset of R³? Note h(U) is defined as the set of all outputs of h where the input ranges over all points in U. (2) Section 2.5, #24.

   

• Week 6: Monday, March 8 - Friday, March 12. Note: First Midterm: March 10 (Wed) in class
  • Daily planner for the week (summarizes reading, HW, and gives the first midterm problem).
  • Read: Chapter 2: Section 2.6. Chapter 3: 3.1, 3.2 and see also my lecture notes on Taylor's Theorem in one-variable.
  • Homework:
    • Due Friday, March 12: Section 2.6: #2ab, #4a, #6a. DO ONE OF #16 (the famous Captain Ralph problem) OR #18.
    • Due Monday, March 15: Do ONE OF Page 176: #23 (homogenous functions) OR #47 (ideal gas law from Chemistry / Physics). Also do: Section 3.1: #1, #8a, #11.
  • Suggested Problems (not to be turned in): Section 2.6: #5a, #12, #17, #21, #23. Page 176: #26, #41, #42.
  • Extra credit problems: Section 2.5, #18.

• Week 7: Monday, March 15 - Friday, March 19.  Note: Second Midterm will be Wednesday, April 14 in class
  • Daily planner for the week (summarizes reading, HW, and gives the quiz).
  • Slides for Wednesday's lecture: no tab version        tabbed version
  • Read: Chapter 3: 3.2, 3.3, 3.4 and see also my lecture notes on Taylor's Theorem in one-variable.
  • Homework:
    • Due Wednesday, March 17: Section 3.2: #2, #3.
    • Due Friday, March 19: Section 3.3: #7 (just find the critical points), #13 (just find the critical point), #22 (hint: minimize the square of the distance).
    • No HW Due Monday, April 4.
  • Suggested Problems (not to be turned in): Section 3.2: #7. Section 3.3: #18, #23, #28, #41. Section 3.4: #20, #27
  • Extra credit problems: Let f(x) = exp(-1/x²) if |x| > 0 and 0 if x = 0. Prove that f⁽ⁿ⁾(0) = 0 (i.e., that all the derivatives at the origin are zero). Show this implies the Taylor series approximation to f(x) is the function which is identically zero. As f(x) = 0 only for x=0, this means the Taylor series (which converges for all x) only agrees with the function at x=0, a very unimpressive feat (as it is forced to agree there).

   

• Week 8: Monday, April 4 to Friday, April 8. Note: Second Midterm will be Wednesday, April 14 in class
  • Daily planner for the week (summarizes reading, HW, and gives the quiz).
  • Read: Chapter 3: 3.4 and TBD
  • Homework:
    • Due Wedesday, April 6: Section 3.4: #2, #10, #21.
    • Due Friday, April 8:
    • Due Monday, April 11: Section 3.4: #2, #10, #21.
  • Suggested Problems (not to be turned in):
  • Extra credit problems: