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 5 - Friday, April 9. 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 my notes on optimization (Lagrange multipliers; includes the four methods to solve the problem of points on a sphere) and my notes on the Method of Least Squares.
  • Mathematica programs:  Battle of Trafalgar (differential equation program written for Math 209); Deploying forces on the Earth (Lagrange Multipliers); for more on the military problem, see my notes on the problem (click here for a clean comparison of the answers using the four different models); Mathematica program on Zipf's law.
  • Homework:
    • Due Wednesday, April 7: Section 3.4: #2, #3, #10. Review Exercises from Chapter 2: Page 176: #21. Review Exercises from Chapter 1: Page 91: #28.
    • Due Friday, April 9: From my notes on the Method of Least Squares: Exercise 3.3, Exercise 3.9. Review Exercises from Chapter 2: Page 177: #33b. Review Exercises from Chapter 1: Page 93: #42.
    • Due Monday, April 12: Section 5.1: #1ac. Section 5.2: #1b. Review Exercises from Chapter 2: Page 174: #7e. Review Exercises from Chapter 1: Page 90: Do either 18a or 18b.
  • Suggested Problems (not to be turned in): Section 3.4: #20, #27. From my notes on the Method of Least Squares: Exercise 3.7.
  • Extra credit problems: From my notes on the Method of Least Squares: Exercise 3.8.

   

• Week 9: Monday, April 12 - Friday, April 16. Second midterm on Wednesday; first question available here.
  • Daily planner for the week (summarizes reading, HW; no quiz as midterm on Wednesday). We will do the Fundamental Theorem of Calculus on Monday (this is supposed to be a review), and then 5.3 and if time permits some of 5.4 on Friday.
  • Read: Chapter 5: Sections 5.3, 5.4.
  • Homework:
    • No HW Due Wednesday, April 14 and no HW Due Friday, April 16.
    • Due Monday, April 19: Section 5.3: #2bd, #8, and do exactly one of #9 and #15.
  • Suggested Problems (not to be turned in): Section 5.3: #3, #12, #16.
  • Extra credit problems: Assume f(x,y) = g(x)h(y); can we always interchange the order of integration, no matter what f and g are and no matter what the region equals?

   

• Week 10: Monday, April 19 - Friday, April 23. NOTE: no class on Monday, April 26th, office hours Tuesday the 27th canceled.
• Daily planner for the week (summarizes reading, HW; instead of a quiz look over the practice midterm on similar material as the last test, but do not write up solutions without talking to me first). We will do mathematical modeling on Monday, finish Chapter 5 on Wednesday, and do Monte Carlo Integration on Friday.
• Slides for Monday's lecture on sabermetrics (applications of mathematics to baseball) are available here.
• Mathematica program for Monte Carlo experiments (for Friday).
• Read: Chapter 5: Sections 5.4 and the sections of my Chapter 5 notes on Monte Carlo Integration (this is pages 36 - 38 in the numbering of the notes). Skim Section 5.5.
• For your enjoyment: Papers on the Monte Carlo Method: Metropolis (beginnings of method) and Metropolis-Ulam (the Monte Carlo Method), as well as my paper on the Pythagorean Won-Loss formula.
• Homework: Click here for a special set of problems and solutions to finding bounds for regions of integration
  • Due Wednesday, April 21: Assume the probability that X equals x is 2 exp(-2x) if x >= 0 and 0 otherwise, and the probability that Y equals y is 3 exp(-3x) if y >= 0 and 0 otherwise. Show both of these densities are, in fact, probability distributions (this means showing they are non-negative and integrate to 1), and calculate the probability that X >= Y. Note exp(u) = e^u. Also do: Page 257, #15 (hint: minimize distance squared instead of distance; this makes the algebra simpler).
  • Due Friday, April 23: Section 5.4: #1bc, #4 (hint: see equation (6) on page 353). Also do: Page 364, #14.
  • Due Wednesday, April 28: Page 192, #20abf. Page 366: #14. Page 367: #25.
• Suggested Problems (not to be turned in): Section 5.4: #6, #14, #15
• Extra credit problem: Show Integral_{x = 0 to 1} Integral_{y = 0 to 1} n (1 - xy)^n-1 = 1 + 1/2 + 1/3 + ... + 1/n whenever n is a positive integer.
• Extra credit problem: Let X be a random variable with density exp(-x) if x >= 0 and 0 otherwise. Calculate the mean and the standard deviation of X, and verify Chebyshev's inequality when k = 10 (in other words, show that Prob(|X - μ| >= 10σ)  <= 1/100 by computing Prob(|X - μ| >= 10σ).

Extra Credit: (1: 1 point) Prove Newton's result that you may assume all the mass of a sphere of radius 1 with uniform density is concentrated at the center. As the book proves this using potentials, you must prove this by direct integration of the force. (2: 2 points) Assume the force of gravity is given by (GMm/r^n-1) e_r in n-dimensional space. Here e_r is the unit vector in the r-direction. Thus the magnitude of the force is GMm/r^n-1 and it is radial. Prove or disprove: we may assume all the mass of a sphere of radius 1 with uniform density is concentrated at the center (ie, the force this exerts is the same as the force of the sphere).