Mathematics 226 practice problems, Fall 2014

Chapter 10:

• Section 10.1: 1-40
• Section 10.2: 1-26. Questions 9-12 are easy applications of vectors, similar to Example 3 (p. 573). Please read that example on your own. Questions 27-33 cover material that you should expect to see in your co-requisite linear algebra class. We are skipping "Hanging cables and chains," pp. 574-576 and Questions 34-37.
• Section 10.3: 1-28
• Section 10.4: 1-19
• Section 10.5: 1-22
• Section 10.6: 1-14

Chapter 12:

• Section 12.1: 1-26, 37-42
• Section 12.2: 1-16
• Section 12.3: 1-31, 36-39
• Section 12.4: 1-19
• Section 12.5: 1-24. Question 31 was done today in class; if you would like to try your hand at using a similar method to derive the general solution to the wave equation, this is outlined in Questions 32 and 33.
• Section 12.6: 1-19. Please make sure to read this section carefully on your own. The point here is not only to learn (several version of) the Chain Rule, but also to understand the distinctions between derivatives of composite functions with respect to different variables, and to learn to choose the notation that conveys your meaning clearly and unambiguously. We talked about some of these issues in class, but you should also study the textbook examples and practice problems on your own until you are confident that you can match the notation to the right derivatives. (Skip "Homogeneous functions", pp. 700-701. "Higher order derivatives", pp. 701-703, involves repeated applications of the Chain Rule.)
• Section 12.7: 1-30
• Section 12.8: 1-16
• Section 12.9: 5, 7, 8, 11, 12. The second order Taylor expansion (as in Example 2) is particularly important in applications, including the classification of critical points. You do not have to remember the general n-th order formulas from page 738. We also skipped "Approximating implicit functions", pp. 741-742. If you'd like to try Exercises 1-4 or 6, these are best done by using Taylor's expansion of a function of one variable as an intermediate step, as in Example 3. In Exercise 5, you should get a polynomial that, after simplifying, should be equal to the original polynomial.

Chapter 13:

• Section 13.1: 1-26, 29. We are skipping the proof of the test for positive or negative definiteness (Theorem 8 of Section 10.7) but a simple proof for n=2 (not using eigenvalues or linear algebra) is given in Exercise 30.
• Section 13.2: 1-12
• Section 13.3: 1-21
• Section 13.5: 2, 3, 4, 6, 9, 11, 12, 13, 14. If you would like to try something more difficult (that we have not covered in class), Questions 7, 8, 10 can be done using the method of least squares, but first you have to rewrite the equation so that the dependence on parameters is linear. For example, the equation in #7 can be rewritten as ln(y) = ln(p) + qx, and then you can look for the values of q and ln(p) that best fit the data for ln(y).

Chapter 14:

• Section 14.1: 13-22. These all can be done either by interpreting the integral as an area or volume, or by using symmetry and cancellation, or some combination of both.
• Section 14.2: 1-30.
• Section 14.3: 1-10. We are skipping "A Mean-Value Theorem for Double Integrals", pp. 822-823. I will try to say something about using integrals to compute average values (Definition 3) at the end of the course.
• Section 14.4: 1-18 (use polar coordinates), 32, 34 (find a oordinate system in which the region of integration has a simpler form).
• Section 14.5: 1-11
• Section 14.6: 1-6, 10-12
• Please note that some of the questions in the indicated sections but not on the above list can be very difficult computationally. In this class, we have emphasized the theory and proofs, and spent less time on computational practice. The practice problems reflect that. Any multiple integration problems on the final exam should not be more difficult than the practice problems here.