Mathematics 226 practice problems, Fall 2016
These problems are for your own practice, not to be handed in. Do as many as you need, until
you are confident that you know how to solve problems of this type. Solutions will not be
provided here, but you can find the answers to odd-numbered questions at the end of the textbook,
and complete solutions to even-numbered problems in the Student Solutions Manual.
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.
(You can skip "Homogeneous functions" and Euler's Theorem, pp. 700-701.)
- Section 12.6: 1-20.
(You can skip "Differentials in Applications" and "Differentials and Legendre Transformations",
pp. 712-714. These sections provide information about applications of differentials in physics,
for those interested, but they will not be required on tests or homework.)
- Section 12.7: 1-20. ("Rates Perceived by a Moving Observer", pp. 722-723, will not be required, but it provides a nice
interpretation of directional derivatives.)
- Section 12.8: 1-4, 6, 13, 14, 15, 16. We will only focus on two important special cases: functions of two variables
defined by an equation F(x,y,z)=0 (Example 1), and coordinate transformations (Examples 3, 8). We will also need the concept of the Jacobian
(Definition 8). We will not try to cover the general theory for arbitrary systems of equations (as in Theorem 8).
- Section 12.9: 5, 7, 8, 11, 12. This will not be required on Midterm 2. 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 will also skip "Approximating implicit functions", pp. 74
1-742.
Chapter 13:
- Section 13.1: 1-26
- Section 13.2: 1-12
- Section 13.3: 1-21
Chapter 14:
- Section 14.1: 13-22. All of these can be done either by interpreting the integral as an area or volume, or by using symmetr
y and cancellation, or some combination of both.
- Section 14.2: 1-30.
- Section 14.3: 1-10. (We skipped "A Mean Value Theorem for Double Integrals", pp. 822-824.)
- Section 14.4: 1-18 (use polar coordinates), 32, 34 (find a coordinate system in which the region of integration has a sim
pler form). There was no time in class for a full proof of the conversion formula with the Jacobian, but the textbook has a more detailed explanation on pages 830-831.
- Section 14.5: 1-10. (In some of the problems, you may have to try different orders of integration.)
- Section 14.6: 1-4, 10-12
[Mathematics Department]
[University of British Columbia]