Home for Calculus II (Spring, 2010)

Week 1

1. Power series, radius of convergence, interval of convergence.

2. Differentiation and integration term by term, product of two power series.

3. Taylor and Maclaurin series.

4. Taylor's theorem and analytic functions.

Week 2

1.Homework #1(due 3/9):§11.7,16,26,32,35,39,40,44,46;§11.8,17,28,29,32;§11.9,6,14,23,24,30(a).

2. Fourier series.

3. Calculus of vector-valued functions.

4. Tangent vectors, normal vectors, binormal vectors.

5. Curvature, torsion.

Week 3

1. Quiz #1, 3/16.

2. Functions of several variables.

3. Graph of a function, level set.

4. Limits of a function, continuity.

5. Partial derivatives, differentiability.

Week 4

1. Differentiability, the chain rule.

2. Directional derivatives, gradient.

3. Normal direction, tangent plane, linear approximation.

4. Necessary condition for an extremum, saddle points.

5. Homework #2(due 3/25):§14.2,35,38,41,46,47,55,58;§14.3,58,65,74,76,77;§14.4,42,48;§14.5,16,18,21;§14.6,5,21,29,42,60,62,63.

Week 5

1. Finding extreme values, second order derivative test.

2. Optimization under constraints.

3. Method of Lagrange multipliers.

4. Quiz #2, 3/30.

Week 6

1. Taylor's formula for a function of multivariable.

2. Definition of double integrals.

3. Iterated integrals, Fubini's theorem for computing double integrals.

4. Area, moments, centers of mass.

5. Double integral in polar coordinates.

6, Homework #3 (due 4/8): §14.7,27,28,39,41,46,52,53;§14.8,16,29,38,42,43;§14.10,6,11

Week 7

1. No class on 4/6. The midterm is on 4/27.

2. Double integral in polar coordinates.

3. Triple integrals, Fubini's theorem.

4. Quiz #3, 4/13.

Week 8

1. Homework #4 (due 4/20): §15.1,36,38,52,59,61,66;§15.2,33,43;§15.3,13,14,42;§15.4,28,29,32,42,44.

2. Applications of triple integrals.

3. Triple integrals in the cylindrical and spherical coordinates.

4. Integrals in terms of a general change of coordinates, Jacobian.

Week 9

1. Some examples of calculating integrals using suitable change of coordinates.

2. Line integrals.

3. Mathematical foundation of the CT scan.

4. Line integrals involving vector fields.

5. Midterm on 4/27 ( §11.7 ― end of Chapter 15).

6. 準備考試，放鬆心情，聽個歌吧。

Week 10
1. Line integrals.
2. Green's theorem, divergence theorem.
3. Surface integrals, divergence theorem.
4. Stokes theorem.

Week 11
1. Homework #5 (due 5/18): §16.2,17,24,45,46;§16.3,25,32;§16.4,24,26,35;§16.6,33,43,53;§16.8,15,19,29,30,31.
2. Quiz #4 (5/18).
3. Introduction to probability.

Week 12
1. Counting principles.
2. Permutations, combinations.
3. Basic probability, conditional probability.
4. Bayes formula.

Week 13
1. Probability space.
2. Random variables: discrete, continuous.
3. Probability mass function, (cumulative) distribution function.
4. Expected value.
5. Homework #6 (due 5/27): p.806, 28, 29, 30, 33; p.817, 11, 12, 24, 25, 31, 32, 33 (problems are in your lecture notes).

Week 14
1. Variance of a random variable.
2. Independence of random variables.
3. Covariance.
4. Binomial distribution.

Week 15
1. Sampling with or without replacement. Hypergeometric distribution.
2. Geometric distribution.
3. Poisson distribution.
4. Poisson approximation to the binomial distribution.
5. Quiz #5 (6/8)

Week 16
1. Lecture note #3.
2. Homework #7 (due 6/17): p.848-852, 8, 9, 25, 40, 56, 62, 77 (這部分在第二次講義）;p.314-315,3.6,3.7,3.8,3.9,3.10,3.11; p.320, 4.6,4.7,4.10; p.325,5.2,5.3; p.327,5.7,5.8. ( 這部分在Lecture note #3).
3. Continuous random variables.
4. Probability density function, probability distribution function.
4. Uniform distribution, exponential distribution, Gamma distribution, normal distribution.

Week 17
1. Joint probability density function.
2. Law of large numbers.
3. Central limit theorem.

(千萬別罵我，祝各位暑假愉快，後會有期。）