Home for Calculus II (Spring, 2011)

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Week 1

1. Parametric equations.

2. Cycloids, Brachistochrones, Tautochrones.

3. Arclength, tagents, area of surface of revolution.

4. Polar coordinates, graph in polar coordinates.

5. Areas and lengths in polar coordinates.

6. Homewrok#1(due 3/8): p.635-636, 35,36,37;p.644,34,43,44;p.649,36,42,62,65;p.653,21;p.657,19,31,32;p.666,81;p.671,6,11,14,22,28,32,33,41,44.

Week 2

1. Conic sections: parabolas, ellipses, hyperbolas.

2. Conic sections in Euclidean coordinates.

3. Conic sections in polar coordinates, eccentricity, directrix.

4. Three or two-dimensional vectors.

5. Dot product (inner product), cross product.

6. Surfaces in three-dimensions.

7. 上課時間注意事項：3/8第九節我要上課，3/15第九節助教演習課（請至各教室），3/17第九節第一次小考（範圍3/10公布）

Week 3

1. Vector-valued function and curves in space.

2. Tangent vectors, unit tangent vectors.

3. Parametric equations in terms of arclength.

4. Curvature, unit normal vectors

5. Binormal vectors, torsion.

6. TNB frame.

7. Acceleration and velocity in polar coordinates.

8. Homewrok#2(due 3/22): p.741, 37;p.745-746,18,19;p.751-752,7,8,9,10,19,20;p.756-757,9,10,21,23;p.760,2,4;p.764,10.

9. Quiz#1(3/17): Chapter 11 and Chapter 13 (without 13.6)。

Week 4

1. Functions of several variables.

2. Graph of a function: level sets.

3. Limits of a function.

4. Continuity.

5. Quiz#2(3/29).

6. Homewrok#3(due 4/7): p.772, 52,59;p.780-782,49,50,51,60,65,66;p.791-793,60,66,71.72,91,90,92;p.800-801,29,42,44,47,50,52;p.809,28,30,36,38.

Week 5

1. Partial derivatives.

2. Differentiability of a function of several variables.

3. The gradient of a function.

4. Differentiability => Continuity.

5. Directional derivatives.

6. The meaning of the gradient.

7. Tangent planes.

Week 6

1. Chain rule, implicit differentiation.

2. Optimization: finding the extrema.

3. Critical points: zero gradient points.

4. Maximum points, minimum points, saddle points.

5. Taylor's formula.

6. Second derivative test for local extrema.

Week 7

1. Second derivative test.

2. Eigenvalues and eigenvectors of a matrix.

3. Second derivative test from the viewpoint of a symmetric matrix.

4. Optimization under constraints.

5. Quiz#3(4/12).

6. Homewrok#4(due 4/19):p.817-820, 24,31,52,66,67;p.827-828,29,37,39,46,47,65;p.836-837,29,40,43,44.

Week 8

1. Midterm on 4/19 (10:20-12:30).

2. Method of Lagrange multiplier.

3. Double integrals.

4. Iterated integrals, Fubini's theorem.

5. Double integrals in polar coordinates.

6. Triple integrals.

Week 9

1. Method of Lagrange multiplier.

2. Double integrals.

3. Iterated integrals, Fubini's theorem.

4. Double integrals in polar coordinates.

5. Triple integrals.

6. Center of mass, centroid.

7. Homewrok#5(due 5/3):p.866-868, 51,54,76,77,78,79,84;p.876-877,39,42;p.884-885,28,29,31,45,48;p.892-893,30,31,35,36;p.903-904,38,48,59,81;p.912-913,1,6,14,16,23.

Week 10

1. Triple integrals in cylindrical coordinates.

2. Triple integrals in spherical coordinates.

3. Multiple integrals under the change of coordinates.

4. Definition of Jacobian.

5. Quiz#4(5/10).

Week 11

1. Line integrals.

2. Application to medical image.

3. Line integral of a vector field.

4. Work, circulation, and flux.

5. Path independence, conservative fields, gradient fields.

Week 12

1. Component test for conservative vector fields.

2. Green's theorem in the plane, divergence.

3. Surface integrals.

4. Stokes' theorem, curl.

5. Homewrok#6(due 5/24):p.925, 26,40,42;p.937-938,51,52,53;p.948-949,25,30,38;p.959-960,17,22,30,37,38,39;p.978-979,15,39,46;p.988-989,3,7,21;p.999-1001,16,27,31.

Week13

1. Divergence theorem.

2. Deriving the continuity equation from the divergence theorem.

3. Introduction to probability.

4. Counting principles.

5. Sample space, events, probability function.

Week14

1. Definition of probability function.

2. Conditional probability.

3. Independence.

4. Bayes formula.

5. Random variables.

6. Quiz#5(5/31).

7. Homewrok#7(due 6/7):p.793-794, 26,32,34;p.806,28,29,30,40;p.817-818,20,31,32,36,37.

Week15

1. Discrete random variables, probability mass function, (cumulative) probability distribution function.

2. Expected value, variance, standard deviation.

3. Joint probability distribution.

4. Binomial distribution, hypergeometric distribution.

5. Geometric distribution.

Week16

1. Poisson distribution.

2. Poisson approximation to the binomial distribution.

3. Continuous random variables, probability density functions.

4. Normal distribution.

5. Uniform distribution.

6. Exponential distribution.

7. Quiz#6(6/14).

Week17

1. Homewrok#8: p.850-853,26,39,40,56,60,61,77,79,80,83,84;p.884-886,9,10,14,35,41,42,45,46,51,55,58.

2. Poisson process, waiting time.

3. Law of Large Numbers.

4. Central Limit Theorem.

5. Estimate the sample size.

6. Final exam on 6/21.