MATH 2433.005 - Calculus and Analytic Geometry III (Honors) - Spring 2009
TR 10:30-11:45 p.m., 122 PHSC

Instructor: Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu

Office Hours: Tue 3:30-4:30 p.m., Wed 2:30-3:30 p.m., or by appointment.

Prerequisite: MATH 2423 (Calculus and Analytic Geometry II).

Course catalog description: Polar coordinates, parametric equations, sequences, infinite series, vector analysis. (F, Sp, Su)

Text: J. Stewart, Calculus, 6th edition, Brooks/Cole, 2007. The course will cover major parts of chapters 11-13.

Homework:

• Homework 1 (problems given on Jan 20, 22, 29, Feb 3), due Feb 5 (Thu).
• Homework 2 (problems given on Feb 5, 10, 12, 17), due Feb 26 (Thu).
• Homework 3 (problems given on Feb 24, 26, Mar 3, 5), due Mar 10 (Tue).
• Homework 4 (problems given on Mar 10, 12, 24, 31), due Apr 2 (Thu).
• Homework 5 (problems given on Apr 2, 7, 9, 14), due Apr 16 (Thu).
• Homework 6 (problems given on Apr 16, 21, 23, 28), due May 5 (Tue).

Content of the lectures:

• Lecture 1 (Tue, Jan 20): Sequences: sequences, recursively defined sequences (example: Fibonacci sequence), limit of a sequence, convergent and divergent sequences, finding limits of sequences by using limits of functions (Theorem 3), convergence to infinity, limit laws for sequences, the squeeze theorem, convergence of |a_n| and of a_n to 0 (Theorem 6), interchanging the order of a limit and a continuous function (Theorem 7), examples (Sec. 12.1).
  Homework: Exercises 12.1/14, 18 (solve it in two ways), 27 (solve it in two ways), 35 (solve it in two ways, see hint), 38, 39, 40 (solve it in two ways), 41.
• Lecture 2 (Thu, Jan 22): Sequences (cont.): limits of rⁿ for different values of r (Example 10), increasing, decreasing, monotonic sequences, bounded above, bounded below, bounded sequences, monotonic sequence theorem (Sec. 12.1).
  Homework: Exercises 12.1/22, 46 (to show that a_n, you can show that |a_n| is decreasing for n>2 and is bounded below; n! is the product of the numbers 1,2,3,...,n), 58, 63, 65 (can you use the properties of this sequence to conclude that it is convergent without doing any calculations?), 69 (hint), 75.
• Lecture 3 (Thu, Jan 29): Series: series, partial sums, convergent and divergent series, geometris series, telescoping series, harmonic series; if Σa_n converges, then lima_n=0 (Theorem 6), the converse is not true, test for divergence; facts about Σca_n, Σ(a_n+b_n), Σ(a_n-b_n) for convergent series Σa_n and Σb_n and a constant c, examples (Sec. 12.2).
  Homework: Exercises 12.2/10, 18, 29, 34, 35, 38, 44, 50.
• Lecture 4 (Tue, Feb 3): The integral test and estimates of sums: the Integral Test (with proof), Remainder Estimate for the Integral Test (do the proof as an exercise), examples (Sec. 12.3).
  The comparison tests: the Comparison Test, examples (pages 741-742 of Sec. 12.4).
  Homework: Exercises 12.3/18, 20, 35; 12.4/12, 26, 28, 29, 44. The complete homework is due on Feb 5 (Thu).
• Lecture 5 (Thu, Feb 5): The comparison tests (cont.): the Limit Comparison Test, examples (pages 743-744 of Sec. 12.4).
  Alternating series: definition, the Alternating Series Test, Alternating Series Estimation Theorem, examples (Sec. 12.5).
  Homework: Exercises 12.3/37, 42; 12.4/30, 39; 12.5/12, 18, 24, 34.
• Lecture 6 (Tue, Feb 10): Absolute convergence and the ratio and root tests: absolutely convergent series, conditionally convergent series, absolute convergence implies convergence (convergence does not imply absolute convergence!); the Ratio Test, the Root Test, examples; rearrangements of series - allowed only for absolutely convergent series (Sec 12.6).
  Homework: Exercises 12.3/40; 12.5/36; 12.6/2 (try both the Ratio and the Root tests), 12, 15, 27, 31 (hint).
• Lecture 7 (Thu, Feb 12): Strategy for testing series (Sec. 12.7).
  Power series: definition of a power series, coefficients of a power series, power series centered at a, theorem about the values of x where a power series converges (pages 759-761 of Sec. 12.8).
  Homework: Exercises 12.7/10, 13, 17, 26, 27, 30, 32, 33.
• Lecture 8 (Tue, Feb 17): Power series (cont.) (Sec. 12.8).
  Homework: Exercises 12.8/7 (hint), 13, 15 (hint), 17, 23 (hint), 29 (hint), 30. The complete homework is due on Feb 26 (Thu).
• Lecture 9 (Thu, Feb 19): Exam 1 (covers Sections 12.1-12.7).
• Lecture 10 (Tue, Feb 24): Representations of functions as power series: examples of using the formula for the sum of a geometric series to derive such representations, differentiation and integration of power series, examples of applications (Sec. 12.9).
  Homework: Exercises 12.9/2, 11, 13 (hint), 20, 24, 28, 38.
• Lecture 11 (Thu, Feb 26): Taylor and Maclaurin series: Taylor series, Maclaurin series, remainder of the Taylor series, condition for convergence of the Taylor polynomials to the value of the function, Taylor's inequality, examples (exponential and trigonometric functions), binomial coefficients, Newton's binomial formula for (a+b)ⁿ for positive integer n (pages 770-778 of Sec. 12.10).
  Homework: Exercises 12.10/16, 18, 20, 21, 34, 48, 54, 64.
• Lecture 12 (Tue, Mar 3): Taylor and Maclaurin series (cont.): the binomial series (Sec. 12.10).
  Applications of Taylor polynomials: read quickly Examples 1 and 2 only and look at the graphs (pages 785-788 of Sec. 12.11).
  Curves defined by parametric equations: parametric equations, parametric curve, "curve" versus parametric curve, examples (pages 657-659 of Sec. 11.1).
  Homework: Exercises 12.11/7, 29, 33, 36; 11.1/14, 24, 31.
• Lecture 13 (Thu, Mar 5): Curves defined by parametric equations (cont.): the cycloid, remarks about its applications - brachistochrone (Johann Bernoulli, 1696), tautochrone (Christiaan Huygens, 1659) (pages 660-662 of Sec. 11.1).
  Calculus with parametric curves: slope of the tangent to a parametric curve, rates of change of the slope of the tangent to a parametric curve, area under a parametric curve, length of a parametric curve (Sec. 11.2).
  Homework: Exercises 11.1/41 (hint); 11.2/7, 16, 27, 34, 40 (just write down the integral), 48, 69.
  Food for thought: Exercise 11.2/74 (the "Food for thought" problems are not a part of the regular homework and do not need to be turned in).
  The complete homework is due on Mar 10 (Tue).
• Lecture 14 (Tue, Mar 10): Polar coordinates: polar coordinates in the plane, relations between polar and Cartesian coordinates, polar curves, tangents to polar curves (Sec. 11.3).
  Homework: Exercises 11.3/12, 14, 24, 36, 52 (hint: see 51), 62, 70.
• Lecture 15 (Thu, Mar 12): Areas and lengths in polar coordinates: derivation of the formula for the area in a polar region, derivation of the formula for the length of a parametric curve in polar coordinates, examples (Sec. 11.4).
  Homework: Exercises 11.4/7 (hint), 23, 33, 41 (hint), 44, 46.
• Lecture 16 (Tue, Mar 24): Areas and lengths in polar coordinates (cont.): more examples (Sec. 11.4). A brief review for Exam 2.
• Lecture 17 (Thu, Mar 26): Exam 2 (covers Sections 12.8-12.11, 11.1-11.4).
• Lecture 18 (Tue, Mar 31): Three-dimensional coordinate systems: remarks about differences between 2- and 3-dimensional spaces; Cartesian coordinates in 3 dimensions, equations of different objects in 3 dimensions (Sec. 13.1).
  Homework: Exercises 13.1/7, 9, 12, 16, 22, 28, 38.
  The complete homework is due on Apr 2 (Thu).
• Lecture 19 (Thu, Apr 2): Vectors: displacement vector, initial and terminal points; what does u=v mean?; addition of vectors ("the triangle rule" and "the parallelogram rule"), "scalar"="number", multiplication of a vector by a scalar, difference of vectors (defined through the other two operations); components of a vector, position vector, magnitude (length) of a vector, vector addition and multiplication of a vector by a scalar in components; fundamental properties of vectors ("machine for adding two vectors" and "machine for multiplying a vector by a scalar", properties of these "machines"); standard basis vectors, unit vectors, construction of a unit vector in the same direction as a vector a≠0 (Sec. 13.2).
  Homework: Exercises 13.2/20, 24, 28, 34, 36, 42, additional problem.
• Lecture 20 (Tue, Apr 7): The dot product: definition of dot product (scalar product, inner product), elementary properties of dot product, angle θ between vectors, a.b=|a||b|cosθ expressing cosθ through a.b, perpendicular (orthogonal) vectors, condition for orthogonality expressed through the dot product, direction angles and direction cosines, vector projection proj_ab of b onto a, scalar projection comp_ab of b onto a, geometric meanings and formal expressions for proj_ab and comp_ab, application to physics - work (Sec. 13.3).
  Homework: Exercises 13.3/11 (hint), 23, 26, 27 (hint), 34, 39, 47, 50.
• Lecture 21 (Thu, Apr 9): The cross product: determinants of 2×2 and 3×3 matrices, definition of cross product (vector product), (a×b)⊥a, (a×b)⊥b, |a×b|=|a||b|sinθ, direction of a×b (the right hand rule), condition for a to be perpendicular to b (a×b=0), geometric interpretation of |a×b| as the area of a parallelogram, properties of the cross product, triple product a.(b×c), geometric interpretation of the triple product as the volume of a parallelopiped, properties of the triple product, application to physics - torque (Sec. 13.4).
  Homework: Exercises 13.4/12, 14, 20, 30, 34, 38, 40, 43 (hint), 49 (hint).
• Lecture 22 (Tue, Apr 14): Equations of lines and planes: lines: vector equation, parameter, parametric equations, direction numbers, symmetric equations, line segments, skew lines; planes: normal vector, vector, scalar, and linear equations of a plane, parallel planes, angle between planes, distance from a point to a plane, distance between parallel planes (Sec. 13.5).
  Homework: Exercises 13.5/18, 22, 34, 38, 56, 70, 73.
  The complete homework is due on Apr 16 (Thu).
• Lecture 23 (Thu, Apr 16): Equations of lines and planes (cont.): discussion and more examples (Sec. 13.5).
  Homework: Exercises 13.4/43 (hint), 44, 46; 13.5/26, 35, 61 (hint), 63.
• Lecture 24 (Tue, Apr 21): Vector functions and space curves: component functions, vector functions, limits and continuity of vector functions, space curves, parametric equations of a space curve, curves obtained as intersections of surfaces, using computers to draw space curves, examples (Sec. 14.1).
  Homework: Exercises 14.1/2, 12, 18, 26, 28, 38, 42 (hint: see exercise 41). (Also look at exercises 19-24, 43, but do not turn them in with the homework.)
• Lecture 25 (Thu, Apr 23): Derivatives and integrals of vector functions: derivative of a vector function, tangent vector to a space curve at a point, unit tangent vector to a space curve at a point, "true" and "false" singularities of a space curve defined as a graph of a vector function, components of the derivative of a vector function, higher-order derivatives, differentiation rules, examples; definite integrals of vector functions (Sec. 14.2).
  Homework: Exercises 14.2/8, 16, 18, 22, 23, 32, 38, 49 (hint).
• Lecture 26 (Tue, Apr 28): Derivatives and integrals of vector functions (cont.): more examples (Sec. 14.2).
  Homework: Exercises 14.2/
  The complete homework is due on May 5 (Tue).
• Lecture 27 (Thu, Apr 30): Exam 3 (covers Sections 13.1-14.2).
• Lecture 28 (Tue, May 5): Arc length and curvature: arc length, arc length function, parameterizing a curve using the arc length function, smooth curves, definition of curvature, expressing the curvatuve from an arbitrary parametrization of a space curve (pages 866-868 of Sec. 14.3).
  Homework: Exercises 14.3/5 (hint), 13, 17 (hint), 31, 33 (hint).
• Lecture 29 (Thu, May 7): Motion in space: velocity and acceleration: position, velocity, speed, acceleration vector, example: motion of a projectile; derivation of Kepler's law of planetary motion saying that the planets move on ellipses (Sec. 14.4).

Attendance: You are required to attend class on those days when an examination is being given; attendance during other class periods is also expected. You are fully responsible for the material covered in each class, whether or not you attend. Periodically I will collect it to be graded (these days will be announced in advance). Make-ups for missed exams will be given only if there is a compelling reason for the absence, which I know about beforehand and can document independently of your testimony (for example, via a note or a phone call from a doctor or a parent).

Homework: It is absolutely essential to solve a large number of problems on a regular basis!
Homework will be assigned and due every class period. You should be prepared to present any of the homework problems due on a given day. Periodically I will collect it to be graded (these days will be announced in advance).
You are allowed (and encouraged) to work in small groups. However, each of you will need to prepare individual solutions written in your own words - this is the only way to achieve real understanding! Please write the problems in the same order in which they are given in the assignment.
All homework should be written on a 8.5"×11" paper with your name clearly written, and should be stapled. No late homework will be accepted!

Exams: There will be three in-class midterms and a (comprehensive) final.
Tentative dates for the midterms are Feb 17 (Tue), Mar 26 (Thu), Apr 28 (Tue).
The final is scheduled for May 11 (Mon), 8:00-10:00 a.m.
All tests must be taken at the scheduled times, except in extraordinary circumstances.
Please do not arrange travel plans that prevent you from taking any of the exams at the scheduled time.

Grading: Your grade will be determined by your performance on the following coursework:

Academic calendar for Spring 2009.

Course schedule for Spring 2009.

Policy on W/I Grades : Through February 27, you can withdraw from the course with an automatic "W". In addition, From March 2 to May 8, you may withdraw and receive a "W" or "F" according to your standing in the class. Dropping after April 6 requires a petition to the Dean. (Such petitions are not often granted. Furthermore, even if the petition is granted, I will give you a grade of "Withdrawn Failing" if you are indeed failing at the time of your petition.) Please check the dates!

The grade of "I" (Incomplete) is not intended to serve as a benign substitute for the grade of "F". I only give the "I" grade if a student has completed the majority of the work in the course (for example everything except the final exam), the coursework cannot be completed because of compelling and verifiable problems beyond the student's control, and the student expresses a clear intention of making up the missed work as soon as possible.

Academic Misconduct: All cases of suspected academic misconduct will be referred to the Dean of the College of Arts and Sciences for prosecution under the University's Academic Misconduct Code. The penalties can be quite severe. Don't do it!
For details on the University's policies concerning academic integrity see the Student's Guide to Academic Integrity. See also the Academic Misconduct Code, which is a part of the Student Code .

Students With Disabilities: The University of Oklahoma is committed to providing reasonable accommodation for all students with disabilities. Students with disabilities who require accommodations in this course are requested to speak with the instructor as early in the semester as possible. Students with disabilities must be registered with the Office of Disability Services prior to receiving accommodations in this course. The Office of Disability Services is located in Goddard Health Center, Suite 166: phone 405-325-3852 or TDD only 405-325-4173.

Good to know: The Greek alphabet.