MATH 2423 - Calculus and Analytic Geometry II (Honors), Section 001 - Fall 2008
TR 12:00-1:15 p.m., 120 PHSC

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

Office Hours: Mon 1:30-2:30 p.m., Wed 1:30-2:30 p.m., or by appointment.

Prerequisite: MATH 1823 (Calculus and Analytic Geometry I).

Course catalog description: Integration and its applications; the calculus of transcendental functions; techniques of integration; and the introduction to differential equations. A student may not receive credit for this course and 2123. (F, Sp, Su) [I-M]

Text: J. Stewart, Calculus, 6th edition, Brooks/Cole, 2007. The course will cover (parts of) chapters 5-9.
Optional supplementary text: C. Adams, J. Hass, and A. Thompson, How to Ace Calculus: The Streetwise Guide, Freeman, 1998 (about one third of this book is relevant to this course).

Homework (solutions are deposited after the due date in the Chemistry-Mathematics Library, 207 PHSC):

• Homework 1 (problems given on Aug 26 & 28, Sep 2 & 4), due Sep 9 (Tue).
• Homework 2 (problems given on Sep 9, 11, 16 & 18), due Sep 23 (Tue).
• Homework 3 (problems given on Sep 30, Oct 2, 7 & 9), due Oct 14 (Tue).
• Homework 4 (problems given on Oct 14, 16, 21, 23), due Oct 28 (Tue).
• Homework 5 (problems given on Nov 4, 6, 11, 13), due Nov 18 (Tue).
• Homework 6 (problems given on Nov 18, 20, Dec 4, 9), due Dec 11 (Thu).

Content of the lectures:

• Lecture 1 (Tue, Aug 26): Basic theorems from Calculus I: Intermediate Value Theorem, Mean Value Theorem, examples of applications, discussion of the importance of all conditions in the statements of the theorems; antiderivatives, basic trigonometric identities (pages 104-105 of Sec. 2.5, pages 214-219 of Sec. 4.2, Sec. 4.9).
  Homework: Problems 2.5/48, 54; 4.2/12, 16, 19 (see the hint at the Hints), 27; 4.9/33.
• Lecture 2 (Thu, Aug 28): Sigma notation: definition, examples, basic theorems, derivation of formulas for sums of first and second powers of the first n natural numbers, telescoping sums (Appendix E).
  Areas and distances: Approximating areas by rectangles, computing the area under a parabola, distance traveled by a moving object (Sec. 5.1).
  Homework: Problems 5.1/4, 14, 18, 20, 22, 26, additional problems.
• Lecture 3 (Tue, Sep 2): The definite integral: definition as a limit of Riemann sums, integrable functions, upper and lower limits of integration, integrand; relation with areas under/over the graph of the integrand, examples of evaluating integrals by computing Riemann sums and by using geometric ideas, properties of definite integrals (Sec. 5.2).
  Homework: Problems 5.2/11, 18, 21, 40, 41, 54, 62.
• Lecture 4 (Thu, Sep 4): The Fundamental Theorem of Calculus (FTC): derivative of a definite integral as a function of its upper limit, FTC - parts 1 and 2, examples of application (Sec. 5.3).
  Read: the proof of Part 1 of FTC (pages 315-316).
  Homework: Problems 5.3/2, 12, 32, 34, 48, 52, 60. The complete homework is due on Sep 9 (Tue).
• Lecture 5 (Tue, Sep 9): Indefinite integrals and the net change theorem: antiderivatives, indefinite integrals, definite integrals, making a table of indefinite integrals, the net change theorem, examples, velocity and speed, displacement and traveled distance, linear density of a rod and mass of a rod segment, concentration of a chemical reactant and change of the mass of the reactant, power and energy produced/consumed (Sec. 5.4).
  Homework: Problems 5.4/4, 10, 40, 42, 45, 50, 56.
• Lecture 6 (Thu, Sep 11): The substitution rule: derivation and examples, connection with the chain rule, the substitution rule for definite integrals, using symmetry to solve integrals (Sec. 5.5).
  Homework: Problems 5.5/18, 24, 30, 38, 42, 54, 62.
• Lecture 7 (Tue, Sep 16): Areas between curves: using integrals to compute the area between two curves when the area is between the straight lines x=a and x=b and the curves y=f(x) and y=g(x); computing the area between two curves when the area is between the straight lines y=c and y=d and the curves x=f(y) and x=g(y); examples (Sec. 6.1).
  Homework: Ch. 5 Review (pp. 341-343): exercises 2(b,c,d), 3, 8. Problems 6.1/4, 10 (using both methods), 32, 40.
• Lecture 8 (Thu, Sep 18): Volumes: computing the volume of a 3-dimensional object as an integral of the area of the cross-sections with a family of parallel planes, examples (Sec. 6.2).
  Homework: Ch. 5 Review (pp. 341-343): exercises 22, 26, 42, 50(a,b). Problems 6.2/21, 32, 50.
  The complete homework is due on Sep 23 (Tue).
  Exam 1 will be in class on Sep 25 (Thu). It will cover the material from Sections 5.1-5.5, 6.1, 6.2.
• Lecture 9 (Tue, Sep 23): Review for Exam 1: problems from Sections 5.1-5.5, 6-1, 6.2.
• Lecture 10 (Thu, Sep 25): Exam 1.
• Lecture 11 (Tue, Sep 30): Volumes by cylindrical shells: idea, computing the volume of a cylindrical shell "honestly" and by "unfolding" it, examples (Sec 6.3).
  Homework: Problems 6.2/41, 51 (hint), 63 (hint). Problems 6.3/44, 45 (hint).
• Lecture 12 (Thu, Oct 2): Work: definition, work done to stretch a spring (Hooke's law), work done to pump the water out of a conical tank (Sec 6.4).
  Homework: Problems 6.3/25 (hint). Problems 6.4/9 (hint), 13 (hint), 17 (hint), 22, 24, 29.
• Lecture 13 (Tue, Oct 7): Average value of a function: definition, the Mean Value Theorem for Integrals, applications: computing the mean age of a population (histogram, probability density function), computing the average speed of a fluid through a cylindrical pipe based on Problem 22 on p. 377 (Sec 6.5).
  Homework: Problems 6.5/ 9 (hint), 13 (hint), 17 (hint), 20, 23 (hint), 24, additional problems.
• Lecture 14 (Thu, Oct 9): Inverse functions: one-to-one function, domain and range of a function, conditions for the invertibility of a function, examples, cancellation equations f^-1(f(x))=x, f(f^-1(y))=y, derivative of an inverse function - derivation using the chain rule, examples (Sec 7.1).
  Homework: Problems 7.1/19 (hint), 21 (hint), 28, 32, 39, 41, 46.
  The complete homework is due on Oct 14 (Tue).
• Lecture 15 (Tue, Oct 14): The natural logarithmic function: definition of the natural logarithm through a definite integral, derivative of ln(x), laws of logarithms (with derivation), examples, definition of the number e, derivative of ln|x|, examples, logarithmic differentiation (Sec 7.2^*).
  Homework: Problems 7.2^*/14, 30, 38, 46, 50, 62, 72.
• Lecture 16 (Thu, Oct 16): The natural exponential function: definition of the number e as the unique solution of ln(x)=1, definition of the function exp as the inverse of the function ln, continuous functions, properties of exp, derivative of exp, examples (Sec 7.3^*).
  Homework: Problems 7.3^*/22, 26, 32, 46, 51, 78, 97.
• Lecture 17 (Tue, Oct 21): General logarithmic and exponential functions: definition of a^x, laws of exponents, derivative and integral of a^x, graphs of the exponential function, derivation of the power rule, general logarithmic functions, change of base formula, derivative and integral of log_ax, the number e as a limit (Sec. 7.4^*).
  Homework: Problems 7.4^*/8, 24, 34, 38, 42, 50, 52.
• Lecture 18 (Thu, Oct 23): Exponential growth and decay: differential equation describing the growth/decay of a population, initial condition, solution of the initial value problem, including more terms in the differential equation to make it more realistic by preventing the unbounded growth, carrying capacity, behavior of the solutions of the modified equation, radioactive decay, Newton's law of cooling, continuously compounded interest (Sec. 7.5).
  The principle of mathematical induciton: description, example of application: proof that 1+nx never exceeds (1+x)ⁿ for x>0 and n=0,1,2,3,... (pages 55, 58-59).
  Homework: Problems 7.5/4, 9, 12, 16, additional problems.
  The complete homework is due on Oct 28 (Tue).
  Exam 2 will be in class on Oct 30 (Thu). It will cover the material from Sections 6.3-6.5, 7.1, 7.2^*-7.4^*, 7.5.
• Lecture 19 (Tue, Oct 28): Review for Exam 2: problems from Sections 6.3-6.5, 7.1, 7.2^*-7.4^*, 7.5, and Mathematical Induction (pages 55, 58-59).
• Lecture 20 (Thu, Oct 30): Exam 2.
• Lecture 21 (Tue, Nov 4): Inverse trigonometric functions: definition, domains and ranges, derivatives of inverse trigonometric functions, examples (Sec. 7.6).
  Homework: Problems 7.6/21, 27, 34, 46, 60, 63, 74.
• Lecture 22 (Thu, Nov 6): Hyperbolic functions: definition, domains and ranges, derivatives of hyperbolic functions, inverse hyperbolic functions, expressing the inverse hyperbolic functions through logarithms, derivatives of inverse hyperbolic functions, examples (Sec. 7.7).
  Homework: Problems 7.7/12, 19, 25, 29(a), 49 (additional question: find an approximate expression for v for shallow water), 52, 68.
• Lecture 23 (Tue, Nov 11): Indeterminate forms and L'Hospital's rule: L'Hospital's rule for indeterminate forms of type ∞/∞ and 0/0, indeterminate products (of type 0⋅∞), indeterminate differentces (of type ∞-∞), indeterminate powers (of type 0⁰, ∞⁰, 1^∞), examples (Sec. 7.8).
  Homework: Problems 7.8/9, (8, 21, 28, 33), 49, (56, 62, 64), 77(a,b,c), 88, 91.
• Lecture 24 (Thu, Nov 13): Integration by parts: derivation of the formula for integration by parts from the product rule, examples (Sec. 8.1).
  Trigonometric integrals: solving integrals of functions of the form sinⁿ(x)cos^m(x) (pages 495-498 of Sec. 8.2).
  Homework: Problems 8.1/8, 10, 15 (hint), 17 (hint), 22, 32, 35 (hint), 44.
  The complete homework is due on Nov 18 (Tue).
• Lecture 25 (Tue, Nov 18): Trigonometric integrals (cont.): more trigonometric integrals (Sec. 8.2).
  Trigonometric substitution: ideas and examples of trigonometric substitution (Sec. 8.3).
  Homework: Problems 8.2/6, 12, 15, 23 (hint), 25, 44.
  Homework: Problems 8.3/4, 13 (hint), 31 (hint).
• Lecture 26 (Thu, Nov 20): Integration of rational functions by partial fractions: reducing a rational function P(x)/Q(x) to its proper form, S(x)+R(x)/Q(x) [where S, R, and Q are polynomials with deg(R) smaller than deg(Q)] partial fraction expansion when the denominator is a product linear factors (pages 509-513 of Sec. 8.4).
  Homework: Problems 8.3/22 (hint), 24, 28, 39.
  Homework: Problems 8.4/17 (hint), 22, 51, 55 (hint), 59.
  Exam 3 will be in class on Dec 2 (Tue). It will cover the material from Sections 7.6-7.8, 8.1-8.4.
• Lecture 27 (Tue, Nov 25): Review for Exam 3: problems from Sections 7.6-7.8, 8.1-8.4.
• Lecture 28 (Tue, Dec 2): Exam 3.
• Lecture 29 (Thu, Dec 4): Improper integrals: improper integrals of type I and type II, convergent and divergent improper integrals, comparison theorem (Sec. ).
  Homework: Problems 8.8/7 (hint), 21 (hint), 26, 32, 41, 49 (hint), 50, 70 (hint).
  
• Lecture 30 (Tue, Dec 9): Arc length: derivation of the arc length formula, examples, the arc length function (Sec. 9.1).
  Area of a surface of revolution: derivation of the formula formula for the area of a surface of revolution, examples (Sec. 9.2).
  Homework: Problems 9.1/13 (hint), 32, 40(a).
  Homework: Problems 9.2/5 (hint), 15 (hint), 25 (hint), 29.
  The complete homework is due on Dec 11 (Thu).
• Lecture 31 (Thu, Dec 11): Review.

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!
Shortly after a homework assignment's due date, solutions to the problems from that assignment will be placed on restricted reserve in the Chemistry-Mathematics Library in 207 PHSC.

Exams: There will be three in-class midterms and a (comprehensive) final.
Tentative dates for the midterms are September 23, October 23 and November 25.
The final is scheduled for Thursday, December 18, 1:30-3:30 p.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 Fall 2008.

Course schedule for Fall 2008.

Policy on W/I Grades : Through October 3, you can withdraw from the course with an automatic "W". In addition, From October 6 to December 12, you may withdraw and receive a "W" or "F" according to your standing in the class. Dropping after November 3 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.) 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 more details on the University's policies concerning academic misconduct see http://www.ou.edu/provost/integrity/. See also the Academic Misconduct Code, which is a part of the Student Code and can be found at http://www.ou.edu/studentcode/.

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.