MAC 2312 (sec 12586): Calculus - Analytic Geometry II
The course provides an introduction to standard techniques from calculus with a single variable. The main focus is on techniques and applications of integration, differential equations, parametric equations, and series. After completion of the course, you should be acquainted with common integration techniques like the substitution rule, integration by parts or using partial fractions. You should know how to apply integration and differential equations to solve problems in physics and engineering.
Also, after completion of the course, you should be acquainted with polar coordinates, parametric equations and the concept of an infinite series.
The lectures cover Chapter 7-Chapter 11 of the book Single Variable Calculus, Early Transcendentals (James Stewart, 5th edition, Thomson Brooks/Cole, 2003), subsequently referred to as [Ste03]. The material will be presented in the same order as in the textbook:
- Techniques for exact and for approximate integration
- Applications of integration
- Differential equations
- Parametric equations and polar coordinates
- Infinite sequences and series
In the first week, a short review of the substitution rule (Chapter 5 in the textbook) is given, and in the last week of classes a review of the covered material will be provided.
More information on the course is available in the syllabus, and comments are welcome.
If you are interested, there is also tutoring for this class available.
Topics Discussed in Class
- 08/27/07: review of the substitution rule, partial integration
Literature: [Ste03, Ch. 5.5, Ch. 7.1]
- 08/29/07: examples of partial integration
Literature: [Ste03, Ch. 7.1]
- 08/30/07: trigonometric integrals; Quiz 1
Literature: [Ste03, Ch. 7.2]
- 08/31/07: trigonometric substitutions
Literature: [Ste03, Ch. 7.3]
- 09/05/07: trigonometric substitutions
Literature: [Ste03, Ch. 7.3]
- 09/06/07: integration of rational functions if the denominator is a product of distinct linear factors; Quiz 2
Literature: [Ste03, Ch. 7.4]
- 09/07/07: integrational of rational functions if the denominator splits into linear factors
Literature: [Ste03, Ch. 7.4]
- 09/10/07: integration of rational functions with quadratic factors of multiplicity one; Homework 1
Literature: [Ste03, Ch. 7.4]
- 09/12/07: integration of arbitrary rational functions
Literature: [Ste03, Ch. 7.4]
- 09/13/07: approximate integration - midpoint and trapezoidal rule
Literature: [Ste03, Ch. 7.7]
- 09/14/07: approximate integration - applying error bounds
Literature: [Ste03, Ch. 7.7]
- 09/17/07: Simpson's rule; improper integrals
Literature: [Ste03, Ch. 7.7, Ch. 7.8]
- 09/19/07: improper integrals; Quiz 3
Literature: [Ste03, Ch. 7.8]
- 09/20/07: improper integrals; comparison theorem
Literature: [Ste03, Ch. 7.8]
- 09/21/07: arc length
Literature: [Ste03, Ch. 8.1]
- 09/24/07: solutions for Homework 1
- 09/26/07: area of a surface of revolution
Literature: [Ste03, Ch. 8.2]
- 09/27/07: area of a surface of revolution
Literature: [Ste03, Ch. 8.2, Ex. 25]
- 09/28/07: moments and centers of mass
Literature: [Ste03, Ch. 8.3]
- 10/01/07: integration and probability; Homework 2
Literature: [Ste03, Ch. 8.5]
- 10/03/07: integration and probability
Literature: [Ste03, Ch. 8.5]
- 10/04/07: differential equations: direction fields and Euler's method
Literature: [Ste03, Ch. 9.1, Ch. 9.2]
- 10/05/07: separable differential equations
Literature: [Ste03, Ch. 9.3]
- 10/08/07: models for population growth, logistic equation
Literature: [Ste03, Ch. 9.4, Ch. 9.5]
- 10/10/07: linear differential equations
Literature: [Ste03, Ch. 9.6]
- 10/11/07: curves defined by parametric equations
Literature: [Ste03, Ch. 10.1]
- 10/12/07: tangents to curves defined by parametric equations
Literature: [Ste03, Ch. 10.2]
- 10/15/07: arc length of parametric curves
Literature: [Ste03, Ch. 10.2]
- 10/17/07: arc length of parametric curves
Literature: [Str03, Ch. 10.2]
- 10/18/07: computing the arc length of a cycloid; Quiz 4
Literature: [Ste03, Ch. 10.2]
- 10/19/07: introduction to polar coordinates
Literature: [Str03, Ch. 10.3]
- 10/22/07: switching between polar and Cartesian coordinates; Exam 1
Literature: [Ste03, Ch. 10.3]
- 10/24/07: symmetry of and tangents to polar curves
Literature: [Ste03, Ch. 10.3]
- 10/25/07: area and arc length in polar coordinates
Literature: [Ste03, Ch. 10.4]
- 10/26/07: introduction to conic sections
Literature: [Ste03, Ch. 10.5]
- 10/29/07: conic sections in polar coordinates
Literature: [Ste03, Ch. 10.6]
- 10/31/07: solutions for the problems in Exam 1
- 11/01/07: introduction to sequences
Literature: [Ste03, Ch. 11.1]
- 11/02/07: limits of sequences; Makeup version of Exam 1
Literature: [Ste03, Ch. 11.1]
- 11/05/07: intorduction to series
Literature: [Ste03, Ch. 11.2]
- 11/07/07: computing with series
Literature: [Ste03, Ch.11.2]
- 11/08/07: examples of convergent and divergent series
Literature: [Ste03, Ch. 11.2]
- 11/09/07:integral test for series; Quiz 5
Literature: [Ste03, Ch. 11.3, Ex. 65 in Ch. 11.2]
- 11/14/07: comparison tests for series; Homework 3
Literature: [Str03, Ch. 11.4]
- 11/15/07: alternating series
Literature: [Str03, Ch. 11.5]
- 11/16/07: absolute convergence of series
Literature: [Str03, Ch. 11.6]
- 11/19/07: introduction to power series
Literature: [Str03, Ch. 11.8]
- 11/21/07: radius of convergence of a power series
Literature: [Str03, Ch. 11.8]
- 11/23/07: representations of functions as power series
Literature: [Str03, Ch. 11.9]
- 11/26/07: differentiation and integration of power series
Literature: [Str03, Ch. 11.9]
- 11/28/07: Taylor and Maclaurin series
Literature: [Str03, Ch. 11.10]
- 11/29/07: Taylor and Maclaurin series
Literature: [Str03, Ch. 11.10. Ch. 11.12]
- 11/30/07: Binomial series
Literature: [Str03, Ch. 11.11]
- 12/03/07: solution of Homework 3; review for the final exam
- 12/05/07: review for the final exam
For questions or comments, please feel free to contact me anytime
(see my homepage for email, phone number, etc.).
Dec 14, 2007