Math 122 Calculus III

Math 122 Calculus III
Fall 2012
Prof. D. Joyce, BP 322, 793-7421
Department of Mathematics and Computer Science
Clark University

Please bookmark this page, http://aleph0.clarku.edu/~djoyce/ma130/, so you can readily access it.

General description from Clark's Academic Catalog
Math 120, 121 and 122 (Calculus I, II, and III) / Lecture
Calculus is essential for majors in biology, chemistry, computer science, mathematics, physics, and environmental science and policy. Part I includes functions, limits, continuity, differentiation of algebraic and trigonometric functions, mean value theorem, and various applications. Part II includes; Riemann sums and integrals, techniques and applications of integration improper integrals, transcendental functions; (logarithms, exponential functions, and inverse trigonometric functions). Part III includes further topics from calculus proper (sequences, series, polar coordinates) and introduces linear algebra (vectors, matrices, and linear systems). Though not all results are derived rigorously, care is taken to distinguish intuitive arguments from rigorous proofs. Math 120, 121, and 122 each fulfill the Formal Analysis requirement. Prerequisite for Math 120: appropriate score on the mathematics placement test, or appropriate grade in Math 119. Prerequisite for Math 121: Math 120, or Math 124, or AP credit in Calculus. Prerequisite for Math 122: Math 121. / Offered every fall (120, 122) and spring (121).
Course Hours. MWF 9:00–9:50. BP room 316.
My office hours are MWF 10-12, subject to change.
My office is right across the hall from the classroom. My phone extension is 7421. Email me at djoyce.
Web pages for related courses
Math 120, Calculus I
Math 121, Calculus II
Math 130, Linear Algebra
Math 131, Multivariate Calculus
Math 217, Probability and Statistics
Textbook. About half of the course is covered in the textbook you used in Math 120–121. That's University Calculus part one single variable, by Hass, Weir, & Thomas, second edition. We'll use class notes for the rest of the material. Some of the homework exercises will be taken from the text.
Syllabus
- Numbers.
  - Integers Z. Mathematical induction for positive integers.
    Natural numbers N.
  - See also appendix A.2 of the text.
  - The rational numbers Q, quotients of two integers.
  - The Real numbers R, a complete ordered field.
    See also appendix A.7 of the text.
    A field has operations of addition, subtraction, multiplication, and division and has the usual properties of those operations.
    A field is ordered if it has a distinguished set of positive elements and satisfies three axioms including the axiom of trichotemy.
    It's complete if every cut is made by some element.
  - Completeness of the real numbers, least upper bounds, greatest lower bounds, completeness axiom
  - Assignment 2 on lubs and glbs, due Mon, Sept. 10. Answers
  - Complex numbers C.
    Dave's Short Course on Complex Numbers.
    Dave's Short Trig Course
  - Assignment 3 on complex numbers, due Fri, Sept. 13 Answers
- Infinite sequences and indeterminate forms
  - Survey of sequences and series.
  - Infinite sequences of real numbers, definition and examples
  - Limits of sequences, convergence and divergence of sequences.
    Def: A sequence converges to a limit L if for each ε > 0, there is some N such that beyond the N^th term, every term is within ε of L.
    See also § 9.1 of the text.
  - Assignment 4 on limits of sequences, due Wed, Sept. 19. Answers
  - Convergence theorems
    Uniqueness of limits. Monotone bounded sequences have limits.
    Limits of sums, differences, products, and quotients.
    Continuous versus discrete limits. Limits of continuous functions of convergent sequences converge.
    Pinching theorem.
    Summary of limits of sequences.
  - (1 + 1/n)ⁿ → e. Notes.
  - Assignment 5, more limits of sequences, due Wed, Sept. 26. Answers
  - Quiz Fri, Sept. 28 on limits
  - Indeterminate forms 0/0, ∞/∞, ∞ − ∞, 0⁰, 1^∞, ∞⁰
    See also § 4.5 of the text.
    L'Hôpital's rule.
  - Assignment 6 on L'Hôpital's rule, due Mon. Oct. 1. Answers
  - Improper integrals. Integrals where one or both limits of integration are infinity. Integrals of unbounded functions.
    See also § 8.7 of the text.
- Infinite series
  - See also § 9.2 of the text.
  - Σ notation
    See also § 5.2 of the text
  - Infinite series (sums) of real numbers
    S_n, the n^th partial sum
    Definition: A series converges to a sum S if S_n → S; the series diverges if the sequence of partial sums S_n diverges
    Geometric series. a + ar + ar² + ... converges to a/(1-r) when −1 < r < 1, diverges otherwise.
    The harmonic series 1/2 + 1/3 + 1/4 + ... diverges
    The term test: only series whose terms approach 0 can converge
  - Various tests for convergence for series with positive terms. Notes
    - Integral test: a positive decreasing series converges iff the corresponding integral converges.
      See also § 9.3 of the text.
    - Comparison test: if the terms of one positive series are less than the terms of another series, and the second converges, so does the first, but if the first diverges, so does the second
      See also § 9.4 of the text.
    - Limit comparison test: If the ratio a_n/b_n of the terms of two positive series converges to a finite number, then they either both converge or both diverge
    - Root test: suppose that a_n^1/n approaches a limit L. If L < 1, the series converges; if L > 1, it diverges; but if L = 1, the test is inconclusive, so use another test
      See also § 9.5 of the text
    - Ratio test: suppose that a_n+1/a_n approaches a limit L. If L < 1, the series converges; if L > 1, it diverges; but if L = 1, the test is inconclusive, so use another test
  - Assignment 7. Due Monday, Oct 22. Answers
  - Alternating series, Absolute and conditional convergence, Leibniz' alternating series test. Notes.
    Second quiz, answers.
    See also § 9.6 of the text
  - Assignment 8. Due Friday, Oct 26. Answers
  - Power series, interval and radius of convergence. Notes.
    See also § 9.7 of the text.
    Differentation and integration of power series
  - Taylor polynomials and Taylor series
    See also § 9.8 of the text.
  - Assignment 9. Due Monday, Nov 5. Answers
  - The harmonic series and Euler's constant
  - Second test, Nov 15, on series. Answers.
- Polar coordinates
  - Polar coordinates (r, θ)
    Polar/rectangular conversion
    x = r cos θ. y = r sin θ. x² + y² = r². tan θ = y/x.
    See also § 10.1 of the text
  - Polar coordinates and complex numbers. z = x + yi = |z| (cos &theta + i sin &theta).
  - Curves in polar coordinates
    Cardioids and Limaçons, Julia and Mandelbrot Sets at http://aleph0.clarku.edu/~djoyce/julia/,
    Spiral of Archimedes, Equiangular or logarithmic spiral,
    Roulettes also called Rose curves
    See also § 10.4 of the text
  - Tangents to curves r = f(θ) in polar coordinates.
    dy
    dx = dy/dθ
    dx/dθ = r' sin θ + r cos θ;
    r' cos θ − r sin θ
    
    See also § 10.4 of the text
  - Area in polar coordinates. The area between the rays θ = α and θ = β and inside the curve r = f(θ) is
    ∫_α^β r²/2 dθ.
    See also § 10.5 of the text
- Paths of moving points, parametric equations
  - Curves given parametrically: (x(t),y(t)), that is, paths of points moving in time
    Cycloids.
    See also § 10.1 of the text
  - Tangents to parametric curves:
    dy
    dx = dy/dθ
    dx/dθ
    Velocity, the vector-valued function (x'(t),y'(t)).
    Speed: the square root of (x'² + y'²)
    Acceleration: (x''(t),y''(t)).
    See also § 10.2 of the text
  - Arclength. The path element is ds, the square root of dx² + dy².
    Thus, the speed is ds/dt is the speed, the square root of (dx/dt)² + (dy/dt)².
  - Areas under parametric curves. ∫_a^b y x' dt.
    The length of a path is ∫_a^b ds. = ∫_a^b(speed) dt.
    See also § 10.2 of the text
  - Area of a surface of revolution.. ∫_a^b 2π y ds. = ∫_a^b2π y (speed) dt.
    See also § 10.2 of the text
  - Summary of polar coordinates and parametric equations
- Vectors
  - Vectors v in the plane R² and space R³
    Vector operations: addition v + w, subtraction v − w, and multiplication by scalars av
    Geometric interpretation of vector operations
    Properties of vector operations.
  - Norm (length) ||v|| of a vector v
    Triangle inequality ||v + w|| ≤ ||v|| + ||w||
  - Dot products (a.k.a inner products or scalar products) v · w
    Angles between vectors, parallel and perpendicular vectors. Notes.
  - Unit vectors, standard bases, vectors in n-space Rⁿ, vector operations, coordinates in physical space. Notes.
  - Assignment 10. Due Friday, Nov 30. Answers -->
  - Cauchy-Schwarz inequality. Notes.
  - Determinants. Notes.
  - Cross products (a.k.a. outer products or vector products) v × w, scalar triple products. Notes.
  - Assignment 11. Due Friday, Dec 7.
- Matrices and linear transformations
  - Concept of linear transformation, matrix notation
  - Operations on linear transformations and matrices
Course administration.
There will be quizzes, two midterms, and a final exam.
Homework and Quizzes. I will assign daily practice exercises to help you master the concepts discussed in class. Although the exercises will not be collected regularly, you are expected keep up to date on the problems. Math is not a spectator sport. You learn by doing; therefore, it is only to your advantage to keep abreast of the current topics under discussion. Periodically I will assign a few specific exercises to be collected and graded, and those assignments are due in class on the assigned day; I won't accept late assignments. I'll give short 15–20 minute quizzes periodically throughout the semester.
Exams are in class, closed book, and closed notebook, but you may use one sheet of notes and a calculator. If you have a legitimate, documented excuse for missing an exam, contact me (before the scheduled time if at all possible) to reschedule the exam (at my convenience).
The final exam will be a two-hour, comprehensive exam, given during the final exam period. All students will be required to take the final exam at that time. Therefore do not make plans to leave campus before the final exam.
The course grade is computed as follows
- 20%, Assignments and quizzes
- 25%, Exam 1
- 25%, Exam 2
- 30%, Final Exam
Work expectation. You will work on average about six to eight hours each week outside of class. This will include studying from your notes and the textbook, doing the homework assignments, and studying for quizzes and exams.
Tutoring
Sunday through Thursday evenings, 8-10 p.m., room BP 312.
Past tests from 2011. Note that the material in the course last year was covered in a slightly different order with a slightly different emphasis.
Interesting Web Sites
- Humor in mathematics
- Dave's Short Trig Course, and Dave's Short Course on Complex Numbers.
- The Little Gallery of Roulettes
- Math Archives links for
- The MacTutor History of Mathematics archive including articles by J J O'Connor and E F Robertson on
- Dave Rusin's The Mathematical Atlas describes the subject of linear algebra and its subfields, at Northern Illinois University.
- On-line textbooks on linear algebra:
  - Linear Algebra: An Introduction to Linear Algebra for Pre-Calculus Students. by Tamara Anthony Carter, Richard A. Tapia, and Anne Papakonstantinou, at Rice University.
  - Linear Algebra by Jim Hefferon, at Saint Michael's College.
  - Elementary Linear Algebra, lecture notes by Keith Matthews.
  - Linear Algebra and Applications Textbook by Thomas S. Shores.

This page is located at http://aleph0.clarku.edu/~djoyce/ma122/