SANTA BARBARA CITY COLLEGE

ASSOCIATE DEGREE CREDIT COURSE OUTLINE

Department: Mathematics

Subject Area and Course Number: Mathematics 160

Course Title: Calculus, with Analytic Geometry II

Discipline: Mathematics

Units: 5

Repeatability: None

Catalog Course Description: Techniques of integration; applications of definite integrals; polar and parametric equations; sequences and infinite series; introduction to differential equations and to vectors.

Description for Schedule of Classes: Techniques of integration; applications of definite integrals; polar and parametric equations; sequences and infinite series; introduction to differential equations and to vectors.

Lecture Hours per Week: 5.3 (80-90 Total Semester Hours)

Laboratory Hours per Week: None

Plus Hours: None

Prerequisite: Math 150, with a grade of "C" or better

Co-requisite: None

Skills Advisories: Eligibility for English 100 and English 103

Course Advisories: None

Limitation on Enrollment: None

Course Objectives: By the end of this course, the student will be able to:

1. Apply techniques of integration to indefinite, definite, and improper integrals.

2. Use integrals to solve problems in the physical and mathematical sciences and other disciplines.

3. Analyze behavior of curves given in parametric form, including applications to differential and integral calculus.

4. Analyze behavior of curves given in polar form, including applications to differential and integral calculus.

5. Calculate the Taylor polynomials and series of a function about a point, including error estimates, and use them to solve problems in the physical and mathematical sciences.

6. Apply beginning techniques of differential equations to appropriate situations.

7. Use vector algebra to solve geometric and algebraic problems.

8. Compute areas and lengths using polar coordinates.

Course Content and Scope:

1. Techniques of integration

a. Substitution (Review)

b. Integration by parts (Review)

c. Tables of Integrals

d. Partial Fraction Decomposition

e. Trigonometric Substitution

f. Trigonometric Integrals

g. Numerical Approximations of definite integrals

h. Improper Integrals

2. Applications of the integral

a. Applications to Geometry

1) Areas between Curves (Review)

2) Volumes

3) Arc Length

4) Surface Area of Solids of Revolution

b. Applications to Physics

1) Work

2) Hydrostatic Force

3) Moments and Centers of Mass

c. Applications to Economics (optional)

d. Probability Distributions (optional)

3. Differential Equations

a. Review of first order ODE's and slope fields

b. Introduction to Linear second order differential equations

c. Applications of first and second order differential equations

4. Sequences and infinite series

a. Convergence of Sequences

b. Taylor Polynomials

c. Taylor Series

1) Radius of Convergence

2) Error in Taylor Approximations

3) Binomial Series

4) New Series by Substitution, Differentiation and Integration

5) Deriving Euler's Formula

d. Geometric Series

e. Fourier Series (optional)

5. Vectors

a. Vector Arithmetic

b. Applications to force, work and torque

6. Parametric Equations and Polar Coordinates

a. Calculus with curves in parametric form

b. Polar Coordinates

1) Areas and Lengths

2) Calculus with curves in polar form

3) Conic Sections

Method of Instruction: Lecture is the primary activity, along with student problem-solving. Problem-solving will frequently require the use of graphing calculators or computer software. Students are also expected to work outside of class on assigned exercises and supplemental reading from the text.

Required Assignments:

A. Appropriate Readings: Students are required to read assigned chapters in texts.

B. Writing Assignments: Students must work assigned mathematical problems requiring the manipulation of abstract symbols, and give a clear and logical written presentation of their solutions..

C. Appropriate Outside Assignments: Students will be expected to spend a sufficient amount of time outside of class to practice techniques taught during class time, read assigned materials, and complete frequent homework assignments.

D. Appropriate Assignments that Demonstrate Critical Thinking: Students must demonstrate mathematical skills which involve analyzing information, recognizing concepts in new contexts, and drawing analogies. They must also analyze logical arguments for validity and write proofs of their own using both inductive and deductive reasoning within a logical system.

Methods of Evaluation: A student's grade will be based on multiple measures of performance in the solving of problems, preparation and analysis of graphs, and analysis of logical arguments. Such measures will include at least three exams and a comprehensive final examination requiring demonstrations of problem-solving skills. In addition, instructors may make use of quizzes, written homework assignments, or other appropriate means to judge a student's dexterity with mathematical skills and familiarity with mathematical vocabulary. Calculator (or computer) use is incorporated in the course, but students are expected to perform differentiation and some integration "by hand" and/or using tables.

Appropriate Texts and Supplies:

Stewart, Calculus with Early Transcendentals, 6^th Ed., Cengage Publishing, 2008

Rogawski, Calculus with Early Transcendentals, 1^st Ed., Freeman Publishing, 2008

TI-84 Graphing Calculator, Maple, or equivalent computer algebra system

Student Learning Outcomes:

1. Recognize and apply appropriate techniques to evaluate definite, indefinite, and improper integrals.

2. Approximate definite integrals or a function’s value by appropriate numerical methods, including error estimation.

3. Use definite integrals for geometry, physics, and other applications.

4. Use techniques of calculus to analyze polar and parametric curves.

5. Express functions as power series and analyze convergence of infinite series..

6. Solve separable and linear differential equations.

CO/mej

Revised October 16, 2006; 5/13/09; 8/24/09

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