Program Overview

MATH 227: Calculus II

Prerequisites and Bulletin Description

The main objective of Calculus II is for students to continue learning the basics of the calculus of functions of one variable. They will study both the concepts and techniques of integration, parametric equations, and infinite sequences and series, culminating with Taylor series. They will also apply these ideas to a wide range of problems that include area between curves, volume, work, arc length, and surface area. They will study approximate integration, a topic that will involve an introduction to a computer algebra system. The students should be able to interpret the concepts of Calculus algebraically, graphically, and verbally. More generally, the students will improve their ability to think critically, to analyze a problem, and solve it using a wide array of tools. These skills will be invaluable to them in whatever path they choose to follow, be it as a mathematics major or in pursuit of a career in one of the other sciences.

Course Objectives

Upon successful completion of the course, students should be able to:

• Evaluate definite, indefinite, and improper integrals exactly using basic integration techniques such as substitution, integration by parts, and partial fractions.
• Apply integration to problems such as volume, arc length, surface area, and work or hydrostatic force.
• Approximate definite integrals numerically including error analysis.
• Use calculus to solve basic problems.
• Correctly apply convergence tests to infinite series and determine the interval of convergence of a power series.
• Approximate functions by Taylor polynomials and provide error estimates.
• Sketch graphs and calculate areas in polar coordinates.

Evaluation of Students

Students will be evaluated on their ability to devise, organize, and present complete solutions to problems. While instructors may design their own methods of evaluating student performance, these methods must include in-class examinations, frequent homework assignments, and a final exam.

Course Outline

The course outline includes the following topics, along with their respective durations and text sections:

• Integration | 2 weeks | Text Sections 5.5, 6.2
• Applications of Definite Integrals | 2 weeks | Text Sections 6.3 - 6.5, 6.6*, 6.7
• Exponential Growth and Decay | 1 week | Text Sections 6.8 - 6.9
• Techniques of Integration | 3 weeks | Text Sections 7.1, 7.2, 7.5 - 7.8
• Infinite Sequences and Series | 4 weeks | Text Sections 8.1 - 8.6, 9.1 - 9.4
• Polar Coordinates, Polar Graphs, and Areas | 2 weeks | Text Sections 10.1 - 10.3

Textbooks and Software

• Calculus Early Transcendentals Second Edition by Briggs, Cochran, and Gillet
• Mathematica by Wolfram Research