Calculus II

Discipline: Mathematics

Course Code: 201-NYB-05

Objectives: 00UP, 00UU

Ponderation: 3-2-3

Credits: 2 2/3

Prerequisite: 00UN

Semester: ____________________

Instructor:____________________

Office: ____________________

Telephone: 457-6610 Loc: _______

Office Hours:
Mon:_______________ Tues: _________________ Wed:_______________ Thurs: _________________ Fri:_______________

Introduction

Calculus II is the sequel to Calculus I, and so is the second Mathematics course in the Science Programme. It is generally taken in the second semester. The Science student at John Abbott will already be familiar with the notions of definite and indefinite integration from Calculus I. In Calculus II these notions are studied to a greater depth, and their use in other areas of science, such as Physics and to a lesser extent Chemistry, is explored. In addition, the course introduces the student to the concept of infinite series, and to the representation of functions by power series.

The primary purpose of the course is the attainment of Objective 00UP (“To apply the methods of integral calculus to the study of functions and problem solving”). To achieve this goal, the course must help the student understand the following basic concepts: limits, derivatives, indefinite and definite integrals, improper integrals, sequences, infinite series, power series involving real-valued functions of one variable (including algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions).

Emphasis is placed on clarity and rigour in reasoning and in the application of methods. The student will learn to use the techniques of integration in several contexts, and to interpret the integral both as an antiderivative and as a sum of products. The basic concepts are illustrated by applying them to various problems where their application helps arrive at a solution. In this way, the course encourages the student to apply learning acquired in one context to problems arising in another.

Students will be encouraged to use a scientific graphing calculator; suitable mathematical software programs (such as MAPLE V) are available for student use in the Mathematics Lab. The course uses a standard college-level Calculus textbook, chosen in collaboration by the Calculus I and II course committees.

Integration, Comprehensive Assessment, and the Exit Profile

As partial fulfillment of Objective 00UU, each student will undertake a project for his/her Program Comprehensive Assessment Module, which will use the concepts and techniques learned in the course to solve some problems in another field of science. This activity will also be designed to specifically achieve the following goals of the Exit Profile (at an introductory level):

(5A) to use appropriate data processing technologies.
(7.1A) to communicate effectively, reading scientific material.
(14A) to apply what has been learned to new situations.

In addition, it is expected that the course will also address and evaluate, to some extent, the following exit profile goals:

(4A) to apply a systematic approach to problem solving.
(6A) to reason logically.
(8A) to learn in an autonomous manner.
(9A) to work as members of a team.
(12A) to put in context the emergence and development of scientific concepts.

OBJECTIVES

STANDARDS

Statement of the Competency

General Performance Criteria

To apply the methods of integral calculus to the study of functions and problem solving (00UP).

· Appropriate use of concepts

· Adequate representation of surfaces and solids of revolution

· Correct algebraic operations

· Correct choice and application of integration techniques

· Accurate calculations

· Proper justification of steps in a solution

· Correct interpretation of results

· Appropriate use of terminology

Elements of the Competency
Specific Performance Criteria

1. To determine the indefinite integral of a function.
2. To calculate the limits of functions presenting indeterminate forms.
3. To calculate the definite integral and the improper integral of a function on an interval.
4. To express concrete problems as differential equations and solve differential equations.
5. To calculate volumes, areas, and lengths, and to construct two and three dimensional drawings.
6. To analyze the convergence of series.
7. To undertake an interdisciplinary project which integrates current learning and demonstrates competence in three specific goals of the exit profile at an introductory level (00UU).
[Specific performance criteria for each of these elements of the competency are shown below with the corresponding intermediate learning objectives. For the items in the list of learning objectives, it is understood that each is preceded by: “The student is expected to …”.]

STANDARDS OBJECTIVES

Specific Performance Criteria
Intermediate Learning Objectives

1. Indefinite integrals

1.1 Use of basic substitutions to determine simple indefinite integrals.
1.1.1 Express Calculus I differentiation rules as antidifferentiation rules.
1.1.2 Use these antidifferentiation rules and appropriate substitutions to calculate indefinite integrals.

1.2 Use of more advanced techniques to determine more complex indefinite integrals.
1.2.1 Use algebraic identities to prepare indefinite integrals for solution by substitution.
1.2.2 Evaluate an indefinite integral by integration by parts.
1.2.3 Evaluate an indefinite integral using trigonometric identities.
1.2.4 Evaluate an indefinite integral by partial fractions.
1.2.5 Evaluate an indefinite integral by selecting the appropriate technique.
1.2.6 Evaluate an indefinite integral using a combination of techniques.

2. Limits of indeterminate forms

2.1 Use of l’Hôpital’s rule to determine limits of indeterminate forms.
2.1.1 State l’Hôpital’s rule and the conditions for which it is valid.
2.1.2 Calculate limits of the indeterminate forms 0/0 and ¥/¥ using l’Hôpital’s rule.
2.1.3 For indeterminate forms 0 ´ ¥, ¥ - ¥, 1^¥, 0⁰, ¥⁰, use the appropriate transformation to determine the limit using l’Hôpital’s rule.

3. Definite and improper integrals

3.1 Use of the Fundamental Theorem of Calculus to calculate a definite integral.
3.1.1 Use the Fundamental Theorem of Calculus to calculate definite integrals.

3.2 Use of limits to calculate improper integrals.
3.2.1 Calculate an improper integral where at least one of the bounds is not a real number.
3.2.2 Calculate an improper integral where the integrand is discontinuous at one or more points in the interval of integration.

4. Differential equations

4.1 Use of antidifferentiation to obtain general solutions to simple differential equations.
4.1.1 Express a simple differential equation in the language of integration, and obtain the general solution.

4.2 Use of antidifferentiation to obtain particular solutions to simple initial value problems.
4.2.1 Express a simple initial value problem in the language of integration, and obtain the particular solution.

5. Areas and Volumes

5.1 Use of differentials to set up definite integrals
5.1.1 Analyze a quantity A as a sum åDA over an interval [a,b]. Approximate DA by a product f(x) dx; conclude dA/dx = f(x) and hence A is the definite integral ò_a^bf(x) dx.

5.2 Calculation of areas of planar regions.
5.2.1 Use 5.1.1 to set up a definite integral to calculate an area.
5.2.2 Sketch the area between two functions ( y = f(x), y = g(x) ) and use 5.2.1 to calculate the area..
5.2.3 Sketch the area between two functions ( x = f(y), x = g(y) ) and use 5.2.1 to calculate the area..
5.2.4 Sketch the area between two functions ( y = f(x), x = g(y) ) and determine the most efficient way (5.2.2, 5.2.3) to calculate the area.

5.3 Calculation of volumes of revolution.
5.3.1 Sketch the three dimensional solid obtained by rotating a region (of type 5.2.2 or 5.2.3) around the x axis or the y axis.
5.3.2 Use 5.1.1 to set up a definite integral to calculate the volume of the solid (5.3.1) by disks.
5.3.3 Use 5.1.1 to set up a definite integral to calculate the volume of the solid (5.3.1) by shells.
5.3.4 Determine the most efficient way (disks or shells) to calculate the volume of the solid (5.3.1), and calculate the volume by that method.

6. Infinite Series

6.1. Determination of the convergence or divergence of a sequence.
6.1.1 State the definition of the limit of a sequence.
6.1.2 Use 6.1.1 to calculate the limit of a sequence, or to show that it diverges.
6.1.3 Use the Squeeze theorem to calculate the limit of a sequence.

6.2 Determination of the convergence or divergence of an infinite series of positive terms
6.2.1 State the definition of convergence for an infinite series.
6.2.2 State the n^thterm test for the divergence of a series.
6.2.3 Use 6.2.1 to determine if a telescoping series converges, and if so, calculate the sum.
6.2.4 State the criterion for the convergence of an infinite geometric series.

6.2.5 Calculate the sum of a converging geometric series (6.2.4); use this to solve appropriate problems (e.g. the distance travelled by a bouncing ball).
6.2.6 State the Integral, p series, Direct Comparison, Limit Comparison, Ratio, and n^th Root tests for convergence of an infinite series.
6.2.7 Determine whether an infinite series converges or diverges by choosing (and using) the correct method among (6.2.1 – 6.2.6).

6.3 Determination of the convergence, conditional or absolute, or divergence of an infinite series.
6.3.1 State the definitions of absolute and conditional convergence of infinite series.
6.3.2 State the definition of an alternating series.
6.3.3 State the criterion for the (conditional) convergence of an alternating series.
6.3.2 Determine if an infinite series is absolutely convergent, conditionally convergent, or divergent, using the methods of (6.2.1 – 6.2.7, 6.3.1 – 6.3.3).

6.4 Expression of functions as power series.
6.4.1 Use the methods of (6.2, 6.3) to find the radius and interval of convergence for a power series.
6.4.2 State the definitions of the Taylor and Maclaurin polynomials of degree n for a function f.
6.4.3 State the definitions of the Taylor and Maclaurin series for a function f based at x = c.
6.4.4 Use 6.4.3 to approximate a function f at a given point.

7. Integration, Comprehensive Assessment, and Exit Profile

7.1 Use of appropriate date processing technologies.
7.1.1 Use a computer and its main peripherals.
7.1.2 Use the principal type(s) of data processing software appropriate to the project.

7.2 Effective communication, reading scientific material.
7.2.1 Read one or more passages from a mathematics or scientific text, and summarize its contents.

7.3 The application of what has been learned to new situations.
7.3.1 Use techniques and results from the course and from assigned readings to solve interdisciplinary problems which have not specifically been covered in the course.
7.3.2 Choose from various notions and techniques those that are most suitable for the resolution of new problems, and apply them successfully.

7.4 Clear demonstration of links between mathematics and at least one other science discipline.
7.4.1 Apply knowledge or skills that have been acquired in Calculus II to topic(s) in Physics, Chemistry, or Biology.

Course Information

Methodology

This course will be 75 hours, meeting three times a week for a total of five hours a week. Most teachers of this course rely mainly on the lecture method, although most also employ at least one of the following techniques as well: question-and-answer sessions, labs, problem-solving periods, class discussions, and assigned reading for independent study.

The Program Comprehensive Assessment Module has an interdisciplinary aspect, and is aimed at developing specific Exit Profile skills that students are expected to attain in the course.

The Mathematics Lab (H-203) functions both as a study area and as a centre where students may seek help with their mathematics courses. The Learning Centre (H-117A) offers student skills classes and individual tutoring.

Bibliography

Text: Single Variable Calculus: Early Transcendentals (4^th Ed.)

by James Stewart, (Brooks/Cole Pub. Co.)

Cost: Approximately $100.

Evaluation

A student's Final Grade is a combination of the Class Mark and the mark on the Final Exam. The class mark will include three or more tests, and possibly homework, quizzes, and other assignments. 20% of the class mark will be determined by the Program Comprehensive Assessment Module. The specifics of the class mark will be given by your instructor during the first week of classes. Every effort is made to ensure equivalence between the various sections of the course.

The Final Exam is set by the Course Committee (which consists of all instructors currently teaching this course).

The Final Grade will be whichever is the better of:

50% Class Mark and 50% Final Exam Mark
or
25% Class Mark and 75% Final Exam Mark

A student with a Class Mark of less than 50% MAY CHOOSE NOT TO WRITE the Final Exam, in which case the Class Mark (< 50%) will be assigned as the Final Grade.

Course Costs

In addition to the cost of the text listed in the Bibliography, a handheld scientific calculator ($10-$25) is essential; a graphics calculator (approximately $100 or more) would be useful.

Regulations

1. Student participation in Special Activities for Student Success (SASS) is obligatory if required by the instructor.

2. Regular attendance is expected. Missing six classes is grounds for automatic failure in this course. The enforcement of this regulation is up to each individual instructor.

3. The Mathematics Department considers any form of cheating to be a serious offense. Cheating includes, but is not limited to using unauthorized material, viewing another student's test while the test is being given, copying another person's work, and allowing your own work to be copied. If you are caught cheating you should expect to be penalized.

STANDARDS	OBJECTIVES
Specific Performance Criteria	Intermediate Learning Objectives

1. Indefinite integrals
1.1 Use of basic substitutions to determine simple indefinite integrals.	1.1.1 Express Calculus I differentiation rules as antidifferentiation rules. 1.1.2 Use these antidifferentiation rules and appropriate substitutions to calculate indefinite integrals.
1.2 Use of more advanced techniques to determine more complex indefinite integrals.	1.2.1 Use algebraic identities to prepare indefinite integrals for solution by substitution. 1.2.2 Evaluate an indefinite integral by integration by parts. 1.2.3 Evaluate an indefinite integral using trigonometric identities. 1.2.4 Evaluate an indefinite integral by partial fractions. 1.2.5 Evaluate an indefinite integral by selecting the appropriate technique. 1.2.6 Evaluate an indefinite integral using a combination of techniques.
2. Limits of indeterminate forms
2.1 Use of l’Hôpital’s rule to determine limits of indeterminate forms.	2.1.1 State l’Hôpital’s rule and the conditions for which it is valid. 2.1.2 Calculate limits of the indeterminate forms 0/0 and ¥/¥ using l’Hôpital’s rule. 2.1.3 For indeterminate forms 0 ´ ¥, ¥ - ¥, 1^¥, 0⁰, ¥⁰, use the appropriate transformation to determine the limit using l’Hôpital’s rule.
3. Definite and improper integrals
3.1 Use of the Fundamental Theorem of Calculus to calculate a definite integral.	3.1.1 Use the Fundamental Theorem of Calculus to calculate definite integrals.
3.2 Use of limits to calculate improper integrals.	3.2.1 Calculate an improper integral where at least one of the bounds is not a real number. 3.2.2 Calculate an improper integral where the integrand is discontinuous at one or more points in the interval of integration.
4. Differential equations
4.1 Use of antidifferentiation to obtain general solutions to simple differential equations.	4.1.1 Express a simple differential equation in the language of integration, and obtain the general solution.
4.2 Use of antidifferentiation to obtain particular solutions to simple initial value problems.	4.2.1 Express a simple initial value problem in the language of integration, and obtain the particular solution.
5. Areas and Volumes
5.1 Use of differentials to set up definite integrals	5.1.1 Analyze a quantity A as a sum åDA over an interval [a,b]. Approximate DA by a product f(x) dx; conclude dA/dx = f(x) and hence A is the definite integral ò_a^bf(x) dx.
5.2 Calculation of areas of planar regions.	5.2.1 Use 5.1.1 to set up a definite integral to calculate an area. 5.2.2 Sketch the area between two functions ( y = f(x), y = g(x) ) and use 5.2.1 to calculate the area.. 5.2.3 Sketch the area between two functions ( x = f(y), x = g(y) ) and use 5.2.1 to calculate the area.. 5.2.4 Sketch the area between two functions ( y = f(x), x = g(y) ) and determine the most efficient way (5.2.2, 5.2.3) to calculate the area.
5.3 Calculation of volumes of revolution.	5.3.1 Sketch the three dimensional solid obtained by rotating a region (of type 5.2.2 or 5.2.3) around the x axis or the y axis. 5.3.2 Use 5.1.1 to set up a definite integral to calculate the volume of the solid (5.3.1) by disks. 5.3.3 Use 5.1.1 to set up a definite integral to calculate the volume of the solid (5.3.1) by shells. 5.3.4 Determine the most efficient way (disks or shells) to calculate the volume of the solid (5.3.1), and calculate the volume by that method.
6. Infinite Series
6.1. Determination of the convergence or divergence of a sequence.	6.1.1 State the definition of the limit of a sequence. 6.1.2 Use 6.1.1 to calculate the limit of a sequence, or to show that it diverges. 6.1.3 Use the Squeeze theorem to calculate the limit of a sequence.
6.2 Determination of the convergence or divergence of an infinite series of positive terms	6.2.1 State the definition of convergence for an infinite series. 6.2.2 State the n^thterm test for the divergence of a series. 6.2.3 Use 6.2.1 to determine if a telescoping series converges, and if so, calculate the sum. 6.2.4 State the criterion for the convergence of an infinite geometric series.
	6.2.5 Calculate the sum of a converging geometric series (6.2.4); use this to solve appropriate problems (e.g. the distance travelled by a bouncing ball). 6.2.6 State the Integral, p series, Direct Comparison, Limit Comparison, Ratio, and n^th Root tests for convergence of an infinite series. 6.2.7 Determine whether an infinite series converges or diverges by choosing (and using) the correct method among (6.2.1 – 6.2.6).
6.3 Determination of the convergence, conditional or absolute, or divergence of an infinite series.	6.3.1 State the definitions of absolute and conditional convergence of infinite series. 6.3.2 State the definition of an alternating series. 6.3.3 State the criterion for the (conditional) convergence of an alternating series. 6.3.2 Determine if an infinite series is absolutely convergent, conditionally convergent, or divergent, using the methods of (6.2.1 – 6.2.7, 6.3.1 – 6.3.3).
6.4 Expression of functions as power series.	6.4.1 Use the methods of (6.2, 6.3) to find the radius and interval of convergence for a power series. 6.4.2 State the definitions of the Taylor and Maclaurin polynomials of degree n for a function f. 6.4.3 State the definitions of the Taylor and Maclaurin series for a function f based at x = c. 6.4.4 Use 6.4.3 to approximate a function f at a given point.
7. Integration, Comprehensive Assessment, and Exit Profile
7.1 Use of appropriate date processing technologies.	7.1.1 Use a computer and its main peripherals. 7.1.2 Use the principal type(s) of data processing software appropriate to the project.
7.2 Effective communication, reading scientific material.	7.2.1 Read one or more passages from a mathematics or scientific text, and summarize its contents.
7.3 The application of what has been learned to new situations.	7.3.1 Use techniques and results from the course and from assigned readings to solve interdisciplinary problems which have not specifically been covered in the course. 7.3.2 Choose from various notions and techniques those that are most suitable for the resolution of new problems, and apply them successfully.
7.4 Clear demonstration of links between mathematics and at least one other science discipline.	7.4.1 Apply knowledge or skills that have been acquired in Calculus II to topic(s) in Physics, Chemistry, or Biology.