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 MATH 1036, Calculus I Summer 2016

### Course outline

• Course: MATH-1036: Calculus I
• Credits: 3
• Prerequisites: The normal prerequisite is OAC/4U Calculus (or equivalent, such as MATH-1912); students with a grade below 60% in their OAC/4U Calculus may find MATH-1036 quite difficult and should perhaps consider taking MATH-1912 first.
• Anti-requisites: MATH-1035
1. Description

This is a standard first year level Calculus course, which covers a wide variety of topics. The course content will be based on Chapters 1 - 3 and 6 of the textbook. Topics include: the real number system; algebra of functions; limits and continuity; inverse functions; differentiation; the intermediate value theorem; the mean value theorem; differentiation of transcendental functions; L'Hospital's rules; curve sketching; and applications of the derivative, including linear approximation and optimization.

MATH-1036 is a cornerstone of the Mathematics program. It belongs to the core group of mathematics and computer science courses. It is a required course for the Bachelor of Science in Mathematics and Bachelor of Science in Computer Science. It is also one of the four courses to choose from in Environmental Science and Geography, as well as Concurrent Education programs (with the total of 6 credit of mathematics courses required). It is often chosen by students majoring in Psychology or Biology to fulfill their degree requirement. The central part of the course is differential calculus. It is one of the most applicable branches of mathematics. Many tools and devices that we use in our everyday life would not be possible without calculus. Examples include calculators, computers, microwave ovens, phones, cars, mp3-players . Calculus helps to create and operate planes and ships, medical and aerospace equipment, to build bridges and maximize profits. Differential calculus is essential for analyzing the behaviour of functions, visualization using curve sketching, finding approximate values and optimal solutions in practical problems. It is also crucial for understanding other topics in mathematics, such as integral calculus, differential equations, and probability and statistics. Students will see how calculus is applied to solve real-life problems. They will enhance their problem-solving and analytical skills. The course familiarizes students with proof techniques as well.

3. Specific Objectives

By the end of the course, students will:
• Understand the concepts of a function, its domain and range; use calculus to find domains and ranges of functions
• Know how to create new functions using arithmetic and algebraic operations, and compositions
• Understand the concept of a limit; calculate limits of functions using different techniques
• Understand the concepts of continuity; apply the properties of continuous functions to solve practical problems
• Understand the concept of the derivative; understand the interpretation of derivatives in terms of slopes of tangent lines and related rates; be able to compute derivatives using its definition
• Know how to compute derivatives of complicated functions using various differentiation formulas and rules
• Use derivatives to sketch graphs of functions and solve optimization problems
• Understand the concept of inverse function, use properties of the inverse function in problem solving
• Develop an ability to work with transcendental functions (trigonometric and inverse trigonometric functions, exponential and logarithmic functions)
• Understand the concept of antiderivative and be able to find antiderivatives of simple functions
• Carry out simple mathematical proofs
• Develop problem-solving skills
• Use calculus to solve problems arising from real-life applications

4. Timetable

Instructional time is 12 weeks (schedule below). The course is composed of 4 units, each with individual modules, for a total of 24 modules to be completed prior to completion of the course. Each unit has an associated assignment.

5. Required Textbook "Single Variable Calculus" by James Stewart 7th edition, Brooks / Cole ISBN-10: 0538497831 ISBN-13: 978-0538497831

This textbook was selected because of:
• in-depth coverage of the most important topics of calculus;
• the appropriate sequence of the topics;
• many motivating and clarifying examples;
• excellent diagrams that accompany the material;
• comprehensive discussion of real-life applications;
• numerous exercises at the end of each section, with answers to the odd-numbered questions provided at the end of the textbook, as well as the review exercises at the end of each chapter;
• sufficiently rigorous exposition with accessible proofs of many important theorems;
• convenient system of notations;
• review of high school mathematics in appendixes
• Students are responsible for purchasing their own textbooks. Textbooks can be ordered online from Nipissing University.

6. Instructor

Ihor Stasyuk
I studied at Ivan Franko Lviv National University (Ukraine) at the Department of Mechanics and Mathematics. I obtained my M.Sc. degree in 2002 and my Ph.D. degree in 2007 from Lviv National University. I worked as a postdoctoral fellow and research assistant between 2007 and 2010 at the University of Saskatchewan. I have worked as an assistant professor at Nipissing University since 2013. I have taught various mathematical courses and have been involved in scientific research.
• Email: ihors@nipissingu.ca
• Phone: 705-474-3450 ext. 4442

 Unit 1 Assignment 15% Unit 2 Assignment 15% Unit 3 Assignment 15% Unit 4 Assignment 15% Final Examination 40%

Assignments: 60% Total
There are four assignments in the course; one for each unit. The assignments are to be submitted for grading in the course. Each assignment is worth 15% of the final mark. Questions for the assignments will be available on-line and are similar to the textbook exercises. Assignments must be completed on-line.

Important: The ONLY acceptable formats for assignments are MS Word or PDF. Assignments MUST BE TYPED. Please DO NOT scan handwritten assignments. An exception can be made for diagrams (such as graphs of functions) only.

Final Examination: 40% There is a final examination in the course, comprising 40% of the final grade. The final examination will cover material from each unit. For the final exam students may bring in a simple scientific calculator (no notes, textbooks, formula sheets, graphic or programming calculators are allowed).

Other Resources Students are encouraged to use the Nipissing University Calculus Help Site . It provides an access to additional on-line tutorials, including those reviewing high-school mathematics, additional exercises, and self-assessment tests.

Additional Section Exercises There are exercises at the end of each section in each chapter of the textbook. Students are expected to do the suggested exercises at the end of each section. The list of suggested exercises is included at the end of this course outline. Answers to the odd-numbered exercises are provided in the textbook. However, students should not look up the answers until they have worked out the answers on their own. The student might also consider purchasing a copy of the Student's Solution Manual, ISBN: 0538497831. The following two textbooks are useful for refreshing the knowledge of high school math:
• “Calculus: Fear No More” by Miroslav Lovric (ISBN: 0176500472)
• “The Math Survival Kit” by Jack Weiner (ISBN: 0176500170)

Review of High-School Mathematics The course begins with an introductory module reviewing essential concepts from high school mathematics. To determine the level of their preparedness for the course students should take self-assessment tests and attempt suggested review exercises, the list of which can be found at the end of the course outline.

8. Expected Workload The expected workload for an average student is approximately 5 hours per module. The exceptions are modules 1.v (10 hours), 2.iii (15 hours), 3.iv (10 hours), and 4.iii (10 hours). The total is approximately 145 hours for the course, which is similar to the workload of a 3-credit on-campus course in a university undergraduate program. Students are responsible for watching the video of the lectures, reading the required textbook, solving suggested exercises, submitting all assignments for grading, and writing the final examination.

9. Schedule

 Dates Unit Module Associated Readings May 4 - 16 Review of high school mathematics Number systems, absolute value linear and quadratic functions, polynomials, rational functions, equations and inequalities, roots, trigonometric functions, basics of plane geometry Review Exercises; Self-Assessment Tests (Diagnostic Tests) Appendixes A, B, C, D May 20 - 30 UNIT 1: Chapters 1 and 2 functions and models; limits (definition of a limit, limit laws; methods to calculate limits); continuity (i) Four Ways to Represent a Function (ii) A Catalog of Essential Functions (iii) New Functions from Old Functions (iv) Why Do We Need Limits (v) The Definition of a Limit (vi) Methods to Calculate Limits (vii) Continuity Assignment 1 - Due June 1 (i) Section 1.1 (ii) Section 1.2 (iii) Section 1.3 (iv) Section 1.4 (v) Sections 1.5 and 1.7 (vi) Section 1.6 (vii) Section 1.8 June 2 - 13 UNIT 2: Derivatives (Chapter 3) definition and interpretations of derivative; tangent line; rates of change; rules for differentiation Linear approximation. (i) Tangent Lines and Rates of Change (ii) Definition of Derivative (iii) Rules for Differentiation (constant multiple rule; sum and difference rule; power rule; product rule; quotient rule; chain rule; derivatives of trigonometric functions; higher derivatives) (iv) Implicit Differentiation (v) Related Rates (vi) Linear Approximations and Differentials Assignment 2 – Due June 15 (i) Section 2.1 (ii) Sections 2.1 and 2.2 (iii) Sections 2.3, 2.4, 2.5 (iv) Section 2.6 (v) Section 2.8 (vi) Section 2.9 June 15 - July 11 UNIT 3: Applications of Differentiation (Chapter 4) maximums and minimums; graphs of functions; optimization problems Antiderivatvies. (i) Maximum and Minimum Values (ii) Rolle's Theorem. The Mean Value Theorem (iii) Limits at Infinity and Horizontal Asymptotes (iv) Curve Sketching (v) Optimization Problems (vi) Antiderivatives Assignment 3 – July 11 (i) Section 3.1 (ii) Section 3.2 (iii) Section 3.4 (iv) Sections 3.3 and 3.5 (v) Section 3.7 (vi) Section 3.9 July 13 - 25 Unit 4: Inverse functions (Chapter 7) exponential, logarithmic, and inverse trigonometric functions Hospital's Rule. (i) Inverse functions (ii) Exponential Functions and Their Derivatives (iii) Logarithmic Functions and Their Derivatives (iv) Inverse Trigonometric Functions (v) L'Hospital's Rule Assignment 4 – Due July 27 (i) Section 6.1 (ii) Section 6.2 (iii) Sections 6.3 and 6.4 (iv) Section 6.6 (v) Section 6.8 July 27 - 31 Review Week Study for Final Exam Take review tests Solve the sample final exam Review Tests N/A July 13 - 31 Student Opinion Survey August 1 - 15 Final Examination Period Student Feedback Survey

10. Academic Dishonesty The University takes a very serious view of such offenses against academic honesty such as plagiarism, cheating and impersonation. Penalties for dealing with such offenses will be strictly enforced. The complete policy on Academic Dishonesty can be found in the Policies section of the Academic Calendar.

11. Review Exercises – High School Mathematics
Text: "Single Variable Calculus" by James Stewart, 7th edition

Appendix A: 4,5,6,11,12,16,21,26,32,34,36,38,42,43,46,51,56,62
Appendix B: 4,10,11,15,19,20,22,25,28,31,34,36,42,45,48,52,57,58,59
Appendix C: 1,4,6,9, 12, 14, 16, 23, 24,30,32 - 35, 39, 40
Appendix D: 4,5,7,9,12,14,16,20,24,26,29,30,31, 37,46, 52, 60,63,68,69,72,73,76

Suggested Exercises

Text: "Single Variable Calculus" by James Stewart, 7th edition

Chapter 1

Section 1.1: 2,7,8,25-67, 73-78
Section 1.2: 1-5,6,8,9,13,14,16,17,25
Section 1.3: 9-24, 27-36, 41-46, 54-56, 59-64
Section 1.4: 3,6,7
Section 1.5: 4,6,7,8,11,12,16,18, 29-37, 38a
Section 1.6: 1, 3-9, 11-32, 38-46, 48, 49, 57-63
Section 1.7: 11,13,16,18,21,23,26,28,30,31,32,36,37,39,41,42,44
Section 1.8: 25,30,31,36,38,39,40,42,45,46,49-60,63,64,65,67,69

Chapter 2

Section 2.1: 5-10,15-17,22,23,24,27-32,33,34,38,46,47,48,54
Section 2.2: 20, 22-26,28,29,35,36,41,44,49,53,55
Section 2.3: 2,5,8,9,12,14,16-22,25,27,28,29,32,33,36,38,42,44,45,51,52,55-62,64,67-70,73-84,86,87,93,95-106
Section 2.4: 2,5,6,10,12,13,18,21,22,29,31,33,34,37,39,40,43,45,46,48,50,52
Section 2.5: 3,6,7,8,10,14,17,20,21,22,31,33,34,40,41,51,52,59-62,73,74,80,86,87
Section 2.6: 1-8,12,16,17,18,23,25,28
Section 2.8: 1-46
Section 2.9: 1,4,8,10,13,18,21-28,31-34,36,42

Chapter 3

Section 3.1: 3,5,8,10,12,13,16-21,26,28,29-42,45-57,68-70,72
Section 3.2: 1-6,10-12,15,17-20,23,24,26,27,29,33,34
Section 3.3: 8-17,20,22,23,29-40,42,52-54,57-59,61,62,66,69
Section 3.4: 8,11-15,18,19,20,22,23,27,29,33,36,38,44-47,54,68,70,71,72
Section 3.5: 1-40
Section 3.7: 3-5,12,14,15,16,20,21,27,33,35,37,38,40,44,46,49,52,67,70,72
Section 3.9: 2,5,7,9,12,13,15,18,22,24,26,28,29,35,36,40,42,47,54,56,60,61

Chapter 6

Section 6.1: 9,11,13,17-19,23-28,33,35-42,45
Section 6.2: 7-12,15,16, 23-25,27,29,31,33,36,45,46,49,52,53,54,59,60,65-73
Section 6.3: 3-8,10,12,14,17,23-36,39,40,47,51,52,55,56,62,64,65,66,68,69
Section 6.4: 2,3,6,8,11,13,16,17,25,26,34,38,45,46,49,54,55,58,61-66
Section 6.6: 1-14,22,23,27-30,38,44,45,52
Section 6.8: 12,14,17,22,24,30,31,34,42,44,45,46,49,52,53,54,56,57,61,66,73,75-80,100