Course Overview

Introduction to Calculus of One Variable

This course provides a comprehensive introduction to the fundamental concepts of calculus for functions of one variable. The curriculum is designed to develop students' understanding of limits, continuity, derivatives, integrals, and their applications, as well as essential techniques of integration and problem-solving strategies.

• Course Code: MATH 101

• Credit Hours: 4 TEDU Credits (3+2+0), 7 ECTS Credits

• Prerequisites: None

• Textbook: Calculus Metric Version Seventh Edition, James Stewart

• Supplementary Texts: Adams & Essex, George B. Thomas

Learning Outcomes

Objectives of the Course

Upon successful completion, students will be able to:

• Recall definitions, theorems, and examples related to functions of one variable.

• Calculate limits and understand the concept of continuity.

• Differentiate elementary and transcendental functions using the chain rule and implicit differentiation.

• Integrate functions using substitution, integration by parts, trigonometric substitution, and partial fractions.

• Solve applied problems such as related rates, optimization, curve sketching, and finding areas, volumes, and arc lengths.

• Practice mathematical writing and logical reasoning.

Course Outline

Weekly Topics and Chapters

The following outline details the main topics covered each week, corresponding to chapters in the primary textbook:

• Week 1: Mathematical Models, Essential Functions, New Functions from Old, The Limit of a Function

• Week 2: Calculating Limits, Limit Laws, Limits at Infinity, Horizontal Asymptotes

• Week 3: Continuity, Derivatives and Rates of Change, The Derivative as a Function

• Week 4: Differentiation Formulas, Derivatives of Trigonometric Functions, The Chain Rule

• Week 5: Implicit Differentiation, Linear Approximations, Mean Value Theorem

• Week 6: Increasing/Decreasing Test, Inverse Functions, Exponential Functions and Their Derivatives

• Week 7: Logarithmic Functions, Derivatives of Logarithmic Functions, Inverse Trigonometric and Hyperbolic Functions

• Week 8: Related Rates, Indeterminate Forms, L’Hospital’s Rule, Maximum and Minimum Values

• Week 9: How Derivatives Affect Graphs, Curve Sketching, Optimization Problems

• Week 10: Antiderivatives, The Definite Integral, The Fundamental Theorem of Calculus

• Week 11: Indefinite Integrals, Net Change Theorem, Substitution Rule, Integration by Parts

• Week 12: Trigonometric Integrals, Trigonometric Substitution, Partial Fractions

• Week 13: Areas Between Curves, Improper Integrals, Volumes

• Week 14: Volumes by Cylindrical Shells, Arc Length, Area of a Surface of Revolution

Assessment and Grading

Evaluation Components

• Midterm Exam 1: 30 points

• Midterm Exam 2: 30 points

• Final Exam: 30 points

• Active Participation in Lectures: 10 points

• Active Learning Exercises (ALE): 5 points

• Active Participation in Practice Hours (Lab): 5 points

Letter Grade Catalog

Key Topics in Calculus

Limits and Continuity

Limits and continuity form the foundation of calculus, allowing us to analyze the behavior of functions as inputs approach specific values.

• Limit of a Function: The value that a function approaches as the input approaches a certain point.

• Continuity: A function is continuous at a point if the limit exists and equals the function's value at that point.

• Limit Laws: Rules for calculating limits, such as the sum, product, and quotient laws.

• Limits at Infinity: Used to determine horizontal asymptotes and end behavior of functions.

Example: Calculate .

Solution: Substitute to get .

The Derivative

The derivative measures the instantaneous rate of change of a function with respect to its variable.

• Definition:

• Interpretation: Slope of the tangent line to the curve at a point.

• Rules: Power rule, product rule, quotient rule, chain rule.

• Applications: Related rates, optimization, curve sketching.

Example: Find the derivative of .

Solution:

Integration

Integration is the process of finding the area under a curve or the accumulation of quantities.

• Definite Integral: gives the net area between and the -axis from to .

• Indefinite Integral: represents the family of antiderivatives of .

• Fundamental Theorem of Calculus: Connects differentiation and integration.

• Techniques: Substitution, integration by parts, trigonometric substitution, partial fractions.

Example:

Applications of Calculus

Calculus is used to solve a variety of real-world and theoretical problems.

• Related Rates: Finding the rate at which one quantity changes with respect to another.

• Optimization: Finding maximum and minimum values of functions.

• Curve Sketching: Using derivatives to analyze and graph functions.

• Area and Volume: Calculating areas between curves and volumes of solids of revolution.

• Arc Length and Surface Area: Determining the length of curves and the area of surfaces generated by revolving curves.

Course Policies and Success Tips

Participation and Academic Integrity

• Active Participation: Required in both lectures and lab sessions for full credit.

• Attendance: Essential for success; missing classes may result in loss of participation points.

• Academic Integrity: Cheating and plagiarism are strictly prohibited and subject to disciplinary action.

Tips for Success

• Attend all lectures and labs.

• Engage actively and ask questions.

• Review material regularly and practice problem-solving.

• Take notes and seek help when needed.

Support Services

• Student Development and Psychological Counseling Center: Offers crisis intervention and support.

• TEDU COPeS: Provides psychosocial support for students and staff.

• Specialized Support for Students with Disabilities: Contact the coordinator for accommodations.

Additional info: This syllabus provides a structured overview of the main calculus topics, assessment methods, and academic expectations for students enrolled in MATH 101. For detailed content, students should refer to the primary textbook and supplementary materials.