Course Overview

Introduction to Calculus I

This syllabus outlines the structure, objectives, and content for MATH 1210 Calculus I. The course covers foundational topics in calculus, including functions, limits, derivatives, and integrals, as well as their applications. Students will engage in lectures, labs, assignments, and exams to develop a comprehensive understanding of calculus concepts.

Core Topics and Learning Objectives

Functions and Their Inverses

Understanding functions and their inverses is essential in calculus, especially for exponential and logarithmic functions.

• Definition: A function is a relation that assigns each input exactly one output. The inverse of a function reverses this assignment.

• Exponential Functions: Functions of the form where .

• Logarithmic Functions: The inverse of exponential functions, .

• Example: If , then .

Limits and Limit Laws

Limits are fundamental to calculus, providing the basis for derivatives and continuity.

• Definition: The limit of as approaches is if gets arbitrarily close to as approaches .

• Limit Laws: Rules for evaluating limits, including sum, product, and quotient laws.

• Indeterminate Forms: Limits of the form or often require special techniques.

• Example: .

Derivatives and Differentiation

Derivatives measure the rate of change of a function and are central to calculus.

• Definition: The derivative of at is .

• Derivative Rules: Includes power rule, product rule, quotient rule, and chain rule.

• Implicit Differentiation: Used when functions are not explicitly solved for .

• Example: If , then .

Applications of the Derivative

Derivatives are used to analyze functions, solve optimization problems, and compute related rates.

• Critical Points: Points where or does not exist.

• Inflection Points: Where the concavity of changes, i.e., .

• Optimization: Finding maximum and minimum values of functions.

• Related Rates: Problems involving rates of change of related quantities.

• L'Hôpital's Rule: Used to evaluate indeterminate limits: (when applicable).

• Example: Maximizing area given a fixed perimeter.

Integration and the Fundamental Theorem of Calculus

Integration is the process of finding antiderivatives and calculating areas under curves.

• Indefinite Integral: represents the family of antiderivatives of .

• Definite Integral: gives the net area under from to .

• Fundamental Theorem of Calculus: Connects differentiation and integration: .

• Example: .

Applications of Integration

Integration is used to compute areas, volumes, and solve other geometric problems.

• Area Between Curves: where .

• Volumes of Solids of Revolution: Using the disk or washer method: .

• Example: Find the area between and from to .

Course Schedule

Weekly Topics

The course is structured by week, covering the following chapters and sections:

• Week 1: Introduction, Functions (Sections 1.3, 1.4)

• Weeks 2-3: Limits (Sections 2.1–2.6)

• Weeks 4-7: Derivatives and Differentiation (Sections 3.1–3.11)

• Weeks 9-11: Applications of the Derivative (Sections 4.1–4.8)

• Weeks 12-15: Integration and Applications (Sections 5.1–6.4)

Grading Scale

Letter Grade Distribution

Grades are assigned based on the following scale:

Additional Information

• Required Text: Calculus Early Transcendentals by Briggs, Cochran, Gillett, and Schulz (3rd edition).

• Calculator Policy: Calculators are not allowed on exams; use them sparingly on homework.

• Support: Tutoring, learning assistants, and departmental videos are available for additional help.

• ADA and Safety: The university provides accommodations and resources for safety and well-being.

• Academic Conduct: Students must adhere to university policies regarding academic honesty and ethical conduct.

Summary Table: Main Calculus I Topics

Additional info: The syllabus does not cover advanced topics such as sequences, series, or differential equations, which are typically found in later calculus courses.