Calculus I: Course Overview and Core Topics

Course Introduction

This course, Calculus I (MAT 122), introduces students to the foundational concepts of calculus, including limits, derivatives, and integrals. The course emphasizes understanding through graphical, numerical, and analytical approaches, and applies these concepts to solve real-world problems.

Chapter 1: Functions

Types and Representations of Functions

• Definition: A function is a relation that assigns exactly one output to each input from a specified domain.

• Representations: Functions can be represented by tables, graphs, and formulas.

• Common Types:

  • Linear functions:

  • Exponential functions:

  • Power functions:

  • Logarithmic functions:

  • Trigonometric functions:

  • Polynomial functions:

  • Rational functions:

• Combinations and Inverses: Functions can be combined (addition, subtraction, multiplication, division, composition) and inverted if one-to-one.

Example: The function is a polynomial function of degree 2.

Chapter 2: Limits

Understanding Limits and Continuity

• Idea of Limits: The limit of a function describes the behavior of the function as the input approaches a particular value.

• Definition of Limits: means that as approaches , approaches .

• Techniques for Computing Limits: Direct substitution, factoring, rationalization, and using special limit laws.

• Infinite Limits: Limits where increases or decreases without bound as approaches a value.

• Limits at Infinity: Describes the behavior of as approaches or .

• Continuity: A function is continuous at if .

• Precise Definitions: The - definition formalizes the concept of a limit.

Example:

Chapter 3: Derivatives

Definition and Computation of Derivatives

• Introducing the Derivative: The derivative measures the instantaneous rate of change of a function.

• Derivative as a Function: The derivative itself is a function, denoted or .

• Rules of Differentiation:

  • Product Rule:

  • Quotient Rule:

  • Chain Rule:

• Derivatives of Trigonometric, Logarithmic, and Exponential Functions: For example, , , .

• Implicit Differentiation: Used when functions are not given explicitly as .

• Derivatives of Inverse Trigonometric Functions: For example, .

• Derivative as Rate of Change: Used to model velocity, acceleration, and other rates.

• Related Rates: Problems involving rates at which related variables change.

Example: If , then .

Chapter 4: Applications of the Derivative

Using Derivatives to Analyze Functions

• Maxima and Minima: Points where a function reaches local or global highest/lowest values.

• Mean Value Theorem: If is continuous on and differentiable on , then such that .

• Concavity and Inflection Points: Concavity describes the direction of curvature; inflection points are where concavity changes.

• Graphing Functions: Use derivatives to determine increasing/decreasing intervals and concavity.

• Optimization: Finding maximum or minimum values in applied contexts.

• Applications of Marginality: Marginal cost, revenue, and profit in economics.

• L’Hôpital’s Rule: Used to evaluate indeterminate forms like or : (if the limit exists).

• Antiderivatives: The reverse process of differentiation.

Example: To maximize area with a fixed perimeter, set up an equation, differentiate, and solve for critical points.

Chapter 5: Integration

Definite and Indefinite Integrals

• Approximating Areas: Use Riemann sums to estimate the area under a curve.

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

• Fundamental Theorem of Calculus: Connects differentiation and integration:

  • Part 1: If is an antiderivative of on , then .

  • Part 2:

• Working with Integrals: Properties and techniques for evaluating integrals.

• Indefinite Integrals: , where .

• Substitution Rule: Used for integrating composite functions: where .

Example:

Learning Outcomes and Objectives

General and Specific Goals

• Apply mathematical ideas to specific situations using graphical, numerical, and analytical methods.

• Understand the derivative as a rate of change and local linear approximation.

• Use derivatives and integrals to solve a variety of problems.

• Understand the relationship between derivatives and definite integrals via the Fundamental Theorem of Calculus.

• Model physical situations with functions, differential equations, or integrals.

SUNY General Education Goals & Outcomes

• Draw Inferences from Mathematical Models: Interpret and analyze formulas, graphs, tables, and schematics.

• Represent Mathematical Information: Use symbolic, visual, numerical, and verbal representations.

• Employ Quantitative Methods: Identify and apply appropriate arithmetic, algebraic, or statistical methods.

• Check Mathematical Results for Reasonableness: Estimate and justify results logically.

• Recognize Limits: Understand the limitations of mathematical and statistical models compared to real-life situations.

Course Schedule Overview

Additional Information

• Textbook: Calculus, Single Variable: Early Transcendentals, 3rd ed., by Briggs et al., Pearson, 2019.

• Calculator: A graphing calculator (e.g., TI-84, TI-89) is required.

• Assessment: Grading is based on tests (100%), with an optional cumulative final exam that can replace the lowest test grade.

• Policies: Strict rules on attendance, academic honesty, and use of electronic devices are enforced.