Chapter 1: Functions and Limits
Course
MATH161 (Calculus I)
Overview
This chapter covers Functions and Limits from Stewart Calculus 9th Edition.
Each section contains:
- Skill breakdowns with learning objectives
- Practice problems at multiple difficulty levels
- Links to related skills and prerequisites
Sections
Section 1.1: Four Ways to Represent a Function
| Skill | Description | Difficulty |
|---|---|---|
| Difference Quotient | Functions and Limits | Intermediate |
| Domain and Range | Functions and Limits | Beginner |
| Evaluating Functions | Functions and Limits | Beginner |
| Even and Odd Functions | Functions and Limits | Intermediate |
| Function Definition | Functions and Limits | Beginner |
| Function Representations | Functions and Limits | Beginner |
| Function Symmetry | Functions and Limits | Intermediate |
| Increasing and Decreasing Functions | Functions and Limits | Intermediate |
| Piecewise Functions | Functions and Limits | Intermediate |
Section 1.2: Mathematical Models: A Catalog of Essential Functions
| Skill | Description | Difficulty |
|---|---|---|
| Algebraic Functions | Functions and Limits | Intermediate |
| Exponential Functions | Functions and Limits | Intermediate |
| Linear Models | Functions and Limits | Beginner |
| Linear Models and Rate of Change | Mathematical Models | Beginner |
| Logarithmic Functions | Functions and Limits | Intermediate |
| Polynomial Functions | Functions and Limits | Intermediate |
| Polynomial Functions and Degree | Mathematical Models | Beginner |
| Power Functions | Functions and Limits | Intermediate |
| Power and Root Functions | Mathematical Models | Intermediate |
| Rational Functions | Functions and Limits | Intermediate |
| Recognizing Function Families | Mathematical Models | Intermediate |
| Trigonometric Functions | Functions and Limits | Intermediate |
Section 1.3: New Functions from Old Functions
| Skill | Description | Difficulty |
|---|---|---|
| Combining Transformations | Functions and Limits | Intermediate |
| Function Arithmetic | Functions and Graphs | Beginner |
| Function Composition | Functions and Graphs | Intermediate |
| Function Transformations | Functions and Graphs | Beginner |
| Reflections and Stretches | Functions and Limits | Intermediate |
| Vertical and Horizontal Shifts | Functions and Limits | Intermediate |
Section 1.4: The Tangent and Velocity Problems
| Skill | Description | Difficulty |
|---|---|---|
| From Secant to Tangent | Functions and Limits | Intermediate |
| Instantaneous Velocity | Limits and Rates of Change | Intermediate |
| Secant Lines and the Difference Quotient | Limits and Rates of Change | Beginner |
| Tangent Slope via Limits | Limits and Rates of Change | Intermediate |
| The Tangent Problem | Functions and Limits | Intermediate |
| The Velocity Problem | Functions and Limits | Intermediate |
Section 1.5: The Limit of a Function
| Skill | Description | Difficulty |
|---|---|---|
| Estimating Limits Numerically and Graphically | Limits | Beginner |
| Finding Limits Numerically and Graphically | Functions and Limits | Intermediate |
| How Can a Limit Fail to Exist? | Functions and Limits | Intermediate |
| Infinite Limits and Vertical Asymptotes | Functions and Limits | Intermediate |
| One-Sided Limits | Limits | Beginner |
| Understanding Limits Intuitively | Limits | Beginner |
Section 1.6: Calculating Limits Using the Limit Laws
| Skill | Description | Difficulty |
|---|---|---|
| Algebraic Techniques for Limits | Functions and Limits | Intermediate |
| Basic Limit Laws | Limits | Beginner |
| Evaluating Limits by Direct Substitution | Functions and Limits | Intermediate |
| Properties of Limits | Functions and Limits | Intermediate |
| Resolving Indeterminate Forms Algebraically | Limits | Intermediate |
| The Squeeze Theorem | Limits | Intermediate |
| Using One-Sided Limits | Functions and Limits | Intermediate |
Section 1.7: The Precise Definition of a Limit
| Skill | Description | Difficulty |
|---|---|---|
| Epsilon-Delta Proofs for Linear Functions | Limits | Intermediate |
| Epsilon-Delta Proofs for Polynomial Functions | Limits | Advanced |
| Epsilon-Delta Proofs: Linear Functions | Functions and Limits | Advanced |
| Epsilon-Delta Proofs: Quadratic Functions | Functions and Limits | Advanced |
| Epsilon-Delta Variations: One-Sided and Infinite Limits | Limits | Intermediate |
| Precise Definition: Infinite Limits | Functions and Limits | Advanced |
| Precise Definition: One-Sided Limits | Functions and Limits | Advanced |
| The Epsilon-Delta Definition of a Limit | Limits | Intermediate |
Section 1.8: Continuity
| Skill | Description | Difficulty |
|---|---|---|
| Continuity at a Point | Functions and Limits | Intermediate |
| Continuity of Combined Functions | Continuity | Intermediate |
| Continuity on an Interval | Functions and Limits | Intermediate |
| Families of Continuous Functions | Functions and Limits | Intermediate |
| Intermediate Value Theorem | Continuity | Intermediate |
| Properties of Continuous Functions | Functions and Limits | Intermediate |
| Types of Discontinuities | Continuity | Intermediate |
Prerequisites
Before starting Chapter 1, review relevant prerequisite skills from earlier chapters.
| Previous | Up | Next |
|---|---|---|
| Home | Home | Chapter 2 |