Calculus

 

 

 

Prerequisites - The Cartesian Plane and Functions

Real Numbers and the Real Line

The Cartesian Plane

Graphs of Equations

Lines in the Plane

Functions

Review of Trigonometric Functions

 

Chapter 1 - Limits and Their Properties

Introduction to Limits

Properties of Limits

Evaluating Limits

Continuity and One-Sided Limits

Infinite Limits

 

Chapter 2 - Differentiation

Derivative and the Tangent Line

Basic Differentiation Rules and Rates of Change

Product and Quotient Rules and Higher-Order Derivatives

The Chain Rule

Implicit Differentiation

 

Chapter 3 - Applications of Differentiation

Extrema on an Interval

Rolle’s Theorem

The Mean Value Theorem

Increasing and Decreasing Functions

The First Derivative Test

Concavity

The Second Derivative Test

Limits at Infinity

Optimization

Newton’s Method

Differentials

 

Chapter 4 - Integration

Antiderivatives and Indefinite Integration

Area

Riemann Sums and Definite Integrals

The Fundamental Theorem of Calculus

Integration by Substitution

Numerical Integration

 

Chapter 5 - Logarithmic, Exponential and Other Transcendental Functions

The Natural Logarithmic Function and Differentiation

The Natural Logarithmic Function and Integration

Inverse Functions

Exponential Functions and Differentiation

Exponential Functions and Integration

Bases Other than e and Applications

Differential Equations

Inverse Trigonometric Functions and Differentiation

Inverse Trigonometric Functions: Integration and Completing the Square

 

Chapter 6 - Applications of Integration

Area of a Region Between Two Curves

Volume: The Disc Method

Volume: The Shell Method

Arc Length and Surfaces of Revolution

Work

Fluid Pressure and Fluid Force