RES.18-001 | Fall 2023 | Undergraduate

Calculus Online Textbook


First published in 1991 by Wellesley-Cambridge Press, this updated 3rd edition of the book is a useful resource for educators and self-learners alike. It is well organized, covers single variable and multivariable calculus in depth, and is rich with applications. There is also an online Instructor’s Manual and a student Study Guide.

The complete textbook (PDF) is also available as a single file. 

Highlights of Calculus                   
MIT Professor Gilbert Strang has created a series of videos to show ways in which calculus is important in our lives. The videos, which include real-life examples to illustrate the concepts, are ideal for high school students, college students, and anyone interested in learning the basics of calculus.                   
Watch the videos

Textbook Components

Table of Contents (PDF)

Chapter 0: Highlights of Calculus (PDF)

0.1 Distance and Speed // Height and Slope      
0.2 The Changing Slope of \(y=x^2\) and \(y=x^n\)     
0.3 The Exponential \(y=e^x\)     
0.4 Video Summaries and Practice Problems      
0.5 Graphs and Graphing Calculators

Chapter 1: Introduction to Calculus (PDF)

1.1 Velocity and Distance                          
1.2 Calculus Without Limits                          
1.3 The Velocity at an Instant                          
1.4 Circular Motion                          
1.5 A Review of Trigonometry                          
1.6 A Thousand Points of Light 

Chapter 2: Derivatives (PDF)

2.1 The Derivative of a Function                          
2.2 Powers and Polynomials                          
2.3 The Slope and the Tangent Line                          
2.4 Derivative of the Sine and Cosine                          
2.5 The Product and Quotient and Power Rules                          
2.6 Limits                          
2.7 Continuous Functions

Chapter 3: Applications of the Derivative (PDF)

3.1 Linear Approximation                           
3.2 Maximum and Minimum Problems                           
3.3 Second Derivatives: Bending and Acceleration                          
3.4 Graphs                           
3.5 Parabolas, Ellipses, and Hyperbolas                          
3.6 Iterations \(x_{n+1}=F(x_n)\)                          
3.7 Newton’s Method (and Chaos)                           
3.8 The Mean Value Theorem and 1’Hôpital’s Rule

Chapter 4: Derivatives by the Chain Rule (PDF)

4.1 The Chain Rule                           
4.2 Implicit Differentiation and Related Rates                           
4.3 Inverse Functions and Their Derivatives                           
4.4 Inverses of Trigonometric Functions

Chapter 5: Integrals (PDF)

5.1 The Idea of an Integral                           
5.2 Antiderivatives                           
5.3 Summation versus Integration                           
5.4 Indefinite Integrals and Substitutions                           
5.5 The Definite Integral                           
5.6 Properties of the Integral and Average Value                           
5.7 The Fundamental Theorem and Its Applications                            
5.8 Numerical Integration

Chapter 6: Exponentials and Logarithms (PDF)

6.1 An Overview                           
6.2 The Exponential \(e^x\)                           
6.3 Growth and Decay in Science and Economics                           
6.4 Logarithms                           
6.5 Separable Equations Including the Logistic Equation                           
6.6 Powers Instead of Exponentials                           
6.7 Hyperbolic Functions

Chapter 7: Techniques of Integration (PDF)

7.1 Integration by Parts                           
7.2 Trigonometric Integrals                           
7.3 Trigonometric Substitutions                           
7.4 Partial Fractions                           
7.5 Improper Integrals

Chapter 8: Applications of the Integral (PDF)

8.1 Areas and Volumes by Slices                           
8.2 Length of a Plane Curve                           
8.3 Area of a Surface of Revolution                           
8.4 Probability and Calculus                           
8.5 Masses and Moments                           
8.6 Force, Work, and Energy

Chapter 9: Polar Coordinates and Complex Numbers (PDF)

9.1 Polar Coordinates                           
9.2 Polar Equations and Graphs                           
9.3 Slope, Length, and Area for Polar Curves                           
9.4 Complex Numbers

Chapter 10: Infinite Series (PDF)

10.1 The Geometric Series                           
10.2 Convergence Tests: Positive Series                           
10.3 Convergence Tests: All Series                           
10.4 The Taylor Series for \(e^x\), \(\sin{x}\), and \(\cos{x}\)                       
10.5 Power Series

Chapter 11: Vectors and Matrices (PDF)

11.1 Vectors and Dot Products                           
11.2 Planes and Projections                           
11.3 Cross Products and Determinants                           
11.4 Matrices and Linear Equations                           
11.5 Linear Algebra 

Chapter 12: Motion Along a Curve (PDF)

12.1 The Position Vector      
12.2 Plane Motion: Projectiles and Cycloids                           
12.3 Curvature and Normal Vector                           
12.4 Polar Coordinates and Planetary Motion

Chapter 13: Partial Derivatives (PDF)

13.1 Surface and Level Curves                           
13.2 Partial Derivatives                           
13.3 Tangent Planes and Linear Approximations                           
13.4 Directional Derivatives and Gradients                           
13.5 The Chain Rule                           
13.6 Maxima, Minima, and Saddle Points                           
13.7 Constraints and Lagrange Multipliers

Chapter 14: Multiple Integrals (PDF)

14.1 Double Integrals                           
14.2 Changing to Better Coordinates                           
14.3 Triple Integrals                           
14.4 Cylindrical and Spherical Coordinates

Chapter 15: Vector Calculus (PDF)

15.1 Vector Fields                           
15.2 Line Integrals                           
15.3 Green’s Theorem                           
15.4 Surface Integrals                           
15.5 The Divergence Theorem                           
15.6 Stokes’ Theorem and the Curl of F

Chapter 16: Mathematics after Calculus (PDF)

Index (PDF)

Course Info

As Taught In
Fall 2023
Learning Resource Types
Online Textbook