RES.18-001 | Fall 2023 | Undergraduate

Calculus Online Textbook

Pages

The calculus courses at MIT are 18.01 Single Variable Calculus and 18.02 Multivariable Calculus. Videos of those courses are on OpenCourseWare (OCW) along with lots of other useful materials. This site is about a completely separate calculus textbook by Gilbert Strang, and it will be helpful to viewers of OCW who would like to have online access to a textbook. 

Professor Strang’s many contributions to OCW are mostly about linear algebra:

Each of those courses has a full set of video lectures recorded on the MIT campus. Those videos have been watched by millions of viewers around the world (especially 18.06 Linear Algebra). Calculus and linear algebra are the two principal lead-ins to pure and applied mathematics.

Professor Strang’s Highlights of Calculus course on OCW is a series of short videos. They focus on the main idea of the subject, involving two functions: function 2 is the “derivative” of function 1, and function 1 is the “integral” of function 2. If you are given one of those functions, then this calculus textbook and the Highlights of Calculus videos show how to derive the other function. The heart of calculus is to use those functions to solve real problems, as described in Lecture 1: Big Picture of Calculus.

Below, Professor Strang shares some thoughts on the history of the calculus textbook:

For the textbook itself, its first printing was in 1991. That was a time of active “rethinking” of the course. I remember being in the audience at a discussion organized by the US National Science Foundation, about needed changes in typical calculus courses (to make them more relevant and interesting to students). Sitting at the back, I thought one necessary step would be a new textbook. That is the book you see here on OpenCourseWare, published by Wellesley-Cambridge Press.

Chapter 0 of the book came in a later edition. Its purposes were to add more figures to illustrate the key ideas of calculus, and also to develop in a new way the most important function in this subject. That function is the exponential \(e^x\). It has a very special feature, because it is both function 1 and function 2! In other words, the derivative of \(e^x\)is \(e^x\), and the integral of \(e^x\)is \(e^x\) (plus any constant C). This becomes the most important function in many applications, including the enormous field of “differential equations” (because it solves the most fundamental differential equation dy/dx = y).

I hope the book and the exercises and the videos and even Chapter 0 will be useful to you! Best wishes in all your work.

Gilbert Strang

Instructor’s Manual Components

ChapterS FILES
1: Introduction to Calculus              
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 1 Manual (PDF)
2: Derivatives              
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 2 Manual (PDF)
3: Applications of the Derivative              
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 3 Manual (PDF)
4: Derivatives by the Chain Rule              
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 4 Manual (PDF)
5: Integrals              
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 5 Manual (PDF)
6: Exponentials and Logarithms              
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 6 Manual (PDF)
7: Techniques of Integration              
7.1 Integration by Parts                        
7.2 Trigonometric Integrals                        
7.3 Trigonometric Substitutions                        
7.4 Partial Fractions                        
7.5 Improper Integrals
Chapter 7 Manual (PDF)
8: Applications of the Integral              
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 8 Manual (PDF)
9: Polar Coordinates and Complex Numbers              
9.1 Polar Coordinates                        
9.2 Polar Equations and Graphs                        
9.3 Slope, Length, and Area for Polar Curves                        
9.4 Complex Numbers
Chapter 9 Manual (PDF)
10: Infinite Series              
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 10 Manual (PDF)
11: Vectors and Matrices              
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 11 Manual (PDF)
12: Motion along a Curve              
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 12 Manual (PDF)
13: Partial Derivatives              
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 13 Manual (PDF)
14: Multiple Integrals              
14.1 Double Integrals                        
14.2 Changing to Better Coordinates                        
14.3 Triple Integrals                        
14.4 Cylindrical and Spherical Coordinates
Chapter 14 Manual (PDF)
15: Vector Calculus              
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 15 Manual (PDF)
16: Mathematics after Calculus           Chapter 16 Manual (PDF)

Highlights of Calculus is a series of short videos that introduces the basics of calculus—how it works and why it is important. The intended audience is high school students, college students, or anyone who might need help understanding the subject.

Professor Strang’s foundational course 18.06 Linear Algebra has long been one of the most popular courses on OCW. It has received more than 10 million visits since its first publication in 2002. Professor Strang also has a website dedicated to his linear algebra teaching. In Spring 2023, Professor Strang taught 18.06 Linear Algebra for the final time before retiring.  The final class was live streamed and recorded.

A new version of Professor Strang’s classic Linear Algebra was released in 2011 in the innovative OCW Scholar format designed for independent learners. 18.06SC Linear Algebra includes 35 lecture videos and 36 short (and highly-praised) problem-solving help videos by teaching assistants.

Professor Strang has continued to offer new insights into key mathematics subjects. In 2014, he published the new textbook Differential Equations and Linear Algebra. In 2016, that textbook was developed into a series of 55 short videos supported by MathWorks, with parallel videos about numerical solutions by Dr. Cleve Moler, the creator of MATLAB®. The textbook and video lectures help students in a basic ordinary differential equations course. This new series, Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, is also available on the MathWorks website.

In 2017, Professor Strang launched a new undergraduate course at MIT: 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning. Published on the OCW site in 2019, the course uses linear algebra concepts for understanding and creating machine learning algorithms, especially as applied to deep learning and neural networks. This course reviews linear algebra with applications to probability and statistics and optimization and, above all, a full explanation of deep learning. 

Professor Raj Rao was the inspiration for the course 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, and co-taught it with Professor Strang in the first year it was offered at MIT. Since then, he has developed a very successful course on computational linear algebra.

Professor Strang recorded a new series of videos called A 2020 Vision of Linear Algebra.  These six videos contain ideas and suggestions from Professor Strang about the recommended order of topics in teaching and learning linear algebra.

Strang, Gilbert. Introduction to Linear Algebra. 5th ed. 2016. Wellesley-Cambridge Press. ISBN: 9780980232776.

Strang, Gilbert. Linear Algebra and Learning from Data. 2019. Wellesley-Cambridge Press. ISBN: 9780692196380.

Strang, Gilbert. Linear Algebra for Everyone. 2020. Wellesley-Cambridge Press. ISBN: 9781733146630.

Strang, Gilbert. Differential Equations and Linear Algebra. 2014. Wellesley-Cambridge Press. ISBN: 9780980232790.

ZoomNotes for Linear Algebra (PDF). Professor Strang created these notes in 2020 and 2021 when many MIT classes were moved online (using Zoom) due to the COVID-19 Pandemic. He hopes that faculty who are planning a linear algebra course and students who are reading for themselves will see these notes.

Guest speaker at the weekly OLSUME (Online Seminar on Undergraduate Mathematics Education) on the topic Linear Algebra and Deep Learning (MP4)

“Linear Algebra, Teaching, and MIT OpenCourseWare” (YouTube) on Lex Fridman Podcast

ChapterS FILES
1: Introduction to Calculus                
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 1 Study Guide (PDF)
2: Derivatives                
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 2 Study Guide (PDF)
3: Applications of the Derivative                
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 3 Study Guide (PDF)
4: Derivatives by the Chain Rule                
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 4 Study Guide (PDF)
5: Integrals                
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 5 Study Guide (PDF)
6: Exponentials and Logarithms                
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 6 Study Guide (PDF)
7: Techniques of Integration                
7.1 Integration by Parts                          
7.2 Trigonometric Integrals                          
7.3 Trigonometric Substitutions                          
7.4 Partial Fractions                          
7.5 Improper Integrals
Chapter 7 Study Guide (PDF)
8: Applications of the Integral                
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 8 Study Guide (PDF)
9: Polar Coordinates and Complex Numbers                
9.1 Polar Coordinates                          
9.2 Polar Equations and Graphs                          
9.3 Slope, Length, and Area for Polar Curves                          
9.4 Complex Numbers
Chapter 9 Study Guide (PDF)
10: Infinite Series                
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 10 Study Guide (PDF)
11: Vectors and Matrices                
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 11 Study Guide (PDF)
12: Motion along a Curve                
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 12 Study Guide (PDF)
13: Partial Derivatives                
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 13 Study Guide (PDF)
14: Multiple Integrals                
14.1 Double Integrals                          
14.2 Changing to Better Coordinates                          
14.3 Triple Integrals                          
14.4 Cylindrical and Spherical Coordinates
Chapter 14 Study Guide (PDF)
15: Vector Calculus                
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 15 Study Guide (PDF)

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