RES.18-001 | Fall 2023 | Undergraduate

Calculus Online Textbook

Study Guide

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)

Course Info

Departments
As Taught In
Fall 2023
Learning Resource Types
Online Textbook