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
6.7 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
13.5 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)

Fall 2023
Online Textbook