Supplementary Materials

This page serves as a repository for textbooks and supplementary materials used in Calculus II (math102-a22), taught by Chris Eppolito in the Advent 2022 semester.

I suggest that you also see the study tips on the home page. Do let me know if you have suggestions for this space (or the website at large).

See the schedule of topics for our day-to-day schedule.

I will suggest readings and exercises from Calculus, a free textbook written by Gilbert Strang.

A 600 page book can be a bit daunting to page through on the computer, so here are links to individual chapter files. Our course mainly covers chapters 5 through 10 (though probably not in that order). I've included the full table of contents here for completeness and for review purposes.

1. Introduction to Calculus
   1. Velocity and Distance (1-7)
   2. Calculus Without Limits (8-15)
   3. The Velocity at an Instant (16-21)
   4. Circular Motion (22-28)
   5. A Review of Trigonometry (29-33)
   6. A Thousand Points of Light (34-35)
   7. Computing in Calculus (36-43)
2. Derivatives
   1. The Derivative of a Function (44-49)
   2. Powers and Polynomials (50-57)
   3. The Slope and the Tangent Line (58-63)
   4. Derivative of the Sine and Cosine (64-70)
   5. The Product and Quotient and Power Rules (71-77)
   6. Limits (78-84)
   7. Continuous Functions (85-90)
3. Applications of the Derivative
   1. Linear Approximation (91-95)
   2. Maximum and Minimum Problems (96-104)
   3. Second Derivatives: Minimum vs. Maximum (105-111)
   4. Graphs (112-120)
   5. Ellipses, Parabolas, and Hyperbolas (121-129)
   6. Iterations \( x_{n+1} = F(x_n) \) (130-136)
   7. Newton’s Method and Chaos (137-145)
   8. The Mean Value Theorem and l’Hôpital’s Rule (146-153)
4. The Chain Rule
   1. Derivatives by the Chain Rule (154-159)
   2. Implicit Differentiation and Related Rates (160-163)
   3. Inverse Functions and Their Derivatives (164-170)
   4. Inverses of Trigonometric Functions (171-176)
5. Integrals
   1. The Idea of an Integral (177-181)
   2. Antiderivatives (182-186)
   3. Summation vs. Integration (187-194)
   4. Indefinite Integrals and Substitutions (195-200)
   5. The Definite Integral (201-205)
   6. Properties of the Integral and the Average Value (206-212)
   7. The Fundamental Theorem and Its Consequences (213-219)
   8. Numerical Integration (220-227)
6. Exponentials and Logarithms
   1. An Overview (228-235)
   2. The Exponential \( e^x \) (236-241)
   3. Growth and Decay in Science and Economics (242-251)
   4. Logarithms (252-258)
   5. Separable Equations Including the Logistic Equation (259-266)
   6. Powers Instead of Exponentials (267-276)
   7. Hyperbolic Functions (277-282)
7. Techniques of Integration
   1. Integration by Parts (283-287)
   2. Trigonometric Integrals (288-293)
   3. Trigonometric Substitutions (294-299)
   4. Partial Fractions (300-304)
   5. Improper Integrals (305-310)
8. Applications of the Integral
   1. Areas and Volumes by Slices (311-319)
   2. Length of a Plane Curve (320-324)
   3. Area of a Surface of Revolution (325-327)
   4. Probability and Calculus (328-335)
   5. Masses and Moments (336-341)
   6. Force, Work, and Energy (342-347)
9. Polar Coordinates and Complex Numbers
   1. Polar Coordinates (348-350)
   2. Polar Equations and Graphs (351-355)
   3. Slope, Length, and Area for Polar Curves (356-359)
   4. Complex Numbers (360-367)
10. Infinite Series
    1. The Geometric Series (368-373)
    2. Convergence Tests: Positive Series (374-380)
    3. Convergence Tests: All Series (381-384)
    4. The Taylor Series for e^x, sin x, and cos x (385-390)
    5. Power Series (391-397)
11. Vectors and Matrices
    1. Vectors and Dot Products (398-406)
    2. Planes and Projections (407-415)
    3. Cross Products and Determinants (416-424)
    4. Matrices and Linear Equations (425-434)
    5. Linear Algebra in Three Dimensions (435-445)
12. Motion along a Curve
    1. The Position Vector (446-452)
    2. Plane Motion: Projectiles and Cycloids (453-458)
    3. Tangent Vector and Normal Vector (459-463)
    4. Polar Coordinates and Planetary Motion (464-471)
13. Partial Derivatives
    1. Surface and Level Curves (472-474)
    2. Partial Derivatives (475-479)
    3. Tangent Planes and Linear Approximations (480-489)
    4. Directional Derivatives and Gradients (490-496)
    5. The Chain Rule (497-503)
    6. Maxima, Minima, and Saddle Points (504-513)
    7. Constraints and Lagrange Multipliers (514-520)
14. Multiple Integrals
    1. Double Integrals (521-526)
    2. Changing to Better Coordinates (527-535)
    3. Triple Integrals (536-540)
    4. Cylindrical and Spherical Coordinates (541-548)
15. Vector Calculus
    1. Vector Fields (549-554)
    2. Line Integrals (555-562)
    3. Green’s Theorem (563-572)
    4. Surface Integrals (573-581)
    5. The Divergence Theorem (582-588)
    6. Stokes’ Theorem and the Curl of \( F \) (589-598)
16. Mathematics after Calculus
    1. Linear Algebra (599-602)
    2. Differential Equations (603-610)
    3. Discrete Mathematics (611-615)

This is another free textbook I like. The Apex Calculus homepage is quite nice. This textbook has been expanded recently; there is quite a lot of content not linked here (because the stable version has not been updated yet).

Here are links to chapters organized by topic.

1. Limits
   1. An introduction to Limits
   2. Epsilon-Delta Definition of a Limit
   3. Finding Limits Analytically
   4. One Sided Limits
   5. Continuity
   6. Limits Involving Infinity
2. Derivatives
   1. Instantaneous Rates of Change: The Derivative
   2. Interpretations of the Derivative
   3. Basic Differentiation Rules
   4. The Product and Quotient Rules
   5. The Chain Rule
   6. Implicit Differentiation
   7. Derivatives of Inverse Functions
3. The Graphical Behavior of Functions
   1. Extreme Values
   2. The Mean Value Theorem
   3. Increasing and Decreasing Functions
   4. Concavity and the Second Derivative
   5. Curve Sketching
4. Applications of the Derivative
   1. Newton's Method
   2. Related Rates
   3. Optimization
   4. Differentials
5. Integration
   1. Antiderivatives and Indefinite Integration
   2. The Definite Integral
   3. Riemann Sums
   4. The Fundamental Theorem of Calculus
   5. Numerical Integration
6. Techniques of Integration
   1. Substitution
   2. Integration by Parts
   3. Trigonometric Integrals
   4. Trigonometric Substitution
   5. Partial Fraction Decomposition
   6. Hyperbolic Functions
   7. L'Hopital's Rule
   8. Improper Integration
7. Applications of Integration
   1. Area Between Curves
   2. Volume by Cross-Sectional Area: Disk and Washer Methods
   3. The Shell Method
   4. Arc Length and Surface Area
   5. Work
   6. Fluid Forces
8. Sequences and Series
   1. Sequences
   2. Infinite Series
   3. Integral and Comparison Tests
   4. Ratio and Root Tests
   5. Alternating Series and Absolute Convergence
   6. Power Series
   7. Taylor Polynomials
   8. Taylor Series
9. Curves in the Plane
   1. Conic Sections
   2. Parametric Equations
   3. Calculus and Parametric Equations
   4. Introduction to Polar Coordinates
   5. Calculus and Polar Functions
10. Vectors
    1. Introduction to Cartesian Coordinates in Space
    2. An Introduction to Vectors
    3. The Dot Product
    4. The Cross Product
    5. Lines
    6. Planes
11. Vector-Valued Functions
    1. Vector-Valued Functions
    2. Calculus and Vector-Valued Functions
    3. The Calculus of Motion
    4. Unit Tangent and Normal Vectors
    5. The Arc Length Parameter and Curvature
12. Functions of Several Variables
    1. Introduction to Multivariable Functions
    2. Limits and Continuity of Multivariable Functions
    3. Partial Derivatives
    4. Differentiability and the Total Differential
    5. The Multivariable Chain Rule
    6. Directional Derivatives
    7. Tangent Lines, Normal Lines, and Tangent Planes
    8. Extreme Values
13. Multiple Integration
    1. Iterated Integrals and Area
    2. Double Integration and Volume
    3. Double Integration with Polar Coordinates
    4. Center of Mass
    5. Surface Area
    6. Volume Between Surfaces and Triple Integration

Here are a few additional resources I suggest for this course. See the schedule page for yet more resources, linked by topic.