Further study of real numbers, their real-valued functions, and integration and differentiation of these functions through the analysis of Manifolds.

Table of Contents:
1. Analytic Geometry
2. Instantaneous Rate of Change: The Derivative
3. Rules for Finding Derivatives
4. Transcendental Functions
5. Curve Sketching
6. Applications of the Derivative
7. Integration
8. Techniques of Integration
9. Applications of Integration
10. Polar Coordinates, Parametric Equations
11. Sequences and Series

Part of a collection of resources found here: https://www.whitman.edu/mathematics/multivariable/

This calculus course covers differentiation and integration of functions of one variable, and concludes with a brief discussion of infinite series. Calculus is fundamental to many scientific disciplines including physics, engineering, and economics.

This course is designed to introduce the student to the study of Calculus through concrete applications. Upon successful completion of this course, students will be able to: Define and identify functions; Define and identify the domain, range, and graph of a function; Define and identify one-to-one, onto, and linear functions; Analyze and graph transformations of functions, such as shifts and dilations, and compositions of functions; Characterize, compute, and graph inverse functions; Graph and describe exponential and logarithmic functions; Define and calculate limits and one-sided limits; Identify vertical asymptotes; Define continuity and determine whether a function is continuous; State and apply the Intermediate Value Theorem; State the Squeeze Theorem and use it to calculate limits; Calculate limits at infinity and identify horizontal asymptotes; Calculate limits of rational and radical functions; State the epsilon-delta definition of a limit and use it in simple situations to show a limit exists; Draw a diagram to explain the tangent-line problem; State several different versions of the limit definition of the derivative, and use multiple notations for the derivative; Understand the derivative as a rate of change, and give some examples of its application, such as velocity; Calculate simple derivatives using the limit definition; Use the power, product, quotient, and chain rules to calculate derivatives; Use implicit differentiation to find derivatives; Find derivatives of inverse functions; Find derivatives of trigonometric, exponential, logarithmic, and inverse trigonometric functions; Solve problems involving rectilinear motion using derivatives; Solve problems involving related rates; Define local and absolute extrema; Use critical points to find local extrema; Use the first and second derivative tests to find intervals of increase and decrease and to find information about concavity and inflection points; Sketch functions using information from the first and second derivative tests; Use the first and second derivative tests to solve optimization (maximum/minimum value) problems; State and apply Rolle's Theorem and the Mean Value Theorem; Explain the meaning of linear approximations and differentials with a sketch; Use linear approximation to solve problems in applications; State and apply L'Hopital's Rule for indeterminate forms; Explain Newton's method using an illustration; Execute several steps of Newton's method and use it to approximate solutions to a root-finding problem; Define antiderivatives and the indefinite integral; State the properties of the indefinite integral; Relate the definite integral to the initial value problem and the area problem; Set up and calculate a Riemann sum; Estimate the area under a curve numerically using the Midpoint Rule; State the Fundamental Theorem of Calculus and use it to calculate definite integrals; State and apply basic properties of the definite integral; Use substitution to compute definite integrals. (Mathematics 101; See also: Biology 103, Chemistry 003, Computer Science 103, Economics 103, Mechanical Engineering 001)

This course is the second installment of Single-Variable Calculus. The student will explore the mathematical applications of integration before delving into the second major topic of this course: series. The course will conclude with an introduction to differential equations. Upon successful completion of this course, students will be able to: Define and describe the indefinite integral; Compute elementary definite and indefinite integrals; Explain the relationship between the area problem and the indefinite integral; Use the midpoint, trapezoidal, and Simpson's rule to approximate the area under a curve; State the fundamental theorem of calculus; Use change of variables to compute more complicated integrals; Integrate transcendental, logarithmic, hyperbolic, and trigonometric functions; Find the area between two curves; Find the volumes of solids using ideas from geometry; Find the volumes of solids of revolution using disks, washers, and shells; Find the surface area of a solid of revolution; Compute the average value of a function; Use integrals to compute displacement, total distance traveled, moments, centers of mass, and work; Use integration by parts to compute definite integrals; Use trigonometric substitution to compute definite and indefinite integrals; Use the natural logarithm in substitutions to compute integrals; Integrate rational functions using the method of partial fractions; Compute improper integrals of both types; Graph and differentiate parametric equations; Convert between Cartesian and polar coordinates; Graph and differentiate equations in polar coordinates; Write and interpret a parameterization for a curve; Find the length of a curve described in Cartesian coordinates, described in polar coordinates, or described by a parameterization; Compute areas under curves described by polar coordinates; Define convergence and limits in the context of sequences and series; Find the limits of sequences and series; Discuss the convergence of the geometric and binomial series; Show the convergence of positive series using the comparison, integral, limit comparison, ratio, and root tests; Show the divergence of a positive series using the divergence test; Show the convergence of alternating series; Define absolute and conditional convergence; Show the absolute convergence of a series using the comparison, integral, limit comparison, ratio, and root tests; Manipulate power series algebraically; Differentiate and integrate power series; Compute Taylor and MacLaurin series; Recognize a first order differential equation; Recognize an initial value problem; Solve a first order ODE/IVP using separation of variables; Draw a slope field given an ODE; Use Euler's method to approximate solutions to basic ODE; Apply basic solution techniques for linear, first order ODE to problems involving exponential growth and decay, logistic growth, radioactive decay, compound interest, epidemiology, and Newton's Law of Cooling. (Mathematics 102; See also: Chemistry 004, Computer Science 104, Mechanical Engineering 002)

Table of Contents:
1. Analytic Geometry
2. Instantaneous Rate of Change: The Derivative
3. Rules for Finding Derivatives
4. Trigonometric Functions
5. Curve Sketching
6. Applications of the Derivative
7. Integration
8. Applications of Integration
9. Transcendental Functions
10. Techniques of Integration
11. More Applications of Integration
12. Polar Coordinates, Parametric Equations
13. Sequences and Series

Table of Contents:
1. Analytic Geometry
2. Instantaneous Rate of Change: The Derivative
3. Rules for Finding Derivatives
4. Transcendental Functions
5. Curve Sketching
6. Applications of the Derivative
7. Integration
8. Techniques of Integration
9. Applications of Integration
10. Polar Coordinates, Parametric Equations
11. Sequences and Series
12. Three Dimensions
13. Vector Functions
14. Partial Differentiation
15. Multiple Integration
16. Vector Calculus
17. Differential Equations

Complementary resources available here: https://www.whitman.edu/mathematics/multivariable/

Text/html version available here: https://www.whitman.edu/mathematics/calculus_online/

Table of Contents:
1 Analytic Geometry
2 Instantaneous Rate of Change: The Derivative
3 Rules for Finding Derivatives
4 Trigonometric Functions
5 Curve Sketching
6 Applications of the Derivative
7 Integration
8 Applications of Integration
9 Transcendental Functions
10 Techniques of Integration
11 More Applications of Integration
12 Polar Coordinates, Parametric Equations
13 Sequences and Series
14 Three Dimensions
15 Vector Functions
16 Partial Differentiation
17 Multiple Integration
18 Vector Calculus
19 Differential Equations
20 Useful formulas

Community resources are available here. An HTML version of the text is available here: https://www.whitman.edu/mathematics/calculus_late_online/

This page emphasizes the practical concepts of calculus, and is intended to provide a new context for the student already familiar with much of the material. The emphasis is on how calculus can actually be used outside of the classroom, and how the language of calculus is important in many other disciplines. It features articles for download, on topics from exponential growth and decay to discontinuities, vector fields and differential equations. All of the articles include extensive notes, examples, and figures. This resource is part of the Teaching Quantitative Skills in the Geosciences collection. http://serc.carleton.edu/quantskills/

This is a text on elementary multivariable calculus, designed for students who have completed courses in single-variable calculus. The traditional topics are covered: basic vector algebra; lines, planes and surfaces; vector-valued functions; functions of 2 or 3 variables; partial derivatives; optimization; multiple integrals; line and surface integrals.

Access also available here: http://www.mecmath.net/

Table of Contents
1 Vectors in Euclidean Space
2 Functions of Several Variables
3 Multiple Integrals
4 Line and Surface Integrals

This is a text on elementary multivariable calculus, designed for students who have completed courses in single-variable calculus. The traditional topics are covered: basic vector algebra; lines, planes and surfaces; vector-valued functions; functions of 2 or 3 variables; partial derivatives; optimization; multiple integrals; line and surface integrals.

These open-source mathematics homework problems are programmed for the WeBWorK mathematics platform and correspond to chapters in OpenStax Calculus Volume I. They were created through a Round Eight Textbook Transformation Grant.

An introductory level single variable calculus book, covering standard topics in differential and integral calculus, and infinite series. Late transcendentals and multivariable versions are also available.

Table of Contents
1 Analytic Geometry
2 The Derivative
3 Rules for Finding Derivatives
4 Transcendental Functions
5 Curve Sketching
6 Applications of the Derivative
7 Integration
8 Techniques of Integration
9 Applications of Integration
10 Polar Coordinates, Parametric Equations
11 Sequences and Series

Access also available here: https://www.whitman.edu/mathematics/multivariable/

An introductory level single variable calculus book, covering standard topics in differential and integral calculus, and infinite series. Late transcendentals and multivariable versions are also available.

I intend this book to be, firstly, a introduction to calculus based on the hyperreal number system. In other words, I will use infinitesimal and infinite numbers freely. Just as most beginning calculus books provide no logical justification for the real number system, I will provide none for the hyperreals. The reader interested in questions of foundations should consult books such as Abraham Robinson's Non-standard Analysis or Robert Goldblatt's Lectures on the Hyperreals. Secondly, I have aimed the text primarily at readers who already have some familiarity with calculus. Although the book does not explicitly assume any prerequisites beyond basic algebra and trigonometry, in practice the pace is too fast for most of those without some acquaintance with the basic notions of calculus.

I intend this book to be, firstly, a introduction to calculus based on the hyperreal number system. In other words, I will use infinitesimal and infinite numbers freely. Just as most beginning calculus books provide no logical justification for the real number system, I will provide none for the hyperreals. The reader interested in questions of foundations should consult books such as Abraham Robinson's Non-standard Analysis or Robert Goldblatt's Lectures on the Hyperreals. Secondly, I have aimed the text primarily at readers who already have some familiarity with calculus. Although the book does not explicitly assume any prerequisites beyond basic algebra and trigonometry, in practice the pace is too fast for most of those without some acquaintance with the basic notions of calculus.

Table of Contents
1 Derivatives

1.1 The arrow paradox
1.2 Rates of change
1.3 The hyperreals
1.4 Continuous functions
1.5 Properties of continuous functions
1.6 The derivative
1.7 Properties of derivatives
1.8 A geometric interpretation of the derivative
1.9 Increasing, decreasing, and local extrema
1.10 Optimization
1.11 Implicit differentiation and rates of change
1.12 Higher-order derivatives
2 Integrals

2.1 Integrals
2.2 Definite integrals
2.3 Properties of definite integrals
2.4 The fundamental theorem of integrals
2.5 Applications of definite integrals
2.6 Some techniques for evaluating integrals
2.7 The exponential and logarithm functions
Answers to Exercises
Index

Access also available here: http://www.synechism.org/wp/yet-another-calculus-text/