This is a text for a two-term course in introductory real analysis for junior or senior mathematics majors and science students with a serious interest in mathematics. Prospective educators or mathematically gifted high school students can also benefit from the mathematical maturity that can be gained from an introductory real analysis course.

The book is designed to fill the gaps left in the development of calculus as it is usually presented in an elementary course, and to provide the background required for insight into more advanced courses in pure and applied mathematics. The standard elementary calculus sequence is the only specific prerequisite for Chapters 1–5, which deal with real-valued functions. (However, other analysis oriented courses, such as elementary differential equation, also provide useful preparatory experience.) Chapters 6 and 7 require a working knowledge of determinants, matrices and linear transformations, typically available from a first course in linear algebra. Chapter 8 is accessible after completion of Chapters 1–5.

This is a Calculus I interactive textbook with modules curated and created on The Ohio State University's Ximera platform.

The software upon which this interactive textbook was built is licensed under a GNU General Public License v.2.0, and therefore this resource caries the same license. Pursuant to this license, no warranties are made.

Review the license terms at https://github.com/XimeraProject/server/blob/master/LICENSE

This is a Calculus II interactive textbook with modules curated and created on The Ohio State University's Ximera platform.

The software upon which this interactive textbook was built is licensed under a GNU General Public License v.2.0, and therefore this resource caries the same license. Pursuant to this license, no warranties are made.

Review the license terms at https://github.com/XimeraProject/server/blob/master/LICENSE

This book covers the standard material for a one-semester course in multivariable calculus. The topics include curves, differentiability and partial derivatives, multiple integrals, vector fields, line and surface integrals, and the theorems of Green, Stokes, and Gauss. Roughly speaking the book is organized into three main parts corresponding to the type of function being studied: vector-valued functions of one variable, real-valued functions of many variables, and finally the general case of vector-valued functions of many variables. As is always the case, the most productive way for students to learn is by doing problems, and the book is written to get to the exercises as quickly as possible. The presentation is geared towards students who enjoy learning mathematics for its own sake. As a result, there is a priority placed on understanding why things are true and a recognition that, when details are sketched or omitted, that should be acknowledged. Otherwise the level of rigor is fairly normal. Matrices are introduced and used freely. Prior experience with linear algebra is helpful, but not required.

Optimal, Integral, Likely is a free, open-source textbook intended for UBC’s course MATH 105: Integral Calculus with Applications to Commerce and Social Sciences. It is shared under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

The Pre-Calculus course was developed through the Ohio Department of Higher Education OER Innovation Grant. This work was completed and the course was posted in September 2019. The course is part of the Ohio Transfer Module and is also named TMM002. For more information about credit transfer between Ohio colleges and universities, please visit: www.ohiohighered.org/transfer.Team LeadKameswarrao Casukhela                   Ohio State University LimaContent ContributorsLuiz Felipe Martins                             Cleveland State UniversityIeda Rodrigues                                   Cleveland State UniversityTeri Thomas                                        Stark State CollegeLibrarianDaniel Dotson                                     Ohio State University                     Review TeamAlice Taylor                                        University of Rio GrandeRita Ralph                                          Columbus State Community College

Sine and Cosine Functions - amplitude, period, phase-shift, sinusoidal functions, periodic functionsTMM 002 PRECALCULUS (Revised March 21, 2017)1. Functions: 1a. Analyze functions. Routine analysis includes discussion of domain, range, zeros, general function behavior (increasing, decreasing, extrema, etc.), as well as periodic characteristics such as period, frequency, phase shift, and amplitude. In addition to performing rote processes, the student can articulate reasons for choosing a particular process, recognize function families and anticipate behavior, and explain the implementation of a process (e.g., why certain real numbers are excluded from the domain of a given function).*

Secant and Cosecant Functions - period, phase-shift, periodic functions, asymptotes, sine function, cosine function, domainTMM 002 PRECALCULUS (Revised March 21, 2017)1. Functions: 1a. Analyze functions. Routine analysis includes discussion of domain, range, zeros, general function behavior (increasing, decreasing, extrema, etc.), as well as periodic characteristics such as period, frequency, phase shift, and amplitude. In addition to performing rote processes, the student can articulate reasons for choosing a particular process, recognize function families and anticipate behavior, and explain the implementation of a process (e.g., why certain real numbers are excluded from the domain of a given function).*

Tangent and Cotangent Functions - period, phase-shift, periodic functions, asymptotes, sine and cosine functionsTMM 002 PRECALCULUS (Revised March 21, 2017)1. Functions: 1a. Analyze functions. Routine analysis includes discussion of domain, range, zeros, general function behavior (increasing, decreasing, extrema, etc.), as well as periodic characteristics such as period, frequency, phase shift, and amplitude. In addition to performing rote processes, the student can articulate reasons for choosing a particular process, recognize function families and anticipate behavior, and explain the implementation of a process (e.g., why certain real numbers are excluded from the domain of a given function).*

Inverse Trigonometric Functions - domain, range, graph, one-to-one function, applications, periodic functions TMM 002 PRECALCULUS (Revised March 21, 2017)1. Functions: 1a. Analyze functions. Routine analysis includes discussion of domain, range, zeros, general function behavior (increasing, decreasing, extrema, etc.), as well as periodic characteristics such as period, frequency, phase shift, and amplitude. In addition to performing rote processes, the student can articulate reasons for choosing a particular process, recognize function families and anticipate behavior, and explain the implementation of a process (e.g., why certain real numbers are excluded from the domain of a given function).*

Sinusoidal function, harmonic motion, periodic functions, applications.TMM 002 PRECALCULUS (Revised March 21, 2017)1. Functions: 1a. Analyze functions. Routine analysis includes discussion of domain, range, zeros, general function behavior (increasing, decreasing, extrema, etc.), as well as periodic characteristics such as period, frequency, phase shift, and amplitude. In addition to performing rote processes, the student can articulate reasons for choosing a particular process, recognize function families and anticipate behavior, and explain the implementation of a process (e.g., why certain real numbers are excluded from the domain of a given function).*

Law of CosinesTMM 002 PRECALCULUS (Revised March 21, 2017)2c. Analyze general triangles. Routine analysis of side lengths and angle measurements using trigonometric ratios/functions, as well as other relationships.*Sample Tasks:The student can solve general triangles using trigonometric ratios and relationships including laws of sine and cosine.The student can compare similar triangles.The student can compute length and angle measurements inside complex drawings involving multiple geometric objects.The student can algebraically describe relationships inside complex drawings involving multiple geometric objects.

Law of SinesTMM 002 PRECALCULUS (Revised March 21, 2017)2c. Analyze general triangles. Routine analysis of side lengths and angle measurements using trigonometric ratios/functions, as well as other relationships.*Sample Tasks:The student can solve general triangles using trigonometric ratios and relationships including laws of sine and cosine.The student can compare similar triangles.The student can compute length and angle measurements inside complex drawings involving multiple geometric objects.The student can algebraically describe relationships inside complex drawings involving multiple geometric objects.

Trigonometric Equations, trigonometric identitiesTMM 002 PRECALCULUS (Revised March 21, 2017)4c. Become fluent with conversions using traditional equivalency families.*(e.g., (sin(𝑡))2+(cos(𝑡))2=1; (tan(𝑡))2+1=(sec(𝑡))2; sums/differences; products; double angle; Euler’s Formula (𝑒𝑖𝜃=cos(𝜃)+𝑖sin(𝜃)); etc.)Sample Tasks:The student can prove trigonometric identities.The student solves trigonometric equations.To solve √cos(4𝑡) = √sin(4𝑡), the student solves cos(4𝑡) =sin(4𝑡) and knows this procedure may result in extraneous solutions.The student solves |cos (2𝜃−3)| + 32 = 2 by rewriting the left-hand side as a piecewise-defined function.The student can rewrite formulas involving multiple occurrences of the variable to formulas involving a single occurrence. Write 𝑎sin(𝑤 𝑡)+𝑏cos(𝑤 𝑡) as 𝐴 sin (𝑤 𝑡+𝐵) or 𝐵 cos (𝑤 𝑡+𝐵). The student can rewrite sums as products to reveal attributes such as zeros, envelopes, and phase interference.The student can solve 2 𝑠𝑖𝑛2(𝑡)+7sin(𝑡)−4=0 on a given interval.The student can solve 𝑙𝑜𝑔4(sin (𝑡))+𝑙𝑜𝑔4(2sin(𝑡)+7)=1 on a given interval. 

Complex numbers, Euler's notation, sine-cosine representation, DeMoivre's Theorem, product and quotient of complext numbers, powers and roots of complex numbers, magnitude, polar coordinates,