AP Calculus AB is organized into 6 units (4 units in the first semester and 2 units in the second semester). The lessons in each unit include: Readings, Multimedia (lessons), Assignments, and Assessments. The course covers the principles of functions, derivatives, integrals, limits, approximation, and applications and modeling. Students will be able to: work with functions represented in a variety of ways; understand the connections among graphical, numerical, analytical, or verbal representations; understand the meaning of the derivative in terms of a rate of change and local linear approximation, and be able to use derivatives to solve a variety of problems; understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change, and use integrals to solve a variety of problems; understand the relationship between the derivative and the definite integral as expressed in both parts of the fundamental theorem of calculus.

This course begins with a review of algebra specifically designed to help and prepare the student for the study of calculus, and continues with discussion of functions, graphs, limits, continuity, and derivatives. The appendix provides a large collection of reference facts, geometry, and trigonometry that will assist in solving calculus problems long after the course is over. Upon successful completion of this course, the student will be able to: calculate or estimate limits of functions given by formulas, graphs, or tables by using properties of limits and L’hopital’s Rule; state whether a function given by a graph or formula is continuous or differentiable at a given point or on a given interval and justify the answer; calculate average and instantaneous rates of change in context, and state the meaning and units of the derivative for functions given graphically; calculate derivatives of polynomial, rational, common transcendental functions, and implicitly defined functions; apply the ideas and techniques of derivatives to solve maximum and minimum problems and related rate problems, and calculate slopes and rates for function given as parametric equations; find extreme values of modeling functions given by formulas or graphs; predict, construct, and interpret the shapes of graphs; solve equations using Newton’s Method; find linear approximations to functions using differentials; festate in words the meanings of the solutions to applied problems, attaching the appropriate units to an answer; state which parts of a mathematical statement are assumptions, such as hypotheses, and which parts are conclusions. This free course may be completed online at any time. It has been developed through a partnership with the Washington State Board for Community and Technical Colleges; the Saylor Foundation has modified some WSBCTC materials. (Mathematics 005)

CK-12 Foundation's Single Variable Calculus FlexBook introduces high school students to the topics covered in the Calculus AB course. Topics include: Limits, Derivatives, and Integration.

CK-12's Texas Instrument Calculus Student Edition is a useful companion to a Calculus course, offering extra assignments and opportunities for students to understand course material through their graphing calculator.

CK-12's Texas Instruments Calculus Teacher's Edition is a useful companion to a Calculus course, offering extra assignments and opportunities for students to understand course material through their graphing calculator.

Calculus with Theory, covers the same material as 18.01 (Single Variable Calculus), but at a deeper and more rigorous level. It emphasizes careful reasoning and understanding of proofs. The course assumes knowledge of elementary calculus.

This book covers the following topics: Sequences, limits, and difference equations; functions and their properties; best affine approximations; integration; polynomial approximations and Taylor series; transcendental functions; the complex plane; differential equations.

Highlights of Calculus is a series of short videos that introduces the basics of calculus—how it works and why it is important. The intended audience is high school students, college students, or anyone who might need help understanding the subject. The videos were created by renowned mathematics professor Gilbert Strang who has taught at MIT since 1962.

The video series reviews the key topics and ideas of calculus with applications to real-life situations and problems
and then fully covers the concept of Derivatives.

A basic introduction to Calculus and Linear Algebra. The goal is to make students mathematically literate in preparation for studying a scientific/engineering discipline. The first week covers differential calculus: graphing functions, limits, derivatives, and applying differentiation to real-world problems, such as maximization and rates of change. The second week covers integral calculus: sums, integration, areas under curves and computing volumes. This is not meant to be a comprehensive calculus course, but rather an introduction to the fundamental concepts. The third and fourth weeks introduce some basic linear algebra: vector spaces, linear transformations, matrices, matrix operations, and diagonalization. The emphasis will be on using the results, not on their proofs.

"This course, which is geared toward Freshmen, is an undergraduate seminar on mathematical problem solving. It is intended for students who enjoy solving challenging mathematical problems and who are interested in learning various techniques and background information useful for problem solving. Students in this course are expected to compete in a nationwide mathematics contest for undergraduates."

This course is designed to introduce the student to the rigorous examination of the real number system and the foundations of calculus. Analysis lies at the heart of the trinity of higher mathematics algebra, analysis, and topology because it is where the other two fields meet. Upon successful completion of this course, the student will be able to: Use set notation and quantifiers correctly in mathematical statements and proofs; Use proof by induction or contradiction when appropriate; Define the rational numbers, the natural numbers, and the real numbers, and understand their relationship to one another; Define the well-ordering principle the completeness/supremum property of the real line, and the Archimedean property; Prove the existence of irrational numbers; Define supremum and infimum; Correctly and fluently manipulate expressions with absolute value and state the triangle inequality; Define and identify injective, surjective, and bijective mappings; Name the various cardinalities of sets and identify the cardinality of a given set; Define Euclidean space and vector space and show that Euclidean space is a vector space; Define the complex numbers and manipulate them algebraically; Write equations for lines and planes in Euclidean space; Define a normed linear space, a norm, and an inner product; Define metric spaces, open sets; define open, closed, and bounded sets; define cluster points; define density; Define convergence of sequences and prove or disprove the convergence of given sequences; Prove and use properties of limits; Prove standard results about closures, intersections, and unions of open and closed sets; Define compactness using both open covers and sequences; State and prove the Heine-Borel Theorem; State the Bolzano-Weierstrass Theorem; State and use the Cantor Finite Intersection Property; Define Cauchy sequence and prove that specific sequences are Cauchy; Define completeness and prove that Euclidean space with the standard metric is complete; Show that convergent sequences are Cauchy; Define limit superior and limit inferior; Define convergence of series using the Cauchy criterion and use the comparison, ratio, and root tests to show convergence of series; Define continuity and state, prove, and use properties of limits of continuous functions, including the fact that continuous functions attain extreme values on compact sets; Define divergence of functions to infinity and use properties of infinite limits; State and prove the intermediate value property; Define uniform continuity and show that given functions are or are not uniformly continuous; Give standard examples of discontinuous functions, such as the Dirichlet function; Define connectedness and identify connected and disconnected sets Construct the Cantor ternary set and state its properties; Distinguish between pointwise and uniform convergence; Prove that if a sequence of continuous functions converges uniformly, their limit is also continuous; Define derivatives of real- and extended-real-valued functions; Compute derivatives using the limit definition and prove basic properties of derivatives; State the Mean Value Theorem and use it in proofs; Construct the Riemann Integral and state its properties; State the Fundamental Theorem of Calculus and use it in proofs; Define pointwise and uniform convergence of series of functions; Use the Weierstrass M-Test to check for uniform convergence of series; Construct Taylor Series and state Taylor's Theorem; Identify necessary and sufficient conditions for term-by-term differentiation of power series. (Mathematics 241)

This introductory Calculus course covers differentiation and integration of functions of one variable, with applications. Topics include: Concepts of function, limits, and continuity, Differentiation rules, application to graphing, rates, approximations, and extremum problems; Definite and indefinite integration; Fundamental theorem of calculus; Applications of integration to geometry and science; Elementary functions; Techniques of integration; Approximation of definite integrals, improper integrals, and L'Hôpital's rule.

Differentiation and integration of functions of one variable, with applications. Concepts of function, limits, and continuity. Differentiation rules, application to graphing, rates, approximations, and extremum problems. Definite and indefinite integration. Fundamental theorem of calculus. Applications of integration to geometry and science. Elementary functions. Techniques of integration. Approximation of definite integrals, improper integrals, and L'Hospital's rule.

Differentiation and integration of functions of one variable, with applications. Concepts of function, limits, and continuity. Differentiation rules, application to graphing, rates, approximations, and extremum problems. Definite and indefinite integration. Fundamental theorem of calculus. Applications of integration to geometry and science. Elementary functions. Techniques of integration. Approximation of definite integrals, improper integrals, and L'Hospital's rule.

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)

In this unit we explore questions of access to community services. To make what might be quite a dry task more challenging we use a fictionalised case study of two people for whom access to community services is particularly problematic. Jim and Marianne are both long-term heroin addicts. Additional problems associated with their addiction are homelessness and physical illness. Their situation raises both practical questions, about how services can be accessed, and moral questions, about entitlement to resources when their problems can be regarded as at least in part self-inflicted.