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Single-Variable Calculus II
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This course is the second installment of Single-Variable Calculus. The student will ...

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)

Subject:
Computer Science
Material Type:
Full Course
Provider:
The Saylor Foundation
Vector Calculus
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This is a text on elementary multivariable calculus, designed for students who ...

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.

Subject:
Calculus
Material Type:
Activity/Lab
Homework/Assignment
Textbook
Provider:
Schoolcraft College
Author:
Michael Corral
Mathematics for Materials Scientists and Engineers, Fall 2005
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This course covers the mathematical techniques necessary for understanding of materials science ...

This course covers the mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from the materials science and engineering core courses (3.012 and 3.014) to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, and fourier analysis. Users may find additional or updated materials at Professor Carter's 3.016 course Web site.

Subject:
Calculus
Material Type:
Full Course
Textbook
Provider:
M.I.T.
Provider Set:
M.I.T. OpenCourseWare
Author:
Carter, W. Craig
The Art of Approximation in Science and Engineering, Spring 2008
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This course teaches simple reasoning techniques for complex phenomena: divide and conquer, ...

This course teaches simple reasoning techniques for complex phenomena: divide and conquer, dimensional analysis, extreme cases, continuity, scaling, successive approximation, balancing, cheap calculus, and symmetry. Applications are drawn from the physical and biological sciences, mathematics, and engineering. Examples include bird and machine flight, neuron biophysics, weather, prime numbers, and animal locomotion. Emphasis is on low-cost experiments to test ideas and on fostering curiosity about phenomena in the world.

Subject:
Engineering
Material Type:
Full Course
Textbook
Provider:
M.I.T.
Provider Set:
M.I.T. OpenCourseWare
Author:
Sanjoy Mahajan
Engineering Mechanics II, Spring 2006
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This subject provides an introduction to fluid mechanics. Students are introduced to ...

This subject provides an introduction to fluid mechanics. Students are introduced to and become familiar with all relevant physical properties and fundamental laws governing the behavior of fluids and learn how to solve a variety of problems of interest to civil and environmental engineers. While there is a chance to put skills from Calculus and Differential Equations to use in this subject, the emphasis is on physical understanding of why a fluid behaves the way it does. The aim is to make the students think as a fluid. In addition to relating a working knowledge of fluid mechanics, the subject prepares students for higher-level subjects in fluid dynamics.

Subject:
Environmental Science
Calculus
Material Type:
Full Course
Textbook
Provider:
M.I.T.
Provider Set:
M.I.T. OpenCourseWare
Author:
Madsen, Ole
Geometry and Quantum Field Theory, Fall 2002
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A rigorous introduction designed for mathematicians into perturbative quantum field theory, using ...

A rigorous introduction designed for mathematicians into perturbative quantum field theory, using the language of functional integrals. Basics of classical field theory. Free quantum theories. Feynman diagrams. Renormalization theory. Local operators. Operator product expansion. Renormalization group equation. The goal is to discuss, using mathematical language, a number of basic notions and results of QFT that are necessary to understand talks and papers in QFT and string theory.

Subject:
Calculus
Geometry
Material Type:
Full Course
Textbook
Provider:
M.I.T.
Provider Set:
M.I.T. OpenCourseWare
Author:
Etingof, Pavel I.
Multivariable Calculus, Fall 2010
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This course covers differential, integral and vector calculus for functions of more ...

This course covers differential, integral and vector calculus for functions of more than one variable. These mathematical tools and methods are used extensively in the physical sciences, engineering, economics and computer graphics.

Subject:
Calculus
Material Type:
Full Course
Textbook
Provider:
M.I.T.
Provider Set:
M.I.T. OpenCourseWare
Author:
Auroux, Denis
Numerical Marine Hydrodynamics (13.024), Spring 2003
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Introduction to numerical methods: interpolation, differentiation, integration, systems of linear equations. Solution ...

Introduction to numerical methods: interpolation, differentiation, integration, systems of linear equations. Solution of differential equations by numerical integration, partial differential equations of inviscid hydrodynamics: finite difference methods, panel methods. Fast Fourier Transforms. Numerical representation of sea waves. Computation of the motions of ships in waves. Integral boundary layer equations and numerical solutions.

Subject:
Calculus
Material Type:
Full Course
Textbook
Provider:
M.I.T.
Provider Set:
M.I.T. OpenCourseWare
Author:
Prof. Jerome Milgram
Logic II, Spring 2004
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This course begins with an introduction to the theory of computability, then ...

This course begins with an introduction to the theory of computability, then proceeds to a detailed study of its most illustrious result: Kurt GĚŚdel's theorem that, for any system of true arithmetical statements we might propose as an axiomatic basis for proving truths of arithmetic, there will be some arithmetical statements that we can recognize as true even though they don't follow from the system of axioms. In my opinion, which is widely shared, this is the most important single result in the entire history of logic, important not only on its own right but for the many applications of the technique by which it's proved. We'll discuss some of these applications, among them: Church's theorem that there is no algorithm for deciding when a formula is valid in the predicate calculus; Tarski's theorem that the set of true sentence of a language isn't definable within that language; and GĚŚdel's second incompleteness theorem, which says that no consistent system of axioms can prove its own consistency.

Subject:
Arts and Humanities
Philosophy
Linguistics
Material Type:
Full Course
Homework/Assignment
Lecture Notes
Syllabus
Provider:
M.I.T.
Provider Set:
M.I.T. OpenCourseWare
Author:
McGee, Vann
Introduction to Analysis, Fall 2012
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Analysis I in its various versions covers fundamentals of mathematical analysis: continuity, ...

Analysis I in its various versions covers fundamentals of mathematical analysis: continuity, differentiability, some form of the Riemann integral, sequences and series of numbers and functions, uniform convergence with applications to interchange of limit operations, some point-set topology, including some work in Euclidean n-space. MIT students may choose to take one of three versions of 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible. Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the plane) and its point-set topology. Option C (18.100C) is a 15-unit variant of Option B, with further instruction and practice in written and oral communication.

Subject:
Calculus
Material Type:
Full Course
Textbook
Provider:
M.I.T.
Provider Set:
M.I.T. OpenCourseWare
Author:
Arthur Mattuck
Space Mathematics Website
Conditions of Use:
No Strings Attached
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This Website contains over 200 authentic math problems that cover solar physics, ...

This Website contains over 200 authentic math problems that cover solar physics, space physics, radiation dosimetry, and the human impacts of space weather. The problems range from pre-algebra to calculus and span the math skills appropriate for grade 8-12 students. The problems are taken from authentic applications of arithmetic, graph analysis, pre-algebra, and algebra.

Subject:
Algebra
Calculus
Geometry
Material Type:
Activity/Lab
Provider:
NASA
Provider Set:
NASA
Volumes of Complex Solids
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Challenged with a hypothetical engineering work situation in which they need to ...

Challenged with a hypothetical engineering work situation in which they need to figure out the volume and surface area of a nuclear power plant’s cooling tower (a hyperbolic shape), students learn to calculate the volume of complex solids that can be classified as solids of revolution or solids with known cross sections. These objects of complex shape defy standard procedures to compute volumes. Even calculus techniques depend on the ability to perform multiple measurements of the objects or find functional descriptions of their edges. During both guided and independent practice, students use (free GeoGebra) geometry software, a photograph of the object, a known dimension of it, a spreadsheet application and integral calculus techniques to calculate the volume of complex shape solids within a margin of error of less than 5%—an approach that can be used to compute the volumes of big or small objects. This activity is suitable for the end of the second semester of AP Calculus classes, serving as a major grade for the last six-week period, with students’ project results presentation grades used as the second semester final test.

Subject:
Career and Technical Education
Mathematics
Geometry
Measurement and Data
Material Type:
Activity/Lab
Provider:
TeachEngineering
Author:
Miguel R. Ramirez, Galena Park High School
Miguel R. Ramirez, high school math teacher, Texas, USA
Advanced Calculus
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Some of the topics that this book addresses are: Vector spaces; finite-dimensional ...

Some of the topics that this book addresses are: Vector spaces; finite-dimensional vector spaces; differential calculus; compactness and completeness; scalar product space; differential equations; multilenear functionals; integration; differentiable manifolds; integral calculus on manifolds; exterior calculus.

Note: this is a 57 MB PDF Document.

Subject:
Calculus
Material Type:
Textbook
Provider:
Harvard University
Provider Set:
Individual Authors
Author:
Sternberg Shlomo and Lynn Loomis
Advanced Stochastic Processes, Fall 2013
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This class covers the analysis and modeling of stochastic processes. Topics include ...

This class covers the analysis and modeling of stochastic processes. Topics include measure theoretic probability, martingales, filtration, and stopping theorems, elements of large deviations theory, Brownian motion and reflected Brownian motion, stochastic integration and Ito calculus and functional limit theorems. In addition, the class will go over some applications to finance theory, insurance, queueing and inventory models.

Subject:
Business and Communication
Finance
Material Type:
Assessment
Full Course
Homework/Assignment
Lecture Notes
Syllabus
Provider:
M.I.T.
Provider Set:
M.I.T. OpenCourseWare
Author:
Gamarnik, David
Multithreaded Parallelism: Languages and Compilers, Fall 2002
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Languages and compilers to exploit multithreaded parallelism. Implicit parallel programming using functional ...

Languages and compilers to exploit multithreaded parallelism. Implicit parallel programming using functional languages and their extensions. Higher-order functions, non-strictness, and polymorphism. Explicit parallel programming and nondeterminism. The lambda calculus and its variants. Term rewriting and operational semantics. Compiling multithreaded code for symmetric multiprocessors and clusters. Static analysis and compiler optimizations.

Subject:
Computer Science
Material Type:
Full Course
Textbook
Provider:
M.I.T.
Provider Set:
M.I.T. OpenCourseWare
Author:
Arvind, V.
Mathematics for Computer Science
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A basic introduction to Calculus and Linear Algebra. The goal is to ...

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.

Subject:
Computer Science
Calculus
Material Type:
Assessment
Full Course
Homework/Assignment
Reading
Provider:
ArsDigita University
Provider Set:
ArsDigita University
Author:
Tara Holm
Calculus Revisited, Fall 2010
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Calculus Revisited is a series of videos and related resources that covers ...

Calculus Revisited is a series of videos and related resources that covers the materials normally found in a freshman-level introductory calculus course.  The series was first released in 1970 as a way for people to review the essentials of calculus.  It is equally valuable for students who are learning calculus for the first time.

Subject:
Calculus
Material Type:
Full Course
Textbook
Provider:
M.I.T.
Provider Set:
M.I.T. OpenCourseWare
Author:
Gross, Herbert
Calculus Grapher
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Draw a graph of any function and see graphs of its derivative ...

Draw a graph of any function and see graphs of its derivative and integral. Don't forget to use the magnify/demagnify controls on the y-axis to adjust the scale.

Subject:
Calculus
Material Type:
Simulation
Provider:
University of Colorado Boulder
Provider Set:
PhET Interactive Simulations
Author:
Dubson, Michael
Loeblein, Trish
Michael Dubson
PhET Interactive Simulations
Trish Loeblein
Calculus (CA Textbook)
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This Internet resource provides introductory information, concept or skill development in Mathematics ...

This Internet resource provides introductory information, concept or skill development in Mathematics for grade 9, 10, 11, and 12 students who are at grade level in a single student situation. This text was initially written by David Guichard. The single variable material (not including infinite series) was originally a modification and expansion of notes written by Neal Koblitz at the University of Washington, who generously gave permission to use, modify, and distribute his work. New material has been added, and old material has been modified, so some portions now bear little resemblance to the original.

This digital textbook was reviewed for its alignment with California content standards.

Subject:
Calculus
Functions
Material Type:
Textbook
Provider:
Whitman College