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 Calculus Teacher's Edition covers tips, common errors, enrichment, differentiated instruction and problem solving for teaching CK-12 Calculus Student Edition. The solution guide is available upon request.
Seminar covering topics of current interest in biology. Includes reading and analysis of research papers and student presentations. Contact Biology Education Office for topics. This course is one of many Advanced Undergraduate Seminars offered by the Biology Department at MIT. These seminars are tailored for students with an interest in using primary research literature to discuss and learn about current biological research in a highly interactive setting. In 1971, President Nixon declared the "War on Cancer," but after three decades the war is still raging. How much progress have we made toward winning the war and what are we doing to improve the fight? Understanding the molecular and cellular events involved in tumor formation, progression, and metastasis is crucial to the development of innovative therapy for cancer patients. Insights into these processes have been gleaned through basic research using biochemical, molecular, and genetic analyses in yeast, C. elegans, mice, and cell culture models. We will explore the laboratory tools and techniques used to perform cancer research, major discoveries in cancer biology, and the medical implications of these breakthroughs. A focus of the class will be critical analysis of the primary literature to foster understanding of the strengths and limitations of various approaches to cancer research. Special attention will be made to the clinical implications of cancer research performed in model organisms and the prospects for ending the battle with this devastating disease.
In this video segment from The Secret of Life school video, "Sex and the Single Gene" follow as a single fertilized egg cell divides, differentiates, and assembles into the tissues and organs of a new organism.
This course will serve as both an introduction to contemporary political philosophy and a way to explore issues of pluralism and multiculturalism. Racial and ethnic groups, national minorities, aboriginals, women, sexual minorities, and other groups have organized to highlight injustice and demand recognition and accommodation on the basis of their differences. In practice, democratic states have granted a variety of group-differentiated rights, such as exemptions from generally applicable laws, special representation rights, language rights, or limited self-government rights, to different types of groups. This course will examine how different theories of citizenship address the challenges raised by different forms of pluralism. We will focus in particular on the following questions: - Does justice require granting group-differentiated rights? - Do group-differentiated rights conflict with liberal and democratic commitments to equality and justice for all citizens? - What, if anything, can hold a multi-religious, multicultural society together? Why should the citizens of such a society want to hold together?
This module discusses complex differentiation and shows its similarity and differences to real differentiation.
A short introduction to writing Content MathML by hand. It covers tokens, prefix notation, and applying functions and operators. In addition it introduces writing derivatives, integrals, vectors, and matrices.
Motion is vital to life, and to science. This unit will help you to understand why classical motion is probably the most fundamental part of physics. You will examine motion along a line and the ways in which such motion can be represented, through the use of graphs, equations and differential calculus.
How does a regenerating animal "know" what's missing? How are stem cells or differentiated cells used to create new tissues during regeneration? In this class we will take a comparative approach to explore this fascinating problem by critically examining classic and modern scientific literature about the developmental and molecular biology of regeneration. We will learn about conserved developmental pathways that are necessary for regeneration, and we will discuss the relevance of these findings for regenerative medicine. This course is one of many Advanced Undergraduate Seminars offered by the Biology Department at MIT. These seminars are tailored for students with an interest in using primary research literature to discuss and learn about current biological research in a highly interactive setting. Many instructors of the Advanced Undergraduate Seminars are postdoctoral scientists with a strong interest in teaching.
" During development, the genetic content of each cell remains, with a few exceptions, identical to that of the zygote. Most differentiated cells therefore retain all of the genetic information necessary to generate an entire organism. It was through pioneering technology of somatic cell nuclear transfer (SCNT) that this concept was experimentally proven. Only 10 years ago the sheep Dolly was the first mammal to be cloned from an adult organism, demonstrating that the differentiated state of a mammalian cell can be fully reversible to a pluripotent embryonic state. A key conclusion from these experiments was that the difference between pluripotent cells such as embryonic stem (ES) cells and unipotent differentiated cells is solely a consequence of reversible changes. These changes, which have proved to involve reversible alterations to both DNA and to proteins that bind DNA, are known as epigenetic, to distinguish them from genetic alterations to DNA sequence. In this course we will explore such epigenetic changes and study different approaches that can return a differentiated cell to an embryonic state in a process referred to as epigenetic reprogramming, which will ultimately allow generation of patient-specific stem cells and application to regenerative therapy. This course is one of many Advanced Undergraduate Seminars offered by the Biology Department at MIT. These seminars are tailored for students with an interest in using primary research literature to discuss and learn about current biological research in a highly interactive setting. Many instructors of the Advanced Undergraduate Seminars are postdoctoral scientists with a strong interest in teaching."
An introduction to the formulation, methodology, and techniques for numerical solution of engineering problems. Fundamental principles of digital computing and the implications for algorithm accuracy and stability. Error propagation and stability. The solution of systems of linear equations, including direct and iterative techniques. Roots of equations and systems of equations. Numerical interpolation, differentiation and integration. Fundamentals of finite-difference solutions to ordinary differential equations. Error and convergence analysis. Subject taught first half of term.
This course is offered to undergraduates and introduces students to the formulation, methodology, and techniques for numerical solution of engineering problems. Topics covered include: fundamental principles of digital computing and the implications for algorithm accuracy and stability, error propagation and stability, the solution of systems of linear equations, including direct and iterative techniques, roots of equations and systems of equations, numerical interpolation, differentiation and integration, fundamentals of finite-difference solutions to ordinary differential equations, and error and convergence analysis. The subject is taught the first half of the term. This class was originally listed in Course 13 (Ocean Engineering) as 13.002J.
Explores the difference between service and manufacturing operations, and the degree of distinct management skills and tools required. Analyzes cases selected from a variety of service operations with a particular focus on e-commerce. Guest speakers from specific service industries discuss the essence of managing those operations. This course covers organizational, strategic and operational aspects of managing Supply Networks (SNs) from domestic and international perspectives. Topics include alternative SN structures, strategic alliances, design of delivery systems and the role of third party logistics providers. Many of the activities exchanged among enterprises in a SN are of a service nature, and the final output is often a combination of tangible products and services which the end-customer purchases. A series of concepts, frameworks and analytic tools are provided to better understand the management of service operations. Guest speakers share their experiences in managing SNs and services. Restricted to MIT Sloan Fellows in Innovation and Global Leadership.
Multivariable Calculus is an expansion of Single-Variable Calculus in that it extends single variable calculus to higher dimensions. This course begins with a fresh look at limits and continuity, moves to derivatives and the process of generalizing them to higher dimensions, and finally examines multiple integrals (integration over regions of space as opposed to intervals). Upon successful completion of this course, the student will be able to: Define and identify vectors; Define and compute dot and cross-products; Solve problems involving the geometry of lines, curves, planes, and surfaces in space; Define and compute velocity and acceleration in space; Define and solve Kepler's Second Law; Define and compute partial derivatives; Define and determine tangent planes and level curves; Define and compute least squares; Define and determine boundaries and infinity; Define and determine differentials and the directional derivative; Define and compute the gradient and the directional derivative; Define, determine, and apply Lagrange multipliers to solve problems; Define and compute partial differential equations; Define and evaluate double integrals; Use rectangular coordinates to solve problems in multivariable calculus; Use polar coordinates to solve problems in multivariable calculus; Use change of variables to evaluate integrals; Define and use vector fields and line integrals to solve problems in multivariable calculus; Define and verify conservative fields and path independence; Define and determine gradient fields and potential functions; Use Green's Theorem to evaluate and solve problems in multivariable calculus; Define flux; Define and evaluate triple integrals; Define and use rectangular coordinates in space; Define and use cylindrical coordinates; Define and use spherical coordinates; Define and correctly manipulate vector fields in space; Evaluate surface integrals and relate them to flux; Use the Divergence Theorem (Gauss' Theorem) to solve problems in multivariable calculus; Define and evaluate line integrals in space; Apply Stokes' Theorem to solve problems in multivariable calculus; Properly apply Maxwell's Equations to solve problems. (Mathematics 103)
This set of modules examines the mathematical analysis of noise and its effects on communications. It is a subset of a 1 semester course in communication theory.
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.
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.
The first half of this course provides students with the essential tools and skills that are required for dealing successfully with physics at first-year university level. The three broad areas that are covered are (a) mathematical techniques and their relationship with physical phenomena, (b) experimental procedures and (c) communication skills, in particular report writing. The second half of the course covers material similar to that of the first half of PHY1004W. Second semester: Mechanics: vectors, kinematics, dynamics, work, energy power, conservative and non-conservative forces, friction, impulse, momentum, collisions, rotation, rotational dynamics, torque, rotational inertia, rotational energy, angular momentum, static equilibrium, gravitation. Properties of matter: elasticity, elastic moduli, hydrostatics, hydrodynamics. Thermodynamics: temperature, heat, kinetic theory of gases, thermodynamic laws, entropy. UCT PHY1023H.
In this course, we will address how transcriptional regulators both prohibit and drive differentiation during the course of development. How does a stem cell know when to remain a stem cell and when to become a specific cell type? Are there global differences in the way the genome is read in multipotent and terminally differentiated cells? We will explore how stem cell pluripotency is preserved, how master regulators of cell-fate decisions execute developmental programs, and how chromatin regulators control undifferentiated versus differentiated states. Additionally, we will discuss how aberrant regulation of transcriptional regulators produces disorders such as developmental defects and cancer. This course is one of many Advanced Undergraduate Seminars offered by the Biology Department at MIT. These seminars are tailored for students with an interest in using primary research literature to discuss and learn about current biological research in a highly interactive setting. Many instructors of the Advanced Undergraduate Seminars are postdoctoral scientists with a strong interest in teaching.
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)
