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Use the TI-Nspire CX graphing calculator or software to find numerical derivatives and definite integrals using the calculator and graph options.To try it ou.

A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The development of the computer algebra systems in the second half of the 20th century is part of the discipline of 'computer algebra' or 'symbolic computation', which has spurred work in algorithms over mathematical objects such as polynomials.

Calculus - Mean Value Theorem (MVT) Calculus - Derivative of Sine (Geometric proof) Calculus - Newton's Method; Calculus - The Derivative of an Inverse Function; Derivative of the arctangent geometrically; Calculus - Derivative of exponential functions f(x)=b^x; GeoGebra Calculus Applets; Calculus - Cone Inscribed In Sphere. Fundamental theorem of calculus and definite integrals: Integrals Reverse power rule: Integrals Indefinite integrals of common functions: Integrals Definite integrals of common functions: Integrals Integrating with u-substitution: Integrals Integrating using long division and completing the square: Integrals Integrating using trigonometric. This video makes an attempt to teach the fundamentals of calculus 1 such as limits, derivatives, and integration. It explains how to evaluate a function usi.

Computer algebra systems may be divided into two classes: specialized and general-purpose. The specialized ones are devoted to a specific part of mathematics, such as number theory, group theory, or teaching of elementary mathematics.

General-purpose computer algebra systems aim to be useful to a user working in any scientific field that requires manipulation of mathematical expressions. To be useful, a general-purpose computer algebra system must include various features such as:

• a user interface allowing a user to enter and display mathematical formulas, typically from a keyboard, menu selections, mouse or stylus.
• a programming language and an interpreter (the result of a computation commonly has an unpredictable form and an unpredictable size; therefore user intervention is frequently needed),
• a simplifier, which is a rewrite system for simplifying mathematics formulas,
• a memory manager, including a garbage collector, needed by the huge size of the intermediate data, which may appear during a computation,
• an arbitrary-precision arithmetic, needed by the huge size of the integers that may occur,
• a large library of mathematical algorithms and special functions.

The library must not only provide for the needs of the users, but also the needs of the simplifier. For example, the computation of polynomial greatest common divisors is systematically used for the simplification of expressions involving fractions.

This large amount of required computer capabilities explains the small number of general-purpose computer algebra systems. The main ones are Axiom, Maxima, Magma, Maple, Mathematica and SageMath.

History[edit]

Computer algebra systems began to appear in the 1960s and evolved out of two quite different sources—the requirements of theoretical physicists and research into artificial intelligence.

A prime example for the first development was the pioneering work conducted by the later Nobel Prize laureate in physics Martinus Veltman, who designed a program for symbolic mathematics, especially high-energy physics, called Schoonschip (Dutch for 'clean ship') in 1963. Another early system was FORMAC.

Using Lisp as the programming basis, Carl Engelman created MATHLAB in 1964 at MITRE within an artificial-intelligence research environment. Later MATHLAB was made available to users on PDP-6 and PDP-10 systems running TOPS-10 or TENEX in universities. Today it can still be used on SIMH emulations of the PDP-10. MATHLAB ('mathematical laboratory') should not be confused with MATLAB ('matrix laboratory'), which is a system for numerical computation built 15 years later at the University of New Mexico.

The first popular computer algebra systems were muMATH, Reduce, Derive (based on muMATH), and Macsyma; a popular copyleft version of Macsyma called Maxima is actively being maintained. Reduce became free software in 2008.^[1] As of today,^[when?] the most popular commercial systems are Mathematica^[2] and Maple, which are commonly used by research mathematicians, scientists, and engineers. Freely available alternatives include SageMath (which can act as a front-end to several other free and nonfree CAS).

In 1987, Hewlett-Packard introduced the first hand-held calculator CAS with the HP-28 series, and it was possible, for the first time in a calculator,^[3] to arrange algebraic expressions, differentiation, limited symbolic integration, Taylor series construction and a solver for algebraic equations. In 1999, the independently developed CAS Erable for the HP 48 series became an officially integrated part of the firmware of the emerging HP 49/50 series, and a year later into the HP 40 series as well, whereas the HP Prime adopted the Xcas system in 2013.

The Texas Instruments company in 1995 released the TI-92 calculator with a CAS based on the software Derive; the TI-Nspire series replaced Derive in 2007. The TI-89 series, first released in 1998, also contains a CAS.

Casio released their first CAS calculator with the CFX-9970G and succeeded it with the Algebra FX Series in 1999-2003 and the current ClassPad Series.^{[citation needed]}

More recently, computer algebra systems have been implemented using artificial neural networks.^[4]

Symbolic manipulations[edit]

The symbolic manipulations supported typically include:

• simplification to a smaller expression or some standard form, including automatic simplification with assumptions and simplification with constraints
• substitution of symbols or numeric values for certain expressions
• change of form of expressions: expanding products and powers, partial and full factorization, rewriting as partial fractions, constraint satisfaction, rewriting trigonometric functions as exponentials, transforming logic expressions, etc.
• partial and total differentiation
• some indefinite and definite integration (see symbolic integration), including multidimensional integrals
• symbolic constrained and unconstrained global optimization
• solution of linear and some non-linear equations over various domains
• solution of some differential and difference equations
• taking some limits
• integral transforms
• series operations such as expansion, summation and products
• matrix operations including products, inverses, etc.
• theorem proving and verification which is very useful in the area of experimental mathematics

In the above, the word some indicates that the operation cannot always be performed.

Additional capabilities[edit]

Many also include:

• a programming language, allowing users to implement their own algorithms
• arbitrary-precision numeric operations
• exact integer arithmetic and number theory functionality
• Editing of mathematical expressions in two-dimensional form
• plotting graphs and parametric plots of functions in two and three dimensions, and animating them
• drawing charts and diagrams
• APIs for linking it on an external program such as a database, or using in a programming language to use the computer algebra system
• string manipulation such as matching and searching
• add-ons for use in applied mathematics such as physics, bioinformatics, computational chemistry and packages for physical computation

Some include:

• graphic production and editing such as computer-generated imagery and signal processing as image processing

Some computer algebra systems focus on specialized disciplines; these are typically developed in academia and are free. They can be inefficient for numeric operations as compared to numeric systems.

Types of expressions[edit]

The expressions manipulated by the CAS typically include polynomials in multiple variables; standard functions of expressions (sine, exponential, etc.); various special functions (Γ, ζ, erf, Bessel functions, etc.); arbitrary functions of expressions; optimization; derivatives, integrals, simplifications, sums, and products of expressions; truncated series with expressions as coefficients, matrices of expressions, and so on. Numeric domains supported typically include floating-point representation of real numbers, integers (of unbounded size), complex (floating-point representation), interval representation of reals, rational number (exact representation) and algebraic numbers.

Use in education[edit]

There have been many advocates for increasing the use of computer algebra systems in primary and secondary-school classrooms. The primary reason for such advocacy is that computer algebra systems represent real-world math more than do paper-and-pencil or hand calculator based mathematics.^[5]This push for increasing computer usage in mathematics classrooms has been supported by some boards of education. It has even been mandated in the curriculum of some regions.^[6]

Computer algebra systems have been extensively used in higher education.^[7][8] Many universities offer either specific courses on developing their use, or they implicitly expect students to use them for their course work. The companies that develop computer algebra systems have pushed to increase their prevalence among university and college programs.^[9][10]

CAS-equipped calculators are not permitted on the ACT, the PLAN, and in some classrooms^[11] though it may be permitted on all of College Board's calculator-permitted tests, including the SAT, some SAT Subject Tests and the AP Calculus, Chemistry, Physics, and Statistics exams.

Mathematics used in computer algebra systems[edit]

• Knuth–Bendix completion algorithm^[12]
• Root-finding algorithms^[12]
• Symbolic integration via e.g. Risch algorithm or Risch–Norman algorithm
• Hypergeometric summation via e.g. Gosper's algorithm
• Limit computation via e.g. Gruntz's algorithm
• Polynomial factorization via e.g., over finite fields,^[13]Berlekamp's algorithm or Cantor–Zassenhaus algorithm.
• Greatest common divisor via e.g. Euclidean algorithm
• Gaussian elimination^[14]
• Gröbner basis via e.g. Buchberger's algorithm; generalization of Euclidean algorithm and Gaussian elimination
• Schwartz–Zippel lemma and testing polynomial identities
• Quantifier elimination over real numbers via e.g. Tarski's method/Cylindrical algebraic decomposition
• Landau's algorithm (nested radicals)
• Derivatives of elementary functions and special functions. (e.g. See derivatives of the incomplete gamma function.)

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See also[edit]

References[edit]

1. ^'REDUCE Computer Algebra System at SourceForge'. reduce-algebra.sourceforge.net. Retrieved 2015-09-28.
2. ^Interview with Gaston Gonnet, co-creator of MapleArchived 2007-12-29 at the Wayback Machine, SIAM History of Numerical Analysis and Computing, March 16, 2005.
3. ^Nelson, Richard. 'Hewlett-Packard Calculator Firsts'. Hewlett-Packard. Archived from the original on 2010-07-03.
4. ^Ornes, Stephen. 'Symbolic Mathematics Finally Yields to Neural Networks'. Quanta Magazine. Retrieved 2020-11-04.
5. ^'Teaching kids real math with computers'. Ted.com. Retrieved 2017-08-12.
6. ^'Mathematics - Manitoba Education'. Edu.gov.mb.ca. Retrieved 2017-08-12.
7. ^'Mathematica for Faculty, Staff, and Students : Information Technology - Northwestern University'. It.northwestern.edu. Retrieved 2017-08-12.
8. ^'Mathematica for Students - Columbia University Information Technology'. cuit.columbia.edu. Retrieved 2017-08-12.
9. ^'Mathematica for Higher Education: Uses for University & College Courses'. Wolfram.com. Retrieved 2017-08-12.
10. ^'MathWorks - Academia - MATLAB & Simulink'. Mathworks.com. Retrieved 2017-08-12.
11. ^ACT's CAAP Tests: Use of Calculators on the CAAP Mathematics TestArchived August 31, 2009, at the Wayback Machine
12. ^ ^abB. Buchberger; G.E. Collins; R. Loos (2013-06-29). Computer Algebra: Symbolic and Algebraic Computation. Springer Science & Business Media. ISBN978-3-7091-3406-1.
13. ^Joachim von zur Gathen; Jürgen Gerhard (2013-04-25). Modern Computer Algebra. Cambridge University Press. ISBN978-1-107-03903-2.
14. ^Keith O. Geddes; Stephen R. Czapor; George Labahn (2007-06-30). Algorithms for Computer Algebra. Springer Science & Business Media. ISBN978-0-585-33247-5.

Cas In Calculus

External links[edit]

• Curriculum and Assessment in an Age of Computer Algebra Systems - From the Education Resources Information Center Clearinghouse for Science, Mathematics, and Environmental Education, Columbus, Ohio.
• Richard J. Fateman. 'Essays in algebraic simplification.' Technical report MIT-LCS-TR-095, 1972. (Of historical interest in showing the direction of research in computer algebra. At the MIT LCS website: [1])

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• CAS MA 107: Mathematical Reasoning in the Elementary Grades: Number Systems
  Mathematical Reasoning in the Elementary Grades: Number Systems Required for undergraduates seeking licensure in elementary education, early childhood education,special education, or deaf studies. Focuses on number systems, whole number, decimal and fraction concepts and operations, and number theory. Non-Wheelock students by signed permission only. Does not satisfy CAS Divisional Studies or Math/Stat major. 4 cr. 1st sem. Effective Fall 2019, this course fulfills a single unit in each of the following BU Hub areas: Quantitative Reasoning I, Oral and/or Signed Communication, Teamwork/Collaboration.
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• CAS MA 108: Mathematical Reasoning in the Elementary Grades: Algebra, Geometry, and Statistics
  Undergraduate Prerequisites: CAS MA 107 or SED ME 503.
  Required for undergraduates seeking licensure in elementary education, special education, or deaf studies. Focuses on topics in algebra, geometry, measurement, and statistics. Non-SED students by signed permission only. Does not satisfy CAS Divisional Studies or Math/Stat major. Effective Fall 2019, this course fulfills a single unit in each of the following BU Hub areas: Quantitative Reasoning II, Oral and/or Signed Communication, Teamwork/Collaboration.
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• CAS MA 111: Mathematical Explorations
  Students explore challenging mathematics problems chosen from an array of topics that can be solved in multiple ways. Includes making, testing, and modifying conjectures; constructing proofs; posing new problems; extensive classroom interaction; reflective and other writing. Effective Fall 2019, this course fulfills a single unit in each of the following BU Hub areas: Quantitative Reasoning II, Oral and/or Signed Communication, Creativity/Innovation.
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• CAS MA 113: Elementary Statistics
  CAS MA 113 may not be taken for credit by any student who has completed any MA course numbered 300 or higher. Students may receive credit for not more than one of the following courses: CAS MA 113, MA 115, or MA 213. Basic concepts of estimation and tests of hypotheses, ideas from probability; one-, two-, and multiple-sample problems. Applications will be in social sciences and students will be able to understand the basics of using a sample to predict uncertainty. Carries MCS divisional credit in CAS. Effective Fall 2018, this course fulfills a single unit in the following BU Hub area: Quantitative Reasoning II. Effective Fall 2020, this course fulfills a single unit in each of the following BU Hub areas: Quantitative Reasoning II, Critical Thinking.
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• CAS MA 115: Statistics I
  CAS MA 115 may not be taken for credit by any student who has completed any MA course numbered 300 or higher. Students may receive credit for not more than one of the following courses: CAS MA 113, MA 115, or MA 213. Numerical and graphical summaries of univariate and bivariate data. Basic probability, random variables, binomial distribution, normal distribution. One- sample statistical inference for normal means and binomial probabilities. Primarily for students in the social sciences with limited mathematics preparation. Carries MCS divisional credit in CAS. Effective Fall 2018, this course fulfills a single unit in the following BU Hub area: Quantitative Reasoning II. Effective Fall 2020, this course fulfills a single unit in each of the following BU Hub areas: Quantitative Reasoning II, Critical Thinking.
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• CAS MA 116: Statistics II
  Undergraduate Prerequisites: CAS MA 115; or equivalent.
  CAS MA 116 may not be taken for credit by any student who has completed any MA course numbered 300 or higher. Students may receive credit for not more than one of the following courses: CAS MA 116, MA 214, or MA 614. This course introduces basic statistical modeling techniques. One- or two- sample inference for unknown means, proportions and variances, categorical data analysis, introduction to design of experiments and analysis of variance, analysis of simple and multiple linear regression models, non- parametric methods. Carries MCS divisional credit in CAS. Effective Fall 2018, this course fulfills a single unit in the following BU Hub area: Quantitative Reasoning II. Effective Fall 2020, this course fulfills a single unit in each of the following BU Hub areas: Quantitative Reasoning II, Critical Thinking.
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• CAS MA 119: Applied Mathematics for Personal Finance
  Applications of mathematics for personal financial decision-making. Systems of equations, exponential functions, logarithms, probability, descriptive statistics, and numerical simulation, for modeling saving, borrowing, inflation, purchasing power, taxation, government benefits, risk management, insurance, annuities, and investments. Carries MCS divisional credit in CAS. Effective Fall 2018, this course fulfills a single unit in each of the following BU Hub areas: Social Inquiry I, Quantitative Reasoning I. This course also fulfills the learning outcomes for Life Skills. Effective Fall 2020, this course fulfills a single unit in each of the following BU Hub areas: Quantitative Reasoning I, Social Inquiry I, Critical Thinking.
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• CAS MA 121: Calculus for the Life and Social Sciences I
  Students may receive credit for not more than one of the following courses: CAS MA 121 or CAS MA 123. Differentiation and integration of functions of one variable. Same topics as CAS MA 123, but with less emphasis on mathematical generality and more on applications. Especially suitable for students concentrating in the biological and social sciences. Carries MCS Divisional credit in CAS. Effective Fall 2018, this course fulfills a single unit in the following BU Hub area: Quantitative Reasoning II. Effective Fall 2020, this course fulfills a single unit in each of the following BU Hub areas: Quantitative Reasoning II, Social Inquiry II, Critical Thinking.
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• CAS MA 122: Calculus for the Life and Social Sciences II
  Undergraduate Prerequisites: CAS MA 121 or CAS MA 123.
  Students may receive credit for not more than one of the following courses: CAS MA 122, MA 124, or MA 129. Continuation of CAS MA 121. Review of univariate calculus, calculus of the elementary transcendental functions, elementary differential equations, elementary multivariate calculus. Applications to exponential growth, optimization, equilibrium, and dynamic modeling problems. Carries MCS divisional credit in CAS. Effective Fall 2018, this course fulfills a single unit in the following BU Hub area: Quantitative Reasoning II. Effective Fall 2020, this course fulfills a single unit in each of the following BU Hub areas: Quantitative Reasoning II, Social Inquiry II, Critical Thinking.
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• CAS MA 123: Calculus I
  Students may receive credit for not more than one of the following courses: CAS MA 121 or CAS MA 123. Limits; derivatives; differentiation of algebraic and transcendental functions. Applications to maxima, minima, and convexity of functions. The definite integral; the fundamental theorem of integral calculus. Carries MCS divisional credit in CAS. Effective Fall 2018, this course fulfills a single unit in the following BU Hub area: Quantitative Reasoning II. Effective Fall 2020, this course fulfills a single unit in each of the following BU Hub areas: Quantitative Reasoning II, Critical Thinking.
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• CAS MA 124: Calculus II
  Undergraduate Prerequisites: CAS MA 121 or CAS MA 123.
  Students may receive credit for not more than one of the following courses: CAS MA 122, MA 124, or MA 129. Logarithmic, exponential, and trigonometric functions. Sequences and series; Taylor's series with the remainder. Methods of integration. Calculus I and II together constitute an introduction to calculus of a function of a single real variable. Carries MCS divisional credit in CAS. Effective Fall 2018, this course fulfills a single unit in the following BU Hub area: Quantitative Reasoning II. Effective Fall 2020, this course fulfills a single unit in each of the following BU Hub areas: Quantitative Reasoning II, Scientific Inquiry II, Critical Thinking.
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• CAS MA 129: Honors Calculus
  Undergraduate Prerequisites: a solid background in high school calculus or consent of instructor.
  Students may receive credit for not more than one of the following courses: CAS MA 122, MA 124, or MA 129. Alternative to CAS MA 127 for advanced students with a deep interest in mathematics. Continuity, degrees of differentiability, integrability. Transcendental functions. Techniques of differentiation and integration. Sequences, series, summability, uniform convergence. Complex numbers, Taylor series. Application to mechanics, probability. Carries MCS divisional credit in CAS. Effective Fall 2020, this course fulfills a single unit in the following BU Hub area: Critical Thinking.
• CAS MA 193: Discrete Mathematics for Engineering
  Propositional logic, set theory, relations, combinatorics, graphs. Topics are treated with particular emphasis on applications to the engineering discipline. (Cannot be taken for credit in addition to CAS MA 293.) Effective Fall 2020, this course fulfills a single unit in the following BU Hub area: Critical Thinking.
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• CAS MA 213: Basic Statistics and Probability
  Undergraduate Prerequisites: good background in high school algebra.
  Students may receive credit for not more than one of the following courses: CAS MA 113, MA 115, or MA 213. Elementary treatment of probability densities, means, variances, correlation, independence, the central limit theorem, confidence intervals, and p-values. Students will be able to answer questions such as how can a pollster use a sample to predict the uncertainty of an election? Carries MCS divisional credit in CAS. Effective Fall 2018, this course fulfills a single unit in each of the following BU Hub areas: Quantitative Reasoning II, Teamwork/Collaboration. Effective Fall 2020, this course fulfills a single unit in each of the following BU Hub areas: Quantitative Reasoning II, Critical Thinking, Teamwork/Collaboration.
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• CAS MA 214: Applied Statistics
  Undergraduate Prerequisites: CAS MA 213; or consent of instructor.
  Students may receive credit for not more than one of the following courses: CAS MA 116, MA 214, or MA 614. Inference about proportions, goodness of fit, student's t-distribution, tests for normality; two-sample comparisons, regression and correlation, tests for linearity and outliers, residual analysis, contingency tables, analysis of variance. Carries MCS divisional credit in CAS. Effective Fall 2018, this course fulfills a single unit in each of the following BU Hub areas: Quantitative Reasoning II, Teamwork/Collaboration. Effective Fall 2020, this course fulfills a single unit in each of the following BU Hub areas: Quantitative Reasoning II, Critical Thinking, Teamwork/Collaboration.
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• CAS MA 225: Multivariate Calculus
  Undergraduate Prerequisites: CAS MA 124 or CAS MA 129.
  Vectors, lines, planes. Multiple integration, cylindrical and spherical coordinates. Partial derivatives, directional derivatives, scalar and vector fields, the gradient, potentials, approximation, multivariate minimization, Stokes's and related theorems. (Cannot be taken for credit in addition to CAS MA 230.) Effective Fall 2018, this course fulfills a single unit in the following BU Hub area: Quantitative Reasoning II. Effective Fall 2020, this course fulfills a single unit in each of the following BU Hub areas: Quantitative Reasoning II, Critical Thinking.
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• CAS MA 226: Differential Equations
  Undergraduate Prerequisites: CAS MA 225 or CAS MA 230.
  First-order linear and separable equations. Second-order equations and first-order systems. Linear equations and linearization. Numerical and qualitative analysis. Laplace transforms. Applications and modeling of real phenomena throughout. (Cannot be taken for credit in addition to CAS MA 231.) Effective Fall 2020, this course fulfills a single unit in the following BU Hub area: Critical Thinking.
• CAS MA 230: Honors-Level Vector Calculus
  Undergraduate Prerequisites: CAS MA 124 or CAS MA 129.
  Linear algebra: linear transformations and matrices, inner products, cross products. Differentiation: tangent spaces, directional derivatives, gradients, vector fields and flow, divergence, curl. Integration: multiple integrals, line integrals, Green's theorem, surface integrals, Stokes's theorem, the divergence theorem. (Cannot be taken for credit in addition to CAS MA 225.) Effective Fall 2020, this course fulfills a single unit in the following BU Hub area: Critical Thinking.
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• CAS MA 231: Honors-Level Differential Equations
  Undergraduate Prerequisites: CAS MA 225 or CAS MA 230.
  Ordinary differential equations with applications to mechanics, biology, and physics. First- and second-order equations, linear and nonlinear systems, Sturm-Louisville problems, vibrating strings, special functions, power series solutions, regular singular points, Laplace transforms, calculus of variations, isoperimetric problems, and Hamilton's principle. (Cannot be taken for credit in addition to CAS MA 226.) Effective Fall 2020, this course fulfills a single unit in the following BU Hub area: Critical Thinking.
• CAS MA 242: Linear Algebra
  Undergraduate Corequisites: CAS MA 122 or CAS MA 124 or CAS MA 129.
  Cannot be taken for credit in addition to CAS MA 442 or ENG EK 103. Matrix algebra, solution of linear systems, determinants, Gaussian elimination, fundamental theory, row-echelon form. Vector spaces, bases, norms. Computer methods. Eigenvalues and eigenvectors, canonical decomposition. Applications. Effective Fall 2019, this course fulfills a single unit in the following BU Hub area: Quantitative Reasoning II. Effective Fall 2020, this course fulfills a single unit in each of the following BU Hub areas: Quantitative Reasoning II, Critical Thinking.
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What Is A Cas In Calculus

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