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In the first two years of university, students typically undergo a diverse range of courses that lay the foundation for their academic journey. The curriculum spans across various disciplines, aiming to provide a well-rounded education. Below, I'll outline a hypothetical example of courses a student might take during their freshman, sophomore, and junior years, including both general education requirements and major-specific courses. Please note that the actual courses may vary based on the university and the chosen major.**Freshman Year:**1. **Introduction to College Writing:**- This course focuses on developing essential writing skills, including critical thinking, research, and proper citation. Students learn how to construct well-organized essays and research papers.2. **Calculus I:**- An introduction to differential calculus, covering topics such as limits, derivatives, and basic integration. This course is fundamental for students majoring in fields such as mathematics, physics, engineering, or economics.3. **General Psychology:**- An overview of fundamental concepts in psychology, including human behavior, cognition, and various psychological theories. This course provides a foundational understanding of the field.4. **Introduction to Computer Science:**- An introduction to programming languages and problem-solving techniques. Students may learn a programming language like Python and gain insights into algorithmic thinking.5. **World History:**- A broad survey of world history, covering key events, civilizations, and developments. This course aims to provide students with a global perspective on historical events.6. **Introduction to Sociology:**-An exploration of sociological concepts and theories, examining the structure of society, social institutions, and the impact of culture on individuals and groups.**Sophomore Year:**1. **Composition and Rhetoric:**- A continuation of the writing skills developed in the freshman year, with an emphasis on rhetorical strategies and advanced composition techniques.2. **Calculus II:**-Building on the concepts from Calculus I, this course delves deeper into integration techniques, sequences, and series. It is a crucial course for students pursuing degrees inmathematics, engineering, or related fields.3. **Introduction to Statistics:**- An introduction to statistical methods, probability, and data analysis. This course is valuable for students in various majors, providing essential skills for interpreting and presenting data.4. **Principles of Microeconomics:**-An introduction to microeconomic theory, covering topics such as supply and demand, market structures, and consumer behavior. It is foundational for students studying economics or business.5. **General Biology:**-An overview of fundamental concepts in biology, including cellular biology, genetics, and evolution. This course serves as a prerequisite for more advanced courses in the biological sciences.6. **Introduction to Environmental Science:**-An interdisciplinary exploration of environmental issues, including ecology, conservation, and sustainability. This course provides an understanding of the interconnections between human activities and the environment.**Junior Year:**1. **Advanced Writing Seminar:**-A more specialized writing course focusing on academic and professional writing skills. Students may work on longer research projects and develop their argumentation and persuasion skills.2. **Linear Algebra:**- A foundational course in linear algebra, covering topics such as vector spaces, matrices, and linear transformations. It is essential for students in mathematics, physics, computer science, and engineering.3. **Principles of Macroeconomics:**-An exploration of macroeconomic theory, including topics such as economic growth, inflation, and monetary policy. This course complements the microeconomics principles learned in the sophomore year.4. **Introduction to Political Science:**- An overview of political systems, political theories, and key concepts in political science. This course provides insights into the structure and functioning of governments and political institutions.5. **Organic Chemistry I:**- A foundational course in organic chemistry, covering the structure, properties, and reactions of organic compounds. This course is critical for students majoring in chemistry, biochemistry, or pre-medical studies.6. **Introduction to Philosophy:**-A survey of philosophical ideas and theories, exploring fundamental questions about existence, knowledge, ethics, and the nature of reality.7. **Elective Courses:**-Junior year often allows students to choose elective courses based on their major or personal interests. These courses could include more specialized topics within the chosen field of study.This hypothetical breakdown provides a general overview of the courses a student might take during their first three years of university, covering general education requirements, major-specific courses, and elective options. It's important to note that the actual courses can vary widely based on the university, the chosen major, and individual academic preferences. Additionally, some students may choose to pursue internships, research opportunities, or study abroad experiences during their university journey, further enhancing their educational experience.。
二、工業管理系、工業工程與管理研究所(碩士班、博士班)、全球運籌管理研究所(碩士班)、健康產業管理研究所(碩士班)工業管理系、工業工程與管理研究所(碩士班、博士班)一、簡介(一) 成立沿革民國80年成立四年制工業管理系民國81年成立二年制工業管理系民國81年成立工業工程與管理研究所碩士班民國89年成立工業工程與管理碩士在職專班民國92年成立工業工程與管理博士班民國94年成立全球運籌管理研究所碩士班民國95年成立全球運籌管理研究所碩士在職專班民國96年核定停招二年制工業管理系民國96年成立四年制雙班民國96年成立健康產業管理研究所碩士在職專班(二) 教學特色為培養紮實的先進產業管理科技人才：以（1）作業研究與資訊系統學程（2）生產製造學程（3）統計品管學程（4）人因工程學程作為基礎學習平台，建構出全球運籌與供應鏈管理及健康產業管理兩大特色領域。
本所之教學資源完整，教師產業經驗豐富，本土與跨國輔導深具實績，重點課程並以英文施教。
(三) 研究發展及特色培育具國際觀與系統觀且理論與實務兼具之工業工程與管理及全球運籌管理人才。
培育具整合人員、物料、設備、資訊方法與科技於產業問題之分析與解決，並能為產業之永續經營做最大貢獻的人才。
針對產業界的實際需求，以及各級學制(四技、二技、一般碩士、在職碩士、博士) 之不同發展重點，加強工業工程與管理的專業課程，以培養配合國家經濟發展、提高產業生產力，具有專業知識之工工管與全球運籌管理人才，並具整合跨領域與跨地域之管理系統建立與改善，而能提高國家競爭力的人才。
簡言之，本系目標在培育產業界的良醫與良相之現代化工業管理人才。
全球運籌與供應鏈管理領域為配合國家發展「全球運籌發展計劃」之政策，本領域師資、設備、課程先進，歷年來除耗資千萬於參與建立製商整合基礎建設外，更於94學年度成立「全球運籌管理研究所」，提供全球運籌資訊系統、全球供應鏈管理、供應鏈模式分析、電腦整合生產與物流系統、整體後勤支援系統分析、企業資源規劃、全球運輸規劃等進階課程，為國家培養高階之全球運籌管理人才。
一类与余弦函数有关的解析函数的三阶Hankel和Toeplitz行列式引言在数学中，解析函数是研究的一个重要领域，它在分析学、复变函数论以及其他数学领域中起着重要的作用。
解析函数可以用级数展开，并在一定的区域内具有无限可微的性质。
在解析函数的研究中，Hankel和Toeplitz行列式是两个重要的概念，它们在研究分析函数的性质和应用中起着重要的作用。
本文将就一类与余弦函数有关的解析函数的三阶Hankel和Toeplitz行列式进行探讨，并给出相应的定理和证明。
一类与余弦函数有关的解析函数我们先来定义一类与余弦函数有关的解析函数。
设f(z)是定义在单位圆内的解析函数，且满足以下形式的级数展开式：f(z)=Σ(c_nz^n+1+ c_nz^n) (1)其中c_n是复数系数，n=0,1,2,……。
根据级数展开式(1)，我们可以得到函数f(z)在单位圆内的解析表达式。
这类函数与余弦函数有着密切的联系，下面我们将讨论这类函数的三阶Hankel和Toeplitz行列式。
三阶Hankel行列式的定义和性质我们来定义三阶Hankel行列式。
设函数f(z)满足式(1)的级数展开式，即f(z)是定义在单位圆内的解析函数。
那么，三阶Hankel行列式H_3(f)定义为：H_3(f)=det[h_ij]=det[f^(i+j-2)]_(i,j=1,3) (2)其中h_ij=f^(i+j-2)，i,j=1,2,3，f^(k)表示f(z)的k阶导数。
根据Hankel行列式的定义，我们可以得到H_3(f)的表达式：H_3(f)=det[f(0) f'(0) f''(0)f'(0) f''(0) f'''(0)f''(0) f'''(0) f''''(0)]= f(0)f'''(0)-f''(0)^2 (3)三阶Hankel行列式H_3(f)的性质如下：1. 对于三阶Hankel行列式H_3(f)，如果f(z)是单位圆内的解析函数，则H_3(f)的值一定是实数。
国外微积分经典著作国外微积分经典著作是学习微积分的重要参考书籍，下面列举了10本经典著作：1.《微积分学原理与应用》（Principles of Mathematical Analysis）- Walter Rudin这本书是微积分教材中的经典之作，被广泛用于大学微积分教学。
书中系统介绍了微积分的基本原理和应用，内容严谨而深入。
2.《微积分》（Calculus）- Michael SpivakSpivak的《微积分》是一本经典的数学教材，对微积分的理论进行了深入的探讨。
书中不仅介绍了微积分的基本概念和技巧，还着重讲解了微积分的证明和推导过程。
3.《微积分学》（Calculus）- James StewartStewart的《微积分学》是一本广受欢迎的微积分教材，适合初学者阅读。
书中以清晰易懂的语言介绍了微积分的概念和方法，并提供了大量的练习题和解答。
4.《微积分与其应用》（Calculus with Applications）- Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey这本书主要面向应用型的微积分学习者，介绍了微积分的基本概念和应用。
书中以实际问题为例，帮助读者理解微积分在科学和工程领域的应用。
5.《微积分学教程》（A Course in Calculus and Real Analysis）- Sudhir R. Ghorpade, Balmohan V. Limaye这本书是一本适合高年级本科生和研究生的微积分教材。
书中深入讲解了微积分的理论和技巧，并提供了大量的例题和习题。
6.《微积分引论》（Introduction to Calculus）- John E. Marsden, Anthony J. Tromba这本书是一本经典的微积分教材，对微积分的基本概念和方法进行了全面介绍。
书中以直观的图形和实例帮助读者理解微积分的概念和原理。
二阶连续可微函数英译-回复Title: Second-Order Continuous Differentiable FunctionsIntroduction:In mathematics, functions are one of the fundamental concepts. They describe the relationship between two sets of numbers, known as the domain and the range. Functions come in various forms, and their properties can be studied to gain a deeper understanding of their behaviors. One important type of function is the second-order continuous differentiable function. This article will delve into the topic, exploring its definition, properties, and applications.I. Definition and Properties:A second-order continuous differentiable function, also known as a twice-differentiable function, is a function that has a continuous first derivative and a continuous second derivative. This implies that the function can be differentiated twice, and these derivatives exist and are continuous over the entire domain.The continuity of the first derivative ensures that the function does not have any abrupt changes or jumps, while the continuity of thesecond derivative indicates how smoothly the function curves and changes direction. These properties make second-order continuous differentiable functions particularly useful in various areas of mathematics, including calculus, optimization, and physics.II. Examples:To understand the concept better, let's consider a few examples of second-order continuous differentiable functions:1. Quadratic Functions:Quadratic functions, such as f(x) = ax^2 + bx + c, where a, b, and c are constants, are second-order continuous differentiable functions. These functions form a family of parabolas that have a single global minimum or maximum point, depending on the leading coefficient.2. Trigonometric Functions:Trigonometric functions, such as sine and cosine, are alsosecond-order continuous differentiable functions. These functions exhibit periodic behavior and are widely used in physics, engineering, and signal processing.3. Exponential and Logarithmic Functions:Exponential and logarithmic functions, such as f(x) = e^x and f(x) = ln(x), respectively, are also examples of second-order continuous differentiable functions. These functions have many applications in growth and decay processes, finance, and probability theory.III. Applications:Second-order continuous differentiable functions play a crucial role in various fields. Here are a few applications where their properties are extensively utilized:1. Optimization:Optimization problems involve finding the maximum or minimum of a given function. Second-order continuous differentiable functions are particularly useful here because of their smoothness and well-defined curvature. Techniques like Newton's method and gradient descent use these functions to efficiently find optimal solutions.2. Physics:Many physical phenomena can be accurately described using second-order continuous differentiable functions. These functions help determine rates of change, acceleration, and other importantquantities in mechanics, electromagnetism, and thermodynamics.3. Regression Analysis:Regression analysis is a statistical method used to model the relationship between variables. Second-order continuous differentiable functions are often employed in regression models because they can accurately capture intricate nonlinear relationships between variables.4. Control Systems:Control systems are widely used in engineering to regulate and stabilize various processes. Second-order continuous differentiable functions play a crucial role in designing controllers and modeling system dynamics, ensuring smooth and precise control.Conclusion:Second-order continuous differentiable functions are a fundamental class of mathematical functions that possess continuous first and second derivatives. These functions provide insights into the smoothness, curvature, and behavior of mathematical models. Their properties make them invaluable invarious fields, including optimization, physics, statistics, and engineering. Understanding and utilizing these functions contribute to the advancement of mathematics and its applications in the real world.。
AP® Calculus BCSyllabus 2Teaching StrategiesClassroom DynamicsBecause of my strong belief that students learn best by discovering new concepts for themselves, I attempt to promote an atmosphere of questioning, exploration, and excitement in the classroom. Rarely does a lesson proceed straight down a prepared path; we take frequent side trips. I encourage students to ask “what if” questions, for which I often do not have ready answers. The objective is to engage students in enjoyable activities that promote interest in mathematics. I try to get them to ask the questions. I rarely, if ever, tell students that some new concept or type of problem is easy. I’d rather they feel a sense of accomplishment from being able to tackle hard concepts and problems than feel frustration at being stumped by even the easy ones.One consequence of calculus reform and of the accessibility of technology is that questions are becoming much more interesting and diverse. The more experience students have with solving interesting and difficult problems, the better, both for the AP® Examinations and in the long run.AssessmentThe issue of assessment in a technology-intensive classroom is one that teachers must resolve intelligently. My own approach is to allow the use of graphing calcu-lators on nearly all unit tests. Before the AP Exam, I make sure the students are proficient at using technology to perform the four basic activities required on it: graphing a function in an arbitrary window, finding roots and points of intersec-tion, finding numerical derivatives, and approximating definite integrals. Students are often directed to use the calculator to investigate concepts such as limits by using the trace and table operations to make conjectures about the answers. They are also frequently asked to use the calculator to approximate answers found alge-braically to see if they are reasonable. [C5]Laboratory ActivitiesFor each major content area, students are introduced to new topics through group work using discovery-learning activities.Calculus JournalStudents are also required to keep a Calculus Journal. Questions are given in class to which students respond in their journals. For instance, one question this year was, “What is the most important concept we’ve learned in calculus so far? Justify your answer.” Another was, “Explain, in your own words, what the first Fundamental Theorem of Calculus says.” Students are encouraged to write fre-quently in their journals. [C4]C4—The course teaches students how to commu-nicate mathematics and explain solutions to prob-lems both verbally and in written sentences.C5—The course teaches students how to use graphing calculatorsto help solve problems, experiment, interpret results, and support con-clusionsMajor ThemesFor each new major idea, I attempt to examine the concept graphically, numeri-cally, and symbolically, and I illustrate connections among the three. I am also attentive to students’ verbal expression of concepts, and make repeated and deter-mined efforts to encourage them to be precise in their use of language. We use graphing calculators throughout the course. [C3] [C4] [C5]AP Calculus BC Course Outline [C2]Preliminary Students who begin Calculus BC have already had experience using graphing calculators. Nonetheless, time is spent at the beginning of the course addressing issues of the limitations of technology, including round-off error, hidden behavior examples, and other issues. Unit I: Functions (12 days)Lab: Exploring function transformations f (x + h ), f (x ) + k, a*f (x ), f (b*x ), f (|x|),|f (x )|Multiple representations of functions • Absolute value and interval notation • Domain and range • Categories of functions, including linear, polynomial, rational, power, • exponential, logarithmic, and trigonometric Even and odd functions• Function arithmetic and composition• Inverse functions• Parametric relations• Unit II. Limits (11 days)Lab: Computing limits graphically and numericallyInformal concept of limit• Language of limits, including notation and one-sided limits• Calculating limits using algebra• Properties of limits• Limits at infinity and asymptotes• C2—The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; Integrals; and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description . C3—The course provides students with the oppor-tunity to work with func-tions represented in a variety of ways—graphi-cally, numerically, analyti-cally, and verbally—and emphasizes the connec-tions among these repre-sentations.C4—The course teachesstudents how to commu-nicate mathematics and explain solutions to prob-lems both verbally and in written sentences. C5—The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support con-clusions.Estimating limits numerically and graphically• Comparing growths of logarithmic, polynomial, and exponential functions • Idea of continuity and the limit definition• Types of discontinuities• The Intermediate Value and Extreme Value Theorems • Local and global behavior• Rate of change concept• Tangent lines, including using the tangent line to approximate a function • Formal definitions of limit and continuity• Unit III. The Derivative (25 days)Lab: The derivative and differentiabilityLinear functions and local linearity• Slope–intercept, point slope, and Taylor forms of linear equations • Difference quotient definition of derivative; computing the derivative at a • point using the definitionEstimating the derivative from tables and graphs• Relationship between differentiability and continuity• Symmetric difference quotient definition• The derivative as a function; computing derivative functions from the • definitionDerivative as a rate of change• Rules for computing derivatives; formulas for all relevant functions, • including implicitly defined functionsUnit IV. Applications of Derivatives (17 days) Lab: An Investigation into the Accuracy of the Tangent Line Approximation•Finding extrema•Increasing and decreasing behavior•The Mean Value Theorem•Critical values and local extrema•The first and second derivative tests•Concavity and points of inflection• ′, and ƒ′′Comparing graphs of ƒ, ƒ•Modeling and optimization•Particle motion; position, velocity, and acceleration functions•Linearization and the Taylor form of the equation of a line•Newton’s method•Related rates problemsReview for Semester Exam (5 days)Unit V. The Definite Integral (22 days)“Car” Lab: Speedometer readings and distance traveledLab:Accumulation Functions (from College Board Professional Development Workshop) Materials Special Focus: The Fundamental TheoremLab: Riemann Sums (from Texas Instruments’ Calculus Activities)Lab: The Fundamental Theorem (from College Board Professional Development Workshop Materials Special Focus: The Fundamental Theorem•Area under a curve and distance traveled•Summation notation and partitions•Riemann sum•Definition of the definite integral as the limit of a Riemann sum•Linearity properties of definite integralsAverage value of a function• Definition of antiderivative• The idea of area function; discovering the fundamental theorem • The First and Second Fundamental Theorems of Calculus and their uses • The Mean Value Theorem for Integrals, and using the Fundamental • Theorem to connect the two Mean Value Theorems Numerical integration techniques: left endpoint, right endpoint, midpoint, • trapezoid, and Simpson’s rules Unit VI. Differential Equations and Mathematical Modeling (24 days)Lab: Using Slope Fields (from Texas Instruments Calculus Activities )Initial value problems• Translating verbal descriptions into differential equations • Antiderivatives and slope fields• Linearity properties of definite integrals• Techniques of antidifferentiation: substitution and integration by parts • Solving separable differential equations analytically • The domain of the solution of a differential equation • Exponential growth problems• The logistic model and antiderivatives by partial fractions • Solving initial value problems by Euler’s method• Solving initial value problems visually using slope fields • Solving initial value problems using the Fundamental Theorem • Unit VII Applications of Definite Integrals (23 days)Integral of a rate of change gives net change• Measuring area under and between functions; Cavalieri’s principle•Measuring volume of solids of known cross-sectional area and solids of • revolutionApplications to particle motion—net and total distance traveled • Arc length of function graphs• Review for Semester Exam (5 Days)Unit VIII. Parametric, Vector, and Polar Functions (17 days)Length of parametrically defined curves• Vectors and vector-valued functions• Calculus of vector functions• Calculus of polar functions, including slope, length, and area • Unit IX. Sequences (12 days)Idea and notation for sequences; arithmetic, harmonic, alternating • harmonic, and geometric sequencesDefinitions of convergence and divergence• Bounded, monotonic, oscillating sequences• Limit properties of sequences• L’H • Ôpital’s Rule and indeterminate formsRelative rates of growth of functions• Improper integrals and the comparison test• Unit X. Series (24 days)Lab: An Investigation into the Accuracy of Polynomial Approximations to Transcendental Functions Definition and notation of series; sequence of partial sums; telescoping, • geometric, harmonic, alternating harmonic seriesRepeating decimals expressed as infinite geometric series; using • substitution and antidifferentiating to calculate series for ln(1+x) and arctan(x) from geometric seriesTerms of series as areas of rectangles; relationship to the integral test•Power series; interval and radius of convergence defined • Taylor series• Maclaurin series for • e x , sin x , cos x , and Functions defined by series• Taylor polynomials• Taylor’s theorem with Lagrange form of the remainder • Alternating series error bound• Linearity properties of series• Radius of convergence: • n th term test; direct comparison test; absolute and conditional convergence; ratio test Interval of convergence and testing endpoints; integral test; • p -series; limit comparison test; alternating series test References and MaterialsMajor textbookFinney, Ross L., Franklin Demana, Bert Waits, and Daniel Kennedy. Calculus:Graphical, Numerical, Algebraic . Reading, Mass.: Addison-Wesley, 2007. 11-x。
