Calculus 4 - Applications in Economics and Management (微積分4-在經濟商管的應用) The goal of this course is to employ tools from Calculus and develop mathematical theory to tackle important Economics problems, specifically constrained optimization problems. We shall begin with a crash course in Linear Algebra. We will define the rank, determinant, eigenvalues, eigenvectors, and definiteness of symmetric matrices. These concepts will be used to solve optimization problems with equality or inequality constraints (or a mix of both). Then we proceed to discuss the Kuhn-Tucker formulation, Economic interpretations of Lagrange multipliers as shadow prices, the Envelope Theorem and the second order test under constraints which determine the nature of a critical point.   Definitions are discussed and the most important theorems are derived in the lectures with a view to help students to develop their abilities in logical deduction and analysis. Practical applications of optimization problems in Economics are illustrated in order to promote a more organic interaction between the theory of Calculus and students' own fields of study. This course also provides discussion sessions in which students are able to make their skills in handling calculations in Calculus more proficient under the guidance of our teaching assistants. 這是一門半學期的課，主題是限制條件下的最佳化問題，目的是裝備學生用微積分工具探討重要的經濟學議題。 課程前三周簡介處理最佳化問題所需的線性代數工具；內容包含矩陣的秩、行列式、特徵值與特徵向量，對稱矩陣的正負定性。 最佳化問題討論在等式限制條件，不等式限制條件，與混和限制條件下求目標函數極值的解法。同時我們還介紹Kuhn-Tucker陳述式，說明 Lagrange 乘子的意義(影子價格)，推導包絡定理，並講解限制條件下的二階微分測試，以判斷臨界點是局部極大或極小值。 課堂上將講解定義並推導重要定理，以培養學生邏輯推理與分析能力；同時會示範最佳化問題在經濟學的應用，幫助學生將微積分與專業科目結合。本課程還設有習題課，學生將在助教的帶領下熟練微積分的計算。

Upon successful completion of this module, students should be able to: 1. define fundamental terminology in linear algebra, such as "linear independence," "bases," and "mutual orthogonality." 2. define the rank of a matrix and calculate it using row echelon form. 3. compute the eigenvalues of symmetric matrices and perform diagonalization of such matrices. 4. determine the definiteness of a symmetric matrix using either its definition or Sylvester’s criterion and apply it to the general second derivative test. 5. state the first-order conditions in various optimization problems. 6. formulate the Non-Degenerate Constraint Qualification (NDCQ) in different optimization problems and use it to verify whether constraints are non-degenerate. 7. state the first-order conditions and formulate NDCQ in the Kuhn-Tucker formulation. 8. state the Envelope Theorem and apply it to estimate changes in extrema when objective functions or constraints are altered. 9. state the second-order conditions for constrained optimization problems. 完成此模組後，同學應能夠 : 1. 定義線性代數中的基本術語，例如「線性獨立」、「基底」和「相互正交性」。 2. 定義矩陣的秩(rank)，並使用row echelon form 計算其值。 3. 計算對稱矩陣的特徵值 (eigenvalues) 並對其進行對角化(diagonalization)處理。 4. 通過定義或 Sylvester’s criterion確定對稱矩陣的正定性 (definiteness)，並將其應用於一般的二階導數測試 (general second derivative test)。 5. 陳述各類最佳化問題中的一階條件 (First order conditions)。 6. 在不同的最佳化問題中構建非退化限制資格條件（NDCQ），並用其判定限制式是否非退化。 7. 在 Kuhn-Tucker 的框架下陳述一階條件並構建 NDCQ。 8. 陳述包絡定理(Envelope Theorem)，並用其估算當目標函數或限制式改變時極值的變化。 9. 陳述最佳化問題的二階條件(Second order conditions)。

Students participating in the course should have taken Calculus 1, 2, and 3. They are expected to attend and participate actively in lectures as well as discussion sessions. 需有「微積分1」「微積分2」「微積分3」的預備知識。 認真參與課堂和習題課的活動與討論。

To ensure maximum engagement and the highest learning outcomes, students are advised to allocate a minimum of 8 hours per week to independent study after class to the following tasks :  1. Assimilate and organize the course materials, including the definitions, theorems, and formulae introduced during lectures. 2. Review and reproduce the examples and problem-solving methodologies demonstrated by the instructor or teaching assistants. 3. Ensure the timely and thorough completion of all assigned work, including WeBWorK exercises, written homework, and Worksheets. 4. Reflect critically on any challenging areas or ambiguities encountered. Proactively pinpoint concepts needing clarification and immediately seek guidance from the instructor or teaching assistants. Students are strongly encouraged to utilize all designated office hours and support sessions. 為了達到最好的學習效果，鼓勵同學每周花 8 小時課後時間，依序完成以下任務 Step 1. 理解、整理並背下課堂中介紹的定義、定理與公式 Step 2. 複習課堂上的重要例題 Step 3. 寫 WeBWorK作業、紙本作業、學習單 Step 4. 回顧寫作業中遇到的瓶頸，如果有不完全理解的內容，盡快尋求助教和老師的協助。 強烈鼓勵同學參加 office hours 和助教習題課。

K.-W. Tsoi, Y.-J. Tsai, Calculus: Applications in Constrained Optimization ISBN 978-626-7768-11-2

1. James Stewart, Daniel Clegg, Saleem Watson, Calculus Early Transcendentals, 9th edition. 2. Carl P. Simon and Lawrence Blume, Mathematics for Economics. 3. Michael W. Klein, Mathematical Methods for Economics. Other useful websites 其他相關資訊  微積分統一教學網站: http://www.math.ntu.edu.tw/~calc/Default.html 台大微積分考古題:  http://www.math.ntu.edu.tw/~calc/cl_n_34455.html 數學知識網站: http://episte.math.ntu.edu.tw/cgi/mathfield.pl?fld=cal  免費線上數學繪圖軟體Desmos Calculator: https://www.desmos.com/calculator  免費知識型計算引擎: https://www.wolframalpha.com