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課程名稱 |
微積分2
CALCULUS (2) |
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開課學期 |
114-2 |
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授課對象 |
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授課教師 |
陳子安 |
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課號 |
MATH4007 |
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課程識別碼 |
201E49820 |
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班次 |
02 |
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學分 |
2.0 |
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全/半年 |
半年 |
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必/選修 |
必修 |
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上課時間 |
第9,10,11,12,13,14,15,16 週
星期一9,10(16:30~18:20)星期四9,10(16:30~18:20) |
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上課地點 |
新505新505 |
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備註 |
本課程以英語授課。密集課程。英文授課
總人數上限：80人 |
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課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
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課程大綱
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為確保您我的權利,請尊重智慧財產權及不得非法影印
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課程概述 |
Calculus 2 (微積分2)
Integration on functions of a single variable together with its profound applications in various subject areas are introduced in this half-semester course. Especially, this course includes the definition of integrals, the Fundamental Theorem of Calculus, techniques of integration, applications of integrals in probability, solving elementary differential equations and more. In addition to these, series and Taylor expansions are briefly introduced in the end of this course to explain how complicated functions are approximated by polynomials.
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 Calculus in various fields 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 improve their skills in handling calculations and explore applications of Calculus under the guidance of teaching assistants.
這是一門半學期的課程，主要介紹單變數函數的積分運算，和積分在各領域豐富的應用。內容涵蓋積分的定義，微積分基本定理，積分技巧，積分在機率的應用，和初步的微分方程等。最後，課程將簡介級數與泰勒展式，解釋如何以多項式逼近複雜的函數。
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；同時會示範微積分在各領域的應用，幫助學生將微積分與其他專業科目結合。本課程還設有習題課，學生將在助教的帶領下熟練計算並探索微積分的應用。
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課程目標 |
Upon completing this course, students are expected to :
1. Be able to define the integral of a continuous function over a closed and bounded interval, and express it using the corresponding Riemann sum.
2. Apply the Fundamental Theorem of Calculus to compute definite integrals.
3. Use techniques such as substitution and integration by parts to compute definite and indefinite integrals.
4. Define improper integrals, determine their convergence, and compute their values.
5. Solve separable and first-order linear differential equations.
6. Apply the basic properties of power series and their calculus.
7. State and apply Taylor’s Theorem to address problems involving smooth functions.
學生修習本課程後，應具備下列能力：
1. 能定義連續函數於封閉有界區間上的積分，並寫出相對的黎曼和 (Riemann sum)。
2. 應用微積分基本定理計算定積分。
3. 能運用變數代換、分部積分等技巧求出函數之(不)定積分。
4. 能定義瑕積分、判斷瑕積分是否收斂，並計算瑕積分。
5. 能解可分離變數型及一階線性微分方程式。
6. 運用冪級數之基本性質及其微積分
7. 陳述並應用泰勒定理以解決關於平滑函數之問題
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課程要求 |
Before taking this course, students should be skilled in high school mathematics and finish the online Precalculus Self Diagnostic Test https://cool.ntu.edu.tw/courses/50879 which is designed for NTU freshmen.
Students are expected to attend and participate actively in lectures as well as discussion sessions.
學生應熟練高中數學，並完成為台大新生預備的線上「微積分學前自我檢測」https://cool.ntu.edu.tw/courses/50879。
學生應出席並積極參與課堂與習題課的討論。
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預期每週課前或/與課後學習時數 |
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。 |
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Office Hours |
每週五 15:00~17:00
每週三 11:00~13:00 備註： 1. 陳子安教授 Prof. Tsz On Mario Chan，地點：天數459
Email: mariochan@ntu.edu.tw
2. 蕭楷熹 SEOW, KAI，地點：天數103
Email: b11504085@g.ntu.edu.tw |
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指定閱讀 |
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參考書目 |
Textbook: James Stewart, Daniel Clegg, and Saleem Watson, Calculus Early Transcendentals, 9th edition, Metric Version.
ISBN: 978-626-7533-06-2
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
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評量方式
(僅供參考) |
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No. |
項目 |
百分比 |
說明 |
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1. |
Exam |
50% |
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2. |
Quizzes |
30% |
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3. |
WeBWorK |
10% |
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4. |
Homework and others |
10% |
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- 本校尚無訂定 A+ 比例上限。
- 本校採用等第制評定成績，學生成績評量辦法中的百分制分數區間與單科成績對照表僅供參考，授課教師可依等第定義調整分數區間。詳見學習評量專區 (連結)。
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週次 |
日期 |
單元主題 |
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第9週 |
4/20, 4/23 |
4.9 Antiderivatives
5.1 The Area and Distance Problems
5.2 The Definite Integral
5.3 The Fundamental Theorem of Calculus
5.4 Indefinite Integrals and the Net Change Theorem |
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第10週 |
4/27, 4/30 |
5.5 The Substitution Rule
7.1 Integration by Parts
7.2 Trigonometric Integrals |
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第11週 |
5/4, 5/7 |
7.3 Trigonometric Substitution
7.4 Integration of Rational Functions by Partial Fractions
7.8 Improper Integrals
[5/7(Thu) 16:30-17:20 Quiz 1 (5.1 - 5.5, 7.1)] |
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第12週 |
5/11, 5/14 |
6.1 Areas Between Curves
6.5 Average Value of a Function
8.5 Probability* |
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第13週 |
5/18, 5/21 |
9.1 Modeling with Differential Equations
9.3 Separable Equations
9.4 Models for Population Growths
9.5 Linear Equations
[5/21(Thu) 16:30-17:20 Quiz 2 (7.2 - 7.4, 7.8, 6.1, 6.5)] |
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第14週 |
5/25, 5/28 |
11.8 Power series (No convergence issues)
11.9 Representations of Functions as Power Series |
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第15週 |
6/1, 6/4 |
11.10 Taylor and Maclaurin Series
[6/4(Thu) 16:30-17:20 Quiz 3 (9.1, 9.3-9.5, 11.8-11.9)] |
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第16週 |
6/8, 6/11 |
[6/11(Thu) 16:30-18:20 Exam] |
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