微積分一 1 課程簡介 (Introduction)

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1 微積分一 1 課程簡介 (Introduction)
陳光琦助理教授 (Kuang-Chi Chen) Office: H603-2

2 Textbooks Textbook : Thomas’ Calculus, 11th ed., Weir, Hass & Giordano, Person, 2005. Reference : Calculus, 8th ed., Larson, Hostetler & Edwards, Houghton Mifflin, 2006.

3 Schedule-1 W1. 09/18 Functions and Their Graphs, Limits and Their Properties W2. 09/25 中秋節放假 W3. 10/02 Continuity and One-Sided Limits, Infinite Limits W4. 10/09 Differentiation (1) – Differentiation Rules, The Chain Rule and Parametric Equations & Lab 1 W5. 10/16 Quiz 1 & Differentiation (2) – Implicit Differentiation, Linearization and Differentials W6. 10/23 Applications of Derivatives (1) – Extreme Values of Functions, The Mean Value Theorem and First Derivative Test W7. 10/30 Applications of Derivatives (2) – Applied Optimization Problems and L’Hospital’s Rule & Lab 2 W8. 11/06 Logarithmic and Exponential Functions – Natural Logarithms and Inverse Functions (more… Other Transcendental Functions) W9. 11/13 -- Midterm --

4 Schedule-2 W10.11/20 Integration (1) – The Definite Integral and The Fundamental Theorem of Calculus W11.11/27 Integration (2) – Indefinite Integrals and the Substitution Rule W12.12/04 Applications of Definite Integrals – Volumes by Slicing, Rotation and Cylindrical Shells & Lab 3 W13.12/11 Integration Techniques (1) – Basic Integration Formulas and Integration by Parts W14.12/18 Quiz 2 & Integration Techniques (2) – Trigonometric Integrals and Improper Integrals W15.12/25 Application of Definite Integrals – Length of Plane Curves Conics and Polar Coordinates & Lab 4 W16.01/01 元旦放假 W17.01/08 Vector-Valued Functions – Vector Functions and Arc Length (more… Functions of Several Variables) W18.01/15 -- Final --

5 Evaluation Assignment: Homework + Attendance 20%, Lab work (Bonus);
Exam: Quiz#1 10%, Quiz#2 10% Midterm 30% , Final 30% .

6 輕鬆拿高分的 n 大通則 用心聽講 - 鋼鐵般的意志 - 坐在前三排 … 注意力集中 做習題 - 真正的學習在課堂之外
- 看清楚題目、找出類似題型、依樣畫葫蘆 (資料來源 :微積分之屠龍寶刀 )

7 輕鬆拿高分的 n 大通則 (cont’d) 找人幫忙 - 才子型的同學 … 把自尊擺一邊，不恥下問， 切記，問過程，不是答案/結果 !!
世界上最好的學習就是，試著講解給別人聽。 - 助教 … 事先準備好要問的觀念與習題 - 老師 … in class / office hour / 上網 - 網路上有許多微積分網站及討論區 考古題 or 其他本教科書

8 輕鬆拿高分的 n 大通則 (cont’d) 熟悉各例題
- 重點就在例題中，教師會把同一套考試題目拿來重複適用 (好比諧星總是重複同樣的笑話 …) 用功 - 有效的學習，學習的很快樂 額外做習題 – huge reward 問問題 in class 避免陰險行為 - no cheat Cheat - high risk, no gain, little reward

9 先修課程 代數 (algebra) 1. 因式分解 : x2 – 6x + 8 … (x - 2)(x - 4)
3. x2 – y2 因式分解為 … (x + y)(x - y) x2 + y2 = ? 4. (x2 + 4)0.5 = ? ≠(x + 4) or (x + 2) 5. (9x)0.5 = ? … 3(x)0.5 6. 找出滿足 (x – 2)/(x + 4) < 7 的所有 x

10 先修課程 (cont’d) 函數 (function), f(x) E.g., f(x) = x2 – 7x + 5
( 類似自動販賣機, 投20元, 掉下伯朗咖啡, 投30元, 掉下Ritz小餅乾一盒) x = 2, f(2) = (2)2 – 7(2) + 5 = -5 x = 3, f(3) = (3)2 – 7(3) + 5 = -7

11 先修課程 (cont’d) 絕對值函數 (absolute function), f(x) E.g., f(x) = |x|
操作型定義 : |x| = x, if x ≥ 0; -x, if x < 0. E.g., f(x) = |x – 2| |x – 2| = x – 2, if x ≥ 2; |x – 2| = -x + 2, if x < 2

12 先修課程 (cont’d) 幾何 (geometry) … OK 三角學 (trigonometry) c a  b
- sin  = a / c , - cos  = b / c , - tan  = a / b = sin  / cos  , - csc  = c / a = 1/ sin  , - sec  = c / b = 1/ cos  , - cot  = b / a = 1/ tan  . c a  b

13 先修課程 (cont’d) 三角學 (trigonometry) c a  b - sin 2 = 2 sin cos ,
- cos 2 = cos2  – sin2  = 1 – 2 sin2  , - sin (+) = sin cos + cos sin , - cos (+) = cos cos – sin sin , - cos2  = (1 + cos 2)/2 , - sin2  = (1 – cos 2)/2 . c a  b

14 先修課程 (cont’d) 合成函數 (compound function)
E.g., f(x) = x2 , g(x) = x + 7 , => g(f(x)) = g(x2) = x2 + 7; => f(g(x)) = f(x + 7) = (x + 7)2 . E.g., h(x) = cos x , k(x) = x + x2 , => h(k(x)) = ? & k(h(x)) = ?