微積分(二)
Calculus II
| 節 | 週日 |
|---|---|
C 20:30–21:20 | 微積分(二) |
* 根據陽明交大上課時間表所列
有別於傳統微積分課程強調嚴謹的數學證明及計算,本課程強調微積分的基本概念及其在生物醫學領域的應用。本課程將延續上學期微積分一的內容,講述積分的應用及微分方程式,此外,機率理論、機率模型等主題也將在本課程中介紹。由於本課程的對象多為生醫領域相關學生,因此課程中的主題將利用生醫相關範例來講解,希望學生可以從課程中學到數學及生醫科學的關聯性。
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無備註
讓學生了解微積分及機率的基本觀念並認識數學模型在生物醫學領域的角色,以培養生醫領域的量化分析能力,此外亦培養學生的邏輯推理能力,建立學生進一步從事生醫領域研究之數學基礎。
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| 週次 | 主題 |
|---|---|
| 第 1 週 | Course introduction Applications of integrals: area, average 2021/02/28 |
| 第 2 週 | Applications of integrals: volume Improper integrals - The comparison test 2021/03/07 |
| 第 3 週 | Review of differential equations Solving autonomous differential equations: separation of variable Solving nonautonomous differential equations: separation of variable 2021/03/14 |
| 第 4 週 | Analysis of autonomous differential equations: equilibria Stability theorem for autonomous differential equations Biological examples for autonomous differential equations - Exponential population growth - Logistic population growthQuiz 1 Analysis of autonomous differential equations: equilibria Stability theorem for autonomous differential equations Biological examples for autonomous differential equations - Exponential population growth - Logistic population growth 2021/03/21 |
| 第 5 週 | Systems of autonomous differential equations Predator-prey model Euler’s method to systems of autonomous differential equations Equilibria of systems of autonomous differential equations Phase plane Nullclines 2021/03/28 |
| 第 6 週 | Phase plane trajectory Direction arrows The dynamics of competition between two species 2021/04/04 |
| 第 7 週 | Limits and continuity of multivariable functions Partial derivatives Double integrals over rectangles Double integral in polar coordinatesQuiz 2 Limits and continuity of multivariable functions Partial derivatives Double integrals over rectangles Double integral in polar coordinates 2021/04/11 |
| 第 8 週 | Midterm exam 2021/04/18 |
| 第 9 週 | Deterministic vs. stochastic Sample space and events Set theory Classical definition of probability Axioms of probability 2021/04/25 |
| 第 10 週 | Conditional probability - Special cases for conditional probability The law of total probability Bayes’ theorem 2021/05/02 |
| 第 11 週 | Independence Random variables Probability mass function (pmf) Cumulative distribution function (cdf) 2021/05/09 |
| 第 12 週 | Expectations of discrete random variables Variances of discrete random variables Bernoulli random variableQuiz 3 Expectations of discrete random variables Variances of discrete random variables Bernoulli random variable 2021/05/16 |
| 第 13 週 | Binomial random variable Geometric random variable 2021/05/23 |
| 第 14 週 | Poisson random variable Continuous random variable Probability density function (pdf) Cumulative distribution function (cdf) 2021/05/30 |
| 第 15 週 | Probability density function of a function of a random variable Expectations of continuous random variables Variances of continuous random variablesQuiz 4 Probability density function of a function of a random variable Expectations of continuous random variables Variances of continuous random variables 2021/06/06 |
| 第 16 週 | Uniform random variable Exponential random variable 2021/06/13 |
| 第 17 週 | Normal (Gaussian) random variable 2021/06/20 |
| 第 18 週 | Final exam 2021/06/27 |
教科書:Modeling the Dynamics of Life: Calculus and Probability for Life Scientists, 3rd edition, Frederick R. Adler, Cengage Learning, 2012. 參考書:Calculus for Biology and Medicine, 3rd edition, Claudia Neuhauser, Pearson, 2010. Biocalculus: Calculus, Probability, and Statistics for the Life Sciences, James Stewart and Troy Day, Brooks Cole, 2015.
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