Calculus I, Fall 2021.

A undergraduate course in Mathematics at the Department of Mathematics, National Tsing Hua University.

Instructor: Professor Wei-Cheng Wang (王偉成).

Office Hours:
TnRn: 綜三721, 分機62231,
or , via Google meet by appointment.

Teaching Assistant:

• B05: 邱志文 (lmp127552@gmail.com).
• B08: 葉曼儀 (yeh60123@gmail.com), 綜三215 分機33076.
• B09: 李長紘 (alex60901@gmail.com).

Lecture: MS building (材料館) classroom 418, T3T4R3R4.

Recitation: Delta building (台達館), classrooms B05, B08, B09. Tuesdays 19:00-21:00.

Grading: 40% quiz(pick n-1 best from n quizzes)+attendance, 20% midterm I, 20% midterm II + 20% final exam. Extra credits for significant contributions.

Textbook: G. B. Thomas, M. D. Weir and J. R. Hass: Thomas's Calculus Early Transcendentals, 13th edition (華通書局有販售,詳見校務資訊系統課程大綱).

Contents:

• Calculus I:
• Limits and Continuity.
• Application of Derivatives.
• Integration.
• Applications of Definite Integrals.
• Integrals and Transcendental Functions.
• Techniques of Integration.
• First-Order Differential Equations.

Syllabus and study guide:

• Calculus I:
• Revised syllabus (Updated Nov 24. Typo corrected Dec 22).
• quiz 01 study guide (v02). quiz 01 solutions (v02).
  
• quiz 02 study guide. quiz 02 solutions.
  
• quiz 03 study guide. quiz 03 solutions.
  
• midterm 01 study guide. midterm 01 solutions.
  
• quiz 04 study guide. quiz 04 solutions.
  
• quiz 05 study guide. quiz 05 solutions (new! v02).
  
• quiz 06 study guide. quiz 06 solutions.
  
• midterm 02 study guide. midterm 02 solutions.
  
• quiz 07 study guide. quiz 07 solutions.
  
• quiz 08 study guide. quiz 08 solutions.
  
• quiz 09 study guide. quiz 09 solutions.
  
• final exam study guide. final exam solutions (New. Problem 4(h) updated).
  
• distribution of semester grade.

Video Links, lecture notes, homework assignments and exam solutions:

• Week 01: No class, no recitation this week.
• Week 02: No recitation this week.
  Lecture 01 (20210923, 68mins): Section 2.2: Review basics of limit (Limit laws); Limits involving quotients; Sandwich Theorem.
  Lecture 01 note.
  Homework week 02.
• Week 03: No recitation this week.
  Lecture 02 (20210930, 98 mins): Section 2.3: Precise definition of limit; How to prove lim_{{x → c}} = L using the ε-δ argument; The ε/2 trick.
  Supplement to Lecture 02.
  Lecture 02 note (with minor corrections).
  Homework week 03. Homework week 02 03 solutions.
  
• Week 04: Recitation starts this week (10/05 19:00).
  Lecture 03 (20211005, 90 mins): Supplements to Section 2.3: How to prove lim_{{x → c}} ≠ L. Section 2.4: One sided limit; Limits involving sinθ/θ. Section 2.5: Definition of continuity.
  Lecture 03 note.
  Lecture 04 (20211107, 99 mins): Section 2.5: Left and right continuity; Basic properties of continuous functions; Composite of continuous functions and generalization; Intermediate Value Theorem and application in root locating. Section 2.6: Limits involving infinity (SKIP asymptotes).
  Lecture 04 note.
  Homework week 04. Homework week 04 solutions.
  
• Week 05: Quiz 01 (10/12 19:00).
  Lecture 05 (20211012, 51 mins): Section 3.2: Definition of derivative; One-sided derivative; Examples of functions not differentiable at a point; Differentiable functions; Differentiability implies continuity.
  Lecture 05 note (v02).
  Lecture 06 (20211014, 88 mins): Section 3.3: Differentiation rules; Derivative of xⁿ for integer n and rational n; Product rule and quotient rule; Derivative of exponential functions;
  Lecture 06 note (v02).
  Homework week 05. Homework week 05 solutions.
  
• Week 06: Quiz 02 (10/19 19:00).
  Lecture 07 (20211019, 89 mins): Section 3.5: Derivative of trigonometric functions; Section 3.6: The chain rule.
  Lecture 07 note (v05, updated Oct 30).
  Lecture 08 (20211021, 57 mins): Section 3.6: The chain rule, continued. Section 3.7: Implicit differentiation.
  Lecture 08 note (v04, updated Oct 31).
  Homework week 06. Homework week 06 solutions.
  
• Week 07: Quiz 03 (10/26 19:00).
  Lecture 09 (20211026, 124 mins): Section 3.8: Derivatives of Inverse Functions and Logarithms.
  Supplement to Lecture 09.
  Lecture 09 note.
  Lecture 10 (20211028): Midterm review. Lecture 10 correction.
  Homework week 07. Homework week 07 solutions.
  
• Week 08: Midterm Exam 01 (11/02 19:00).
  Lecture 11 (20211102, 72 mins): Section 3.9: Inverse Trigonometric Functions: restricted domains and derivation of derivatives of inverse trigonometric functions.
  Lecture 11 note.
  Lecture 12 (20211104, 75 mins): Section 3.11: Linearizations and differentials: definition of linearization (SKIP differentials); approximate function values using linear approximation (linearization); equivalent expression of differentiability at a point; flawless proof of the chain rule.
  Lecture 12 note.
  Homework week 08. Homework week 08 solutions.
• Week 09:
  Lecture 13 (20211109, 70 mins): Section 4.1: Definitions of local and absolute maximum and minimum. The extreme value theorem for continuous functions on closed intervals. The first derivative theorem for local extreme values. How to find absolute extrema for continuous functions on closed intervals.
  Lecture 13 note (with corrections).
  Lecture 14 (20211111, 110 mins): Section 4.2: Rolle's Theorem, Mean Value Theorem and proof. Corollary of f'(x) == 0 on (a,b). Proof of error formula for linear approximation. Section 4.3: First derivative test for local extrema. Examples.
  Lecture 14 note.
  Homework week 09. Homework week 09 solutions.
• Week 10: Quiz 04 (11/16 19:00).
  Lecture 15 (20211116, 100 mins): Section 4.4: Concave upward and downward. The second derivative test for concavity. Definition of a point of inflection. Examples of curve sketching. Section 4.5: L’Hôpital’s Rule and how to use it.
  Lecture 15 note.
  Lecture 16 (20211118, 100 mins): Section 4.5: More on L’Hôpital’s Rule. Application and limitation. Proof of L’Hôpital’s Rule using Cauchy's Mean Value Theorem.
  Lecture 16 note. Supplement to Lecture 16: reference for the proof of L’Hôpital’s rule ("∞/∞" version).
  Homework week 10. Homework week 10 solutions (v03).
• Week 11: Quiz 05 (11/23 19:00).
  Lecture 17 (20211123, 100 mins): Section 4.6: Applied Optimization. Section 4.8: Antiderivative: application to equation of motion.
  Lecture 17 note.
  Lecture 18 (20211125, 90 mins): Section 5.2-5.3: Definite integral as limit of Riemann sum. Explain why use lim_{{||P|| → 0}} as the definition and not the standard lim_{{n → ∞}}. Examples.
  Lecture 18 note.
  Homework week 11. Homework week 11 solutions.
• Week 12: Quiz 06 (11/30 19:00).
  Lecture 19 (20211130, 100 mins): Section 5.4: Fundamental Theorem of calculus, proof and application. Section 5.5: Indefinite integrals and substitution method (reverse engineering of the Chain Rule).
  Lecture 19 note.
  Lecture 20 (20211202, 85 mins): Section 5.5: Substitution method, continued. Section 5.6: Area between curves.
  Lecture 20 note.
  Homework week 12. Homework week 12 solutions.
• Week 13: Midterm 02 (12/07 19:00).
  Lecture 21 (20211209, 100 mins): Section 6.1: : Volumes Using Cross-Sections. Section 6.2: Volumes Using Cylindrical Shells.
  Lecture 21 note.
  Homework week 13.
• Week 14:
  Lecture 22 (20211214, 100 mins): Section 6.3: Arclength. Section 6.4: Areas of Surfaces of Revolution.
  Lecture 22 note.
  Lecture 23 (20211216, 50 mins): Section 6.4, continued. Section 7.2: Separable Differential Equations (only).
  Lecture 23 note.
  Summary of formula in Chap 06 (v02).
  Homework week 14. Homework week 14 solutions.
• Week 15: Quiz 07 (12/21 19:00).
  Lecture 24 (20211221, 85 mins): Section 9.2: Linear Differential Equations.
  Lecture 24 note.
  Lecture 25 (20211223, 95 mins): Section 7.3: Hyperbolic Functions (SKIP inverse hyperbolic functions. Section 7.4: Relative Rates of Growth.
  Lecture 25 note.
  Homework week 15. Homework week 15 solutions.
• Week 16: Quiz 08 (12/28 19:00).
  Lecture 26 (20211228, 110 mins): Section 8.2: Integration by Parts. Section 8.3: Trigonometric Integrals
  Lecture 26 note.
  Lecture 27 (20211230, 100 mins): Section 8.3: Trigonometric Integrals, continued. Section 8.4: Trigonometric Substitutions.
  Lecture 27 note.
  Homework week 16.
• Week 17: Quiz 09 (01/04 19:00).
  Lecture 28 (20220104, 100 mins): Section 8.5: Integration of Rational Functions by Partial Fractions. Half-angle Substitution.
  Lecture 28 note.
  Homework week 17. Homework week 17 solutions.
• Homework assignments in previous years for your reference: Chapter 02. Chapter 03. Chapter 04. Chapter 05. Chapter 06. Chapter 07. Chapter 08. Chapter 09.