Calculus (I)

Calculus I, Fall 2023.

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

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

Office Hours:
T,R: 12:00-13:00 綜三721, 分機62231 or Google meet by Email appointment.

Teaching Assistant:

林明駿, 綜三207, 分機33036. Office hour: Mon 17:00-18:00. Email: lmj111021513@gapp.nthu.edu.tw

Lecture: MXIC building (旺宏館), classroom 243, T3T4R3R4.

Recitation: General III building (綜三館), classroom 201. Thursdays 19:00-21:00.

僑外生微積分輔導:

輔導員:森田展弘(Email：morita880210@gmail.com)。週三、週四晚上在綜三205輔導。

國立清華大學基礎科目個別輔導:

目前還在應徵課輔員，各位同學應該在第四週就可以開始向課輔員諮詢。詳細連結

Grading: 40% quiz(pick n-2 best from n quizzes)+attendance, 20% midterm I, 20% midterm II + 20% final exam. Extra credits for significant contributions to the class (correcting mistakes, etc.).

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

Contents:

Calculus I:
Chapter 2: Limits and Continuity.
Chapter 3: Derivatives.
Chapter 4: Application of Derivatives.
Chapter 5: Integrals
Chapter 6: Applications of Definite Integrals.
Chapter 7: Integrals and Transcendental Functions.
Chapter 8: Techniques of Integration.
Chapter 9: First-Order Differential Equations.

Syllabus, study guide and exam solutions:

Lecture notes, homework assignments and solutions:

Homework policy.
Week 01: Recitation starts this week (Rb,Rc) (Sep 14, 19:00-21:00).
Lecture 01 (20230912, 75 mins) Section 2.2: Review Limit laws; Limits involving quotients; Sandwich Theorem.
Homework 01 (lecture 01). Homework 01 scan. Homework 01 solutions.
Lecture 02 (20230914, 100 mins) Section 2.3: Precise definition of limit; How to prove lim_{{x → c}} = L using the ε-δ argument.
Supplement to Lecture 02.
Homework 02 (lecture 02-03). Homework 02 solutions.
Lecture 02 corresponds to problems 1-2.
Week 02: Quiz 01 (Sep 19, 10:10AM).
Lecture 03 (20230919, 70 mins) Section 2.3: More examples of proving the limit using definition; Using the definition to prove Theorems. Section 2.4: One sided limit; Limits involving sinθ/θ.
Lecture 04 (20230921, 100 mins) Section 2.5: Definition of continuity; Left and right continuity; Basic properties of continuous functions; Composite of continuous functions and generalization; Intermediate Value Theorem and application in root allocating.
Homework 03 (lecture 04-05) (updated with some new problems and (*) problems). Homework 03 solutions.
Week 03: Quiz 02 (Sep 26, 10:10AM).
Lecture 05 (20230926, 70 mins)
Section 2.6: Limits Involving Infinity (SKIP the asymptotes); Section 3.2: Definition of derivative; One-sided derivative; Differentiable functions.
Week 04: No quiz this week.
Lecture 06 (20231003, 100 mins)
Section 3.2: Examples of functions not differentiable at a point; proof of "Differentiability implies Continuity". Section 3.3: Differentiation rules; Derivative of xⁿ for some integer n's and rational n's; proof of Product rule and Quotient rule; Derivative of exponential functions;
Homework 04 (lecture 05-06), updated with (*) problems. Homework 04 solutions (updated with section 3.2, problem 58(a)).
Week 05: No quiz this week.
Lecture 07 (20231012, 100 mins)
Section 3.5: Derivative of trigonometric functions; Section 3.6: The Chain Rule. Examples of applying the Chain Rule; proof of the Chain Rule (naive version)
Homework 05 (lecture 07-08). Homework 05 solutions.
Week 06: Quiz 03 (Oct 17, 10:10AM).
Lecture 08 (20231017, 70 mins)
Section 3.7: Implicit differentiation. Section 3.8: Derivatives of Inverse Functions and Logarithms.
Supplement to Lecture 08.
Lecture 09 (20231019, 50 mins)
Continue on Section 3.8: Derivatives of Inverse Functions and Logarithms.
Homework 06 (lecture 08-09). Homework 06 solutions.
Week 07: Midterm 01 (Oct 24, 10:10AM, Section 2.2-3.7).
Lecture 10 (20231026, 75 mins)
Section 3.9: Inverse Trigonometric Functions: restricted domains and derivation of derivatives of inverse trigonometric functions.
Homework 07 (lecture 10-11). Homework 07 solutions (updated, including the red text part).
Week 08: No quiz this week.
Lecture 11 (20231031, 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 (20231102, 85 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. Section 4.2: Rolle's Theorem, Mean Value Theorem. Proof of Rolle's Theorem and the equivalence of the two Theorems.
Homework 08 (lecture 12-13). Homework 08 solutions.
Week 09: Quiz 04 (Nov 07, 10:10AM).
Lecture 13 (20231107, 60 mins)
Section 4.2: Corollaries of Mean Value Theorem. Corollary 1: f'(x) == 0 implies f is a constant. Corollary 2: f'(x) == g'(x) implies f(x)-g(x) is a constant. Corollary 3: f'(x) > 0 (< 0) implies f(x) is a increasing (decreasing). Section 4.3: First derivative test for local extrema. Examples. Section 4.4: Second derivative test for local extrema. Concave upward and downward. Definition of a point of inflection and examples.
Lecture 14 (20231109, 100 mins)
Section 4.4: Examples of curve sketching. Section 4.5: L’Hôpital’s Rule and how to use it.
Homework 09 (lecture 14-15) (New, updated with definition of asymptotes). Homework 09 solutions.
Week 10: Quiz 05 (Nov 14, 10:10AM).
Lecture 15 (20231114, 60 mins)
Section 4.5: L’Hôpital’s Rule continued.
Lecture 16 (20231116, 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. Supplement to Lecture 16: reference for the proof of L’Hôpital’s rule ("∞/∞" version). Section 4.6: Applied Optimization.
Homework 10 (lecture 16-17) Homework 10 solutions.
Week 11: Quiz 06 (Nov 21, 10:10AM).
Lecture 17 (20231121, 70 mins)
Section 4.8: Antiderivative: Examples (polynomial, trigonometric functions, etc.). Application to equation of motion under gravity. 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 → ∞}}.
Lecture 18 (20231123, 90 mins)
Section 5.3: Examples on how to write definite integrals as limit of Riemann sums and vice versa. How to find the limit of a Riemann sum directly for low order monomials. Section 5.4: Fundamental Theorem of calculus, proof and application. Homework 11 (updated) (lecture 18-19) Homework 11 solutions (s5.3 problem 5 corrected).
Week 12: Quiz 07 (Nov 28, 10:10AM).
Lecture 19 (20231128, 70 mins)
Section 5.5: Indefinite integrals and substitution method (reverse engineering of the Chain Rule).
Lecture 20 (20231130, 40 mins)
Section 5.5: Substitution method, continued. Section 5.6: Area between curves. Additional recitation: section 5.3-5.4.
Week 13: Midterm 02 (Dec 05, 10:10AM).
Lecture 21 (20231207, 90 mins)
Section 6.1: Volumes Using Cross-Sections. Section 6.2: Volumes Using Cylindrical Shells.
Homework 12 (lecture 21-22). Homework 12 solutions.
Week 14:
Lecture 22 (20231212, 95 mins) (New: page 3 corrected)
Section 6.3: Arclength. Section 6.4: Areas of Surfaces of Revolution.
Summary of formula in Chap 06.
Lecture 23 (20231214, 100 mins)
Section 6.4: Areas of Surfaces of Revolution, coninued. Section 7.2: Separable Differential Equations.
Homework 13 (lecture 23-24). Section 9.2: Linear Differential Equations.
Homework 13 solutions.
Week 15: Quiz 08 (Dec 19, 10:10AM).
Lecture 24 (20231219, 70 mins)
Section 7.3: Hyperbolic Functions (SKIP inverse hyperbolic functions).
Lecture 25 (20231221, 95 mins, page 2, page 11 corrected)
Section 7.4: Relative Rates of Growth.
Homework 14 (lecture 25-26). Homework 14 solutions (updated. added page 7-8).
Week 16: Quiz 09 (Dec 26, 10:10AM).
Lecture 26 (20231226, 70 mins. v02: inserted page 8)
Section 8.2: Integration by Parts. Section 8.3: Trigonometric Integrals
Lecture 27 (20231228, 100 mins, page 8 corrected)
Section 8.3: Trigonometric Integrals, continued.
Homework 15 (lecture 27-28). Homework 15 solutions.
Week 17: Quiz 10 (Jan 02, 10:10AM).
Lecture 28 (20240102, 70 mins. page 4, page 7 corrected).
Section 8.3: Trigonometric Integrals, continued. Section 8.4: Trigonometric Substitutions. Section 8.5: Integration of Rational Functions by Partial Fractions.
Lecture 29 (20240104, 80 mins).
Section 8.5: Integration of Rational Functions by Partial Fractions, continued. Half-angle Substitution.
Week 18: Final Exam (Section 6.1-8.5. Jan 09, 10:10AM).
Homework assignments of this semester (temporary, subject to minor revisions): Chapter 02. Chapter 03. Chapter 04. Chapter 05. Chapter 06. Chapter 07. Chapter 08. Chapter 09.