Calculus II, Spring 2026.

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:

• 台達B08: 柯秉辰, 綜三215, 分機33076. Office hour: F13:00-F15:00. Email: f8220219@gmail.com
• 台達B09: 鄭亦展, 綜三209, 分機33070. Office hour: R5R6. Email: a60929069@gmail.com

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

Recitation: Delta building (台達館), classrooms B08(學號末位數:0, 2, 4, 6, 8), B09 (學號末位數: 1, 3, 5, 7, 9). Tuesdays 19:00-21:00.

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 II:
• Chap 08: Section 8.8: Improper Integrals.
• Chap 10: Infinite Sequences and Series.
• Chap 14: Partial Derivatives.
• Chap 15: Multiple Integrals.
• Chap 16: Integration in Vector Fields.

Syllabus and study guide:

• Calculus II:
• Syllabus (New! updated Mar 03).
• quiz 01 (section 8.8) study guide.
• quiz 02 (section 10.1, 10.2, 10.3) study guide.
• quiz 03 (section 10.4, 10.5, 10.6) study guide.
• quiz 04 (section 10.7, 10.8 part I) study guide.
• Midterm 01 (section 8.8, 10.1-10.10) study guide.

Lecture notes, homework assignments and solutions:

• Homework policy.
  
• Week 01:
  Lecture 01 (20260224): A4, 800x700
  Section 8.8: Definition of improper integrals, type I and type II and examples.
  Direct computation to find out convergence and divergence.
  Integrals ∫₀¹ x^-p dx and ∫₁^∞ x^-p dx and related integrals.
  Lecture 02 (20260226): A4, 800x700
  Section 8.8: Direct Comparison Test (Theorem 2). Limit Comparison Test (Theorem 3). Examples.
  Homework 01. Homework 01 solutions.
  
• Week 02: Quiz 01 (Section 8.8), Mar 05, 10:10AM.
  Lecture 03 (20260303): A4, 800x700
  Section 10.1: Definition of limit of sequences, including the limit = +- ∞ cases.
  Find limit of a sequence using the Sandwich Theorem and L’Hôpital’s Rule.
  Section 10.2: Infinite series, definition and example of convergent and divergent series. Geometric Series, telescoping sum. The n-th term test and its proof.
  Examples of divergent series.
  Lecture 04 (20260305): A4, 800x700
  Section 10.3: The integral test and Remainder estimate, Proof and examples.
  Section 10.4: Direct comparison test; Limit comparison test.
  Examples for both Direct and Limit comparison tests.
  Homework 02. Homework 02 solutions.
  
• Week 03: Quiz 02 (Section 10.1-10.3; Mar 12, 10:10AM).
  Lecture 05 (20260310, page 1-2 reorganized, page 10 corrected): A4, 800x700
  Section 10.5: The Absolute Convergence Test and proof.
  Section 10.5; Ratio test and Root test. Examples of both tests.
  Section 10.5: Proof of the Ratio Test.
  Lecture 06 (20260312, added page 5-6): A4, 800x700
  Section 10.6: Definition of Conditional convergence and Absolute convergence;
  The Alternating Series Test (Leibniz Test) and remainder estimate; Examples and proof. Statement and proof of The Monotone Sequence Theorem (Section 10.1, Theorem 6);
  Section 10.7: Definition and examples of power series, including geometric series; Examples of power series whose radius of convergence is finite, zero and ∞, respectively.
  Homework 03. Homework 03 solutions.
  
• Week 04: Quiz 03 (Section 10.4, 10.5, 10.6: Mar 19, 10:10AM).
  Lecture 07 (20260317, v02: added proof of Theorem 19): A4, 800x700
  Section 10.7: The convergence Theorem for power series (Theorem 18) and proof;
  Definition of the "radius of convergence" and the "interval of convergence";
  Finding radius of convergence using ratio and root test; Examples of convergence/divergence when |x-a| = R.
  Section 10.7: Algebraic manipulation of power series: addition, subtraction, multiplication and Division (find the power series of A(x)/B(x) by long division).
  Remark on radius of convergence of A(x)/B(x).
  Lecture 08 (20260319, v02. page 1, 5, 7 revised): A4, 800x700
  Section 10.7: Term by term differentiation and integration and application.
  Section 10.8: Taylor and MacLaurin series. Definition of T_f,a(x). Examples on how to find T_f,a(x) with f(x)= e^x, cos(x), sin(x).
  Homework 04. Homework 04 solutions.
  
• Week 05: Quiz 04 (Section 10.7, 10.8: Mar 26, 10:10AM).
  Lecture 09 (20260324): A4, 800x700
  Section 10.8: Example of T_f,a(x) ≠ f(x) except at x=a.
  Section 10.9: Taylor's Theorem (Taylor's formula) and proof;
  Corollary on how to check whether T_f,a(x) = f(x) from the remainder term in Taylor's formula;
  Sufficient condition on estimates of the remainder term in Taylor's formula;
  Examples of "T_f,a(x) = f(x)" with f(x)= e^x, cos(x), sin(x);
  Theorem A (Section 10.9: Exercise 51): How to find T_f,a(x) if f already has a power series representation (Example: f(x)=1/(1-x)).
  Examples of finding T_f,a(x) using Theorem A.
  Lecture 10 (20260326): A4, 800x700
  Section 10.10:
  Binomial series. Application to Taylor series generated by sin^-1(x). Leibniz's formula for π;
  Remark on variants of geometric series expansion (binomial series with m=-1);
  Applications:
  Evaluation of non-elementary integrals and intermediate forms.
  Find the sum of a power series by identifying it with the Taylor series of basic functions.
  Euler identity of exp(iθ).
  Homework 05 Homework 05 solutions.
  
• Week 06: No Quiz this week (Holiday: April 2).
  Lecture 11 12 (20260331-0409): preclass version
  Section 14.2: Definition of lim_{(x,y)→(x0,y0)} f(x,y) using ε and δ. Definition of continuity. Elementary examples of finding lim_{(x,y)→(x0,y0)} f(x,y). Several examples of lim_{(x,y)→(x0,y0)} f(x,y) does not exist ( fig 14.14, fig 14.15). Two path Theorem. Composition of continuous functions.
  
• Week 07: Midterm 01 (Tuesday April 7, 10:10 AM).
  
• Homework assignments of this semester (temporary): Chapter 10. Chapter 14. Chapter 15. Chapter 16.