Calculus (II)

Calculus II, Spring 2024.

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.

Grading: 40% quiz(pick n-2 best from n quizzes)+attendance (36%+4%), 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 10: Infinite Sequences and Series.
Chap 14: Partial Derivatives.
Chap 15: Multiple Integrals.
Chap 16: Integration in Vector Fields.

Syllabus, study guide and exam solutions:

Lecture notes, homework assignments and solutions:

Homework policy.
Week 01:
Lecture 01 (20240220, 85 mins)
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 (20240222, 75 mins)
Section 8.8: Direct Comparison Test (Theorem 2). Limit Comparison Test (Theorem 3). Examples.
Homework 01 (lectures 01). Homework 01 solutions.
Homework 02 (lectures 02-03) (updated with section 10.2). Homework 02 solutions.
Week 02: Quiz 01 on Lecture 01 (Feb 27, 10:10AM).
Lecture 03 (20240227, 75 mins)
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 (20240229, 75 mins)
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 03 (lectures 04-05). Homework 03 solutions (page 4 corrected).
Week 03: Quiz 02 (Section 8.8, 10.1, 10.2; Mar 05, 10:10AM).
Lecture 05 (20240305, 65 mins)
Section 10.4: More examples of Direct and Limit comparison tests. Section 10.5: The Absolute Convergence Test and proof. Ratio test and Root test. Examples of both tests.
Lecture 06 (20240307, 100 mins)
Section 10.5: Proof of the Ratio Test. Section 10.6: Definition of Conditional convergence and Absolute convergence; The Alternating Series Test (Leibniz Test) and remainder estimate; Examples and proof. 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 04 (lectures 06-07). Homework 04 solutions.
Week 04: Quiz 03 (Section 10.3, 10.4, 10.5: Mar 12, 10:10AM).
Lecture 07 (20240312, 50 mins)
Section 10.7: Definition and examples of power series, including geometric series; The convergence Theorem for power series (Theorem 10) 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,
Lecture 08 (20240314, 90 mins)
Section 10.7: Algebraic manipulation of power series: Division: find the power series of A(x)/B(x) by long division. Remark on radius of convergence of A(x)/B(x). Term by term differentiation and integration and application.
Homework 05 (lectures 08-09) (New: updated with Theorem A typed in) (Newer: add update on Hint for Section 10.8, Problem 35, Method 2). Homework 05 solutions.
Week 05: Quiz 04 (Section 10.6, 10.7 (part I): Mar 19, 10:10AM).
Lecture 09 (20240319, 70 mins)
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). Theorem A: How to find T_f,a(x) if f already has a power series representation (Example: f(x)=1/(1-x)).
Lecture 10 (20240321, 90 mins)
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)" for f(x)= e^x, cos(x), sin(x); Examples of finding T_f,a(x) using Theorem A. <
Homework 06 (lectures 10-12) Homework 06 solutions.
Week 06: Quiz 05 (Section 10.7 (part II), 10.8: Mar 26, 10:10AM).
Lecture 11 (20240326, 60 mins)
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 to evaluation of non-elementary integrals and intermediate forms.
Lecture 12 (20240328, 60 mins)
Section 10.10: More applications to evaluation of non-elementary integrals and intermediate forms. Finding the sum of a power series by identifying it with the Taylor series of basic functions. Euler identity of exp(iθ).
Week 07: Midterm 01 (Section 8.8, 10.1-10.10: Apr 02, 10:10AM).
Week 08: No quiz this week.
Lecture 13 (20240409, 90 mins)
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.
Lecture 14 (20240411, 90 mins) (v03: page 1, 6 corrected)
Section 14.3: Partial derivatives. Implicit partial differentiation. Higher order partial derivatives. Definition of differentiability; Further explanation of tangency of two curves in a plane.
Homework 07 (lectures 13) Homework 07 solutions.
Homework 08 (lectures 14-15) Homework 08 solutions (new: page 3 corrected to "Mean Value Theorem").
Week 09: Quiz 06 (Section 14.2: Apr 16, 10:10AM).
Lecture 15 (20240416, 75 mins) (v02: added remark on page 1)
Section 14.3:
Equivalent Definitions of differentiability.
Further explanation of tangency of two surfaces in space.
Theorem 2: Continuity of second partial derivatives implies interchanging order of two partial derivatives and a counter example.
Theorem 3: Sufficient condition of differentiability in terms of continuity of first partial derivatives and a counter example.
Theorem 4: Differentiability implies continuity. Proof.
Lecture 16 (20240418, 90 mins)
Section 14.4:
Application of differentiability to the Chain rule. Examples of the chain rule.
Implicit partial differentiation revisited;
Section 14.5:
Definition and examples of directional derivative.
Theorem 9: Differentiability leads to formula of directional derivative.
Homework 09 (lectures 16-17) Homework 09 solutions.
Week 10: Quiz 07 (Section 14.3: Apr 23, 10:10AM).
Lecture 17 (20240423, 75 mins)
Section 14.4:
Relation between gradient vector and directional derivatives. Example and counter example (fig 14.13).
Section 14.5:
Properties of ∇f.
Finding tangent and normal lines of a level curve of F(x,y) using the gradient vector ∇F.
Lecture 18 (20240425, 80 mins)
Section 14.6:
Find tangent plane and normal line of a level surface of F(x,y,z).
Find tangent line and normal plane of a curve in space.
Linear approximation of z=f(x,y) near (x0,y0,z0).
Linear approximation = tangent plane.
Error estimate for Linear Approximation.
Homework 10 (lectures 18-19) (update: added hint for Section 14.7, problems 31, 35. Homework 10 solutions.
Week 11: Quiz 08 (Section 14.4-14.5: Apr 30, 10:10AM).
Lecture 19 (20240430, 70 mins)
Section 14.7:
First Derivative Test for local extremes.
Definition of critical points.
Second Derivative Test, Proof and Example.
Introduction of "gradient analysis" as a (better) alternative to Second Derivative Test.
Lecture 20 (20240502, 80 mins)
Section 14.7:
Find absolute extremes on a bounded region using gradient analysis.
Section 14.8:
Constrained Optimization: Method of Lagrangian Multiplier.
Example: Optimize a function of two variables with one constraint.
Homework 11 (lectures 20-21) Homework 11 solutions.
Week 12: Quiz 09 (Section 14.6-14.7: May 07, 10:10AM).
Lecture 21 (20240507, 75 mins)
Section 14.8:
Example: Optimize a function of three variables with one constraint.
Example: Optimize a function of three variables with two constraints. Example.
Section 14.9:
Derivation of Taylor's formula for functions of two variables.
Lecture 22 (20240509, 40 mins)
Section 14.9:
- Generalization to functions of three or more variables.
- Applications: Error estimate for linearization, validity of Second Derivative Test for local extremes.
Week 13: Midterm 02 (Section 14.2-14.9 + Content of Midterm 01, May 14, 10:10AM).
Remark on definition of differentiability
Lecture 23 (20240516, 70 mins)
Section 14.10:
- Partial Derivatives with Constrained variables. Explain the notation and how to compute. Examples.
No recitation this week.
Homework 12 (lectures 23-24 + page 1-3 of Lecture 25) Homework 12 solutions.
Week 14: No Quiz this week.
Lecture 24 (20240521, 75 mins)
Section 15.1:
- Double integrals defined as limit of Riemann sum.
Section 15.2:
- Fubini's Theorem and iterated integrals on rectangular regions.
- Fubini's Theorem (strong form) and iterated integrals on rectangular-like regions.
Section 15.3:
- Area by double integrals (special case of section 15.2).
Lecture 25 (20240523, 95 mins)
Section 15.2-15.3:
- Interchanging order of integration for double (iterated) integrals.
Section 15.4:
- Integration in polar form. Derivation of dA = r dr dθ.
- Examples on identifying limits of integration for dA = r dr dθ and = dθ r dr, respectively.
Homework 13 (lectures 25-26) Homework 13 solutions.
Week 15: Quiz 10 (Section 14.10, 15.1-15.2, May 28, 10:10AM).
Lecture 26 (20240528, 70 mins)
Section 15.5:
- Triple Integrals. Definition. Method of identifying limits of integration for various choice of dV = dz dy dx, or dy dz dx, etc.
Section 15.7:
- Triple Integrals in Cylindrical Coordinates. Definition. dV = dA dz = r dr dθ dz.
Lecture 27 (20240530, 70 mins using slides)
Homework 14 (lectures 27-28) Homework 14 solutions.
Week 16: Quiz 11 (Section 15.4-15.5, June 04, 10:10AM).
Lecture 28 (20240604, 60 mins using slides)
Lecture 29 (20240606, 100 mins)
Homework 15 (lectures 29-30) Homework 15 solutions.
Week 17: Quiz 12 (Section 15.7-15.8, June 11, 10:10AM).
Lecture 30 (20240611, 55 mins. The last lecture.)
Lecture ended.
Week 18: Final Exam (Section 14.10-16.3, Jun 18, 10:10AM).
Note on Conservative Fields and Simply Connected Domains.
Homework assignments of this semester (temporary): Chapter 10. Chapter 14. Chapter 15. Chapter 16.