Calculus II, Spring 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:
- 林文揚 (pingu900818@gmail.com)
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,
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:
- Calculus II:
- Syllabus (revised Feb 08).
- quiz 01 (section 10.1, 10.2) study guide.
Quiz 01 solutions.
- quiz 02 (section 10.3, 10.4) study guide.
Quiz 02 solutions.
- quiz 03 (section 10.5, 10.6) study guide.
Quiz 03 solutions.
- quiz 04 (section 10.7, 10.8) study guide.
Quiz 04 solutions.
- midterm 01 (section 10.1-10.10) study guide.
midterm 01 solutions.
- quiz 05 (section 14.2, 14.3) study guide
New! item 3 revised.
Quiz 05 solutions.
- quiz 06 (section 14.4, 14.5) study guide.
Quiz 06 solutions.
- quiz 07 (section 14.6, 14.7) study guide.
Quiz 07 solutions.
- midterm 02 (section 14.2-14.9) study guide.
midterm 02 solutions.
- quiz 08 (section 15.1-15.4) study guide.
Quiz 08 solutions.
- quiz 09 (section 15.5, 15.7) study guide.
Quiz 09 solutions.
- quiz 10 (section 15.8, 16.1, 16.2) study guide.
- Final Exam (section 15.1-16.4) study guide.
Final Exam solutions.
Lecture notes, homework assignments and solutions:
- Homework policy.
- Week 01: No recitation this week.
Lecture 01 (20230214):
Chapter 10 overview: The role of sequences and infinite series in Calculus.
Section 10.1: Definition of limit of sequences, including the limit = +- ∞ cases.
Lecture 01 note.
Lecture 02 (20230216):
Section 10.1: Verify the limit of a sequences using definition, 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.
Lecture 02 note (v02, revised).
Homework 01 (lectures 01-02).
Homework 01 solutions.
- Week 02: Recitation starts this week (Rb,Rc) (Feb 23, 19:00-21:00).
Lecture 03 (20230221):
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.
Lecture 03 note.
Lecture 04 (20230223):
Section 10.4: More examples for The Comparison Test and Limit Comparison Test.
Section 10.5: The Absolute Convergence Test and proof.
Ratio test and Root test. Examples of both tests.
Lecture 04 note.
Homework 02 (lectures 03-04).
Homework 02 solutions (part 1),
(part 2).
- Week 03: Quiz 01 (Section10.1-10.2, Mar 02, 19:00-19:30).
Lecture 05 (20230302):
Section 10.5: More examples of the Ratio test and the Root test. 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.
Lecture 05 note (v02, page 2-3 revised).
Homework 03 (lectures 05-06).
Homework 03 solutions.
- Week 04: Quiz 02 (Section 10.3-10.4, Mar 07, 10:10AM).
Lecture 06 (20230307):
Section 10.7: Definition and examples of power series, including geometric series;
The convergence Theorem for power series;
Definition of the "radius of convergence"
and the "interval of convergence";
Finding radius of convergence using ratio and root test;
Lecture 06 note.
Lecture 07 (20230309):
Section 10.7: Algebraic manipulation of power series: addition, subtraction, multiplication,
division, term by term differentiation and integration and application.
Lecture 07 note.
Homework 04 (lectures 07-08).
Homework 04 solutions.
- Week 05: Quiz 03 (Section 10.5-10.6, Mar 14, 10:10AM).
Lecture 08 (20230314) (v02)
(Page 11 modified. See additional remark on page 13 for the function given on page 12):
Section 10.8: Taylor and MacLaurin series.
Definition of Tf,a(x). Example of Tf,a(x) ≠ f(x).
Examples on how to find Tf,a(x) with f(x)= ex, cos(x), sin(x).
How to find Tf,a(x) if
f already has a power series representation (Example: f(x)=1/x).
Lecture 09 (20230316):
Section 10.9: Taylor's Theorem (Taylor's formula) and proof;
Corollary on how to check whether Tf,a(x) = f(x);
Sufficient condition on estimates of the remainder term in Taylor's formula;
Examples of "Tf,a(x) = f(x)" for f(x)= ex, cos(x), sin(x);
Application to numerical approximation of nonelementaty integral
using Taylor polynomial of order n and error estimate.
Homework 05 (lectures 09-10).
Homework 05 solutions.
- Week 06: Quiz 04 (Section 10.7-10.8, Mar 21, 10:10AM).
Lecture 10 (20230321)
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 intermediate forms.
Lecture 11 (20230323)
Section 10.10: More applications to evaluation of intermediate forms
and Euler identity of exp(iθ).
Finding the sum of a power series by identifying it with the Taylor series of
basic functions.
- Week 07: Midterm 01 (Section 10.1-10.10, Mar 28, 10:10AM).
Lecture 12 (20230330)
Section 14.2:
Definition of lim(x,y)→(x0,y0) f(x,y).
Definition of continuity using ε and δ.
Example of finding δ from a given ε for a simple f(x,y).
Several examples of lim(x,y)→(x0,y0) f(x,y) does not exist
and two path Theorem.
fig 14.13,
fig 14.14,
fig 14.15.
Homework 06 (lectures 12-13).
Homework 06 solutions.
- Week 08: Spring Break, No class this week.
- Week 09: No quiz this week.
Lecture 13 (20230411)
Section 14.3:
Definition of continuity; Composition of continuous functions;
Partial derivatives. Implicit partial differentiation. Higher order partial derivatives.
Definition of differentiability;
Further explanation of tangency of two curves in a plane.
Lecture 14 (20230413)
Section 14.3:
Defining differentiability in terms of existence of a tangent plane.
Further explanation of tangency of two surfaces in space.
Differentiability implies continuity.
Sufficient condition of differentiability in terms of continuity of first partial derivatives
and a counter example..
Continuity of second partial derivatives implies interchanging
order of two partial derivatives and a counter example.
Section 14.4: Application of differentiability to the Chain rule.
Remark on definition of
differentiability of f(x,y)
Homework 07 (lectures 14-15) (New! Added problem 7. New!! Added hint for problem 1)
.
Homework 07 solutions.
Homework 07 problem 7 solutions.
- Week 10: Quiz 05 (Section 14.2-14.3 (homework 06), Apr 18, 10:10AM).
Lecture 15 (20230418)
Section 14.4: Implicit partial differentiation revisited;
Section 14.5: Definition and examples of directional derivative.
Differentiability leads to formula of directional derivative;
Relation between gradient vector and directional derivatives, example and counter example;
Lecture 16 (20230420)
Section 14.5:
Properties of ∇f. Finding tangent and normal lines of a level curve of F(x,y)
using the gradient vector ∇F.
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.
Relation between Linearization and tangency.
Homework 08 (lectures 16-17) .
Homework 08 solutions.
- Week 11: Quiz 06 (Section 14.4-14.5, Apr 25, 10:10AM).
Lecture 17 (20230425)
Section 14.6: Linearization (Linear Approximation) of f(x,y) near (x0,y0);
Error Estimate for Linear Approximation.
Section 14.7: Extreme Values and Saddle Point: First Derivative Test, Second Derivative Test.
Lecture 18 (20230427) (New. Added page 11)
Section 14.7: Proof of Second Derivative Test; Find absolute extremes using gradient analysis.
Homework 09 (lectures 18-20).
Homework 09 solutions.
- Week 12: Quiz 07 (Section 14.6-14.7(homework 08 only), May 02, 10:10AM).
Lecture 19 (20230502)
Section 14.8: Constrained Optimization: Method of Lagrangian Multiplier.
Lecture 20 (20230504)
Section 14.8: More examples.
Section 14.9: Derivation of Taylor's formula for two variables.
- Week 13: Midterm 02 (Section 14.2-14.9, May 09, 10:10AM).
Lecture 21 (20230511)
Section 15.1:
Double integrals defined as limit of Riemann sum.
Fubini's Theorem and iterated integrals on rectangular regions.
Section 15.2: Fubini's Theorem and iterated integrals on rectangular-like regions.
Interchanging order of integration for double (iterated) integrals.
Section 15.3: Area by double integrals (special case of section 15.2).
Section 15.4: Integration in polar form. Derivation of dA = r dr dθ.
Homework 10 (lectures 21-22).
Homework 10 solutions.
- Week 14:
Lecture 22 (20230516)
Section 15.4: Integration in polar form. More examples on identifying limits of
integration for of dA = r dr dθ and dθ r dr, respectively.
Section 15.5: Triple Integrals. Method of identifying limits of integration for
various choice of dV = dx dy dx, or dy dz dx, etc.
Lecture 23 (20230518)
Section 15.5: Triple Integrals continued.
Section 15.7: Triple Integrals in Cylindrical Coordinates.
Method of identifying limits of integration for
various choice of dV = r dr dθ dz, dz r dr dθ, etc.
Homework 11 (lectures 23-24).
Homework 11 solutions (updated again).
- Week 15: Quiz 08 (Section 15.1-15.4, May 23, 10:10AM).
Lecture 24 (20230523)
Section 15.7: Triple Integrals in Spherical Coordinates.
Method of identifying limits of integration for
various choice of order in dV = ρ^2 sinφ dρ dθ dφ.
Section 15.8: Substitution in Multiple Integrals: Formula of Jacobian for double integrals.
Lecture 25 (20230525)
Section 15.8: Deriving the formula of Jacobian for double and triple integrals. Examples.
Section 16.1: Definition of line integrals as limit of Riemann sum.
Computation of scalar line integrals by means of parametrization of the curve.
Homework 12 (lectures 25-26).
- Week 16: Quiz 09 (Section 15.5 and 15.7, May 30, 10:10AM).
Lecture 26 (20230530)
Section 16.1: Definition of scalar line integral and examples.
Section 16.2: Computation of Work done along the path by means of correctly
oriented parametrization of the curve.
Explanation of simple closed curves.
Notation of circulation and flux over a simple closed curve. Examples.
Lecture 27 (20230601)
More examples of circulation and flux over a simple closed curve.
Fundamental Theorem of Line Integrals. Theorem and proof.
Definition of conservative fields and
equivalent characterization in terms of
closed loop and potential function. Construction of potential function
for conservative fields. Theorem: Conservative fields satisfy the Component
Test. Inverse statement for simply connected domains.
Example on how to find potential function for closed fields on
simply connected domains.
Homework 13 (lectures 25-26).
Homework 13 solutions.
- Week 17: Quiz 10 (Section 15.8, 16.1, 16.2, Jun 06, 10:10AM).
Lecture 28 (20230606)
Lecture 29 (20230608)
- Week 18: Final Exam (Section 15.1-16.4, Jun 13, 10:10AM).
- Homework assignments of this semester (temporary):
Chapter 10.
Chapter 14.
Chapter 15.
Chapter 16.