MATH 170
Calculus II
Mathematical and Physical
Summer Term 2026 |
Instructor: Staff |
Total sessions: 35 Sessions |
Office Hours: TBA |
Session Length: 145 Minutes |
Classroom: TBA |
Credits: 3 Units |
Class Length: 7 Weeks |
Language: English |
Course Overview:
This course builds on foundational calculus and introduces core topics such as advanced integration techniques, sequences and series, and first-order differential equations. Students will learn how to evaluate integrals using a variety of methods, analyze the behavior of sequences and infinite series, and represent functions through power series. The course also explores parametric equations and polar coordinates, extending calculus to new types of curves and applications. Throughout, emphasis is placed on developing both computational fluency and a deeper understanding of key concepts, including convergence and mathematical reasoning.
Required Material:
James Stewart, Daniel K. Clegg and Saleem Watson, Calculus: Early Transcendentals, Cengage Learning, 2020.
Additional supporting materials, including selected notes and problem sets, will be provided throughout the course.
Learning Objectives:
1. Apply advanced integration techniques to evaluate definite and indefinite integrals
2. Determine convergence of sequences and series using appropriate tests
3. Represent and approximate functions using power series
4. Solve and interpret first-order differential equations in applied contexts
Course Outline:
Week 1:
Lecture 1-5: Foundations & Rigorous Calculus
Lecture 1: Continuity and Integral Review
o Review continuity and definite integrals
o Interpret integrals in applied contexts
Lecture 2: Formal Definition of Limits (ε-δ)
o ε-δ definition of limits
o Proving limits rigorously
Lecture 3: Mean Value Theorem (MVT)
o Rolle’s Theorem and MVT conditions
o Applications to function behavior
Lecture 4: Intermediate & Extreme Value Theorems
o IVT and EVT statements
o Existence and optimization problems
Lecture 5: Mathematical Induction
o Proof structure and logic
o Applications in sequences and formulas
Assignment 1
More specific requirements (e.g., formatting requirements, deadlines, etc.) will be provided in the course.
Week 2:
Lecture 6-10: Integration Techniques I
Lecture 6: Substitution Method
o Change of variables in integrals
o Reverse chain rule applications
Lecture 7: Integration by Parts
o Formula and derivation
o Strategy (LIATE rule)
Lecture 8: Trigonometric Integrals I
o Powers of sin and cos
o Even/odd power strategies
Lecture 9: Trigonometric Integrals II
o Tangent and secant integrals
o Identity-based simplification
Lecture 10: Strategy in Integration
o Choosing appropriate techniques
o Multi-method problems
Week 3:
Lecture 11-15: Integration Techniques II
Lecture 11: Trigonometric Substitution
o Substitution for radicals
o Triangle interpretation
Lecture 12: Partial Fractions I
o Linear factor decomposition
o Setup and solving
Lecture 13: Partial Fractions II
o Repeated and quadratic factors
o Advanced decomposition
Lecture 14: Improper Integrals I
o Infinite intervals
o Convergence definition
Lecture 15: Improper Integrals II
o Comparison tests
o Divergence criteria
Midterm Exam: multiple choice and problem-solving questions
(Contains topics in Lecture 1-15)
Week 4:
Lecture 16-20: Applications of Integration
Lecture 16: Constructing Functions via Integration
o Accumulation functions
o Initial value problems
Lecture 17: Area and Volume
o Disk and washer methods
o Physical applications
Lecture 18: Arc Length
o Arc length formula derivation
o Applications to curves
Lecture 19: Parametric Arc Length & Area
o Arc length in parametric form
o Area under parametric curves
Lecture 20: Gamma Function
o Definition and properties
o Connection to factorials
Week 5:
Lecture 21-25: Sequences and Series
Lecture 21: Sequences and ε-N Definition
o Formal definition of sequence limits
o Convergence proofs
Lecture 22: Monotone Convergence Theorem
o Bounded monotone sequences
o Proof ideas and applications
Lecture 23: Infinite Series Basics
o Partial sums and convergence
o Divergence concepts
Lecture 24: Convergence Tests I
o Comparison tests
o Geometric and p-series
Lecture 25: Convergence Tests II
o Ratio and root tests
o Alternating series
Assignment 2
More specific requirements (e.g., formatting requirements, deadlines, etc.) will be provided in the course.
Week 6:
Lecture 26-30: Power Series & Taylor Series
Lecture 26: Power Series Fundamentals
o Definition and manipulation
o Radius and interval of convergence
Lecture 27: Finding Radius of Convergence
o Ratio/root test applications
o Endpoint analysis
Lecture 28: Taylor & Maclaurin Series
o Series construction
o Common expansions
Lecture 29: Taylor Series Applications
o Approximation and error
o Modeling functions
Lecture 30: Advanced Series Applications
o Combining series techniques
o Function representation
Week 7:
Lecture 31-35: Parametric, Polar & Differential Equations
Lecture 31: Parametric Curves
o Graphing and interpretation
o Eliminating parameters
Lecture 32: Polar Coordinates
o Conversion and graphing
o Curve analysis
Lecture 33: Area in Polar Coordinates
o Area formula
o Applications
Lecture 34: First-Order Differential Equations
o Separable equations
o Linear equations
Lecture 35: Applications of Differential Equations
o Growth and decay models
o Real-world modeling
Final Exam: multiple choice and Problem-solving questions
(Contains topics in All Lectures)
Grading Assessment:
Assignment 1 |
15% |
Assignment 2 |
15% |
Midterm |
30% |
Final exam |
30% |
Seminar Participation |
10% |
Total |
100% |
Assignments:
Students are required to complete two assignments during the semester, each assignment will consist of a set of problems that align with the topics discussed in lectures, offering students the opportunity to apply the theories and techniques learned in a practical setting. Problems will focus on key course topics such as integration methods, sequences and series, and differential equations. Students are expected to present clear and well-structured solutions, with logical reasoning and proper mathematical notation throughout. Each solution should include step-by-step explanations that demonstrate understanding of the methods used. Assignments must be submitted as written documents. Late submissions may be subject to penalties unless prior arrangements have been approved.
Attendance:
Students are required to attend a weekly seminar led by TA to focus on the week's topic and deepen understanding. Seminar time assigned by TA. Seminar attendance counts toward the final grade.
Exams:
The examinations in this course consist of multiple choice and Problem-solving questions. The final exam is cumulative.
Final Evaluation:
Letter Grade |
Percentage (%) |
Letter Grade |
Percentage |
A+ |
≥95 |
C+ |
64-67 |
A |
89-94 |
C |
60-64 |
A- |
84-88 |
C- |
56-59 |
B+ |
79-83 |
D+ |
54-56 |
B |
73-78 |
D |
50-53 |
B- |
68-72 |
F |
≤50 |
General Policies:
Academic integrity
Academic integrity is the cornerstone of academia and requires students and researchers to maintain honesty, fairness, trust and responsibility in all academic activities. It includes not only avoiding dishonest behaviors such as plagiarism, cheating, and falsifying data, but also requires taking responsibility for one's own academic actions and ensuring that all work is done independently and accurately cites the research of others. Violations of academic integrity can result in severe academic penalties, such as zero grades, suspension or even expulsion, and can cause serious damage to an individual's reputation and future career. Upholding academic integrity is therefore essential to promoting a fair academic environment and facilitating the authentic dissemination of knowledge.
Accessible Resources Policy
The policy ensures that all students, especially those with disabilities, are able to participate equally in school learning and activities. The school provides a wide range of accessibility resources including, but not limited to, specialized classrooms, hearing aids, Braille textbooks, assistive technology, and flexible testing arrangements. Students are required to apply to the school in advance and provide appropriate medical or psychological evaluations so that an individualized support plan can be developed for them. This policy is designed to remove barriers in the academic environment and to ensure that every student has access to equitable learning opportunities.
Withdrawal Policy
Students may choose to withdraw from a course within a specified period of time, and may not be able to do so after the expiration date. When withdrawing from a course, students are required to fill out a withdrawal form with a reason, which will be reviewed and processed on a case-by-case basis. Withdrawal from a course may not affect the student's academic performance. If a student withdraws from a course with incomplete requirements, a “W” may be assigned instead of a grade, depending on the course.
