MATH 180
Calculus II
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 is the second part of an accelerated single-variable calculus sequence. It is designed to deepen students’ understanding of fundamental calculus concepts through a balance of theoretical analysis and computational techniques. The course covers transcendental functions, methods of integration, sequences and series, and calculus in alternative coordinate systems, with an emphasis on formal definitions, logical reasoning, and clear mathematical communication. Students will develop the ability to perform advanced calculations, construct and justify mathematical arguments, and analyze relationships between key concepts. The course aims to strengthen both technical proficiency and conceptual understanding, providing a foundation for further study in advanced mathematics such as differential equations and multivariable calculus.
Required Material:
James Stewart, Daniel K. Clegg and Saleem Watson, Calculus: Early Transcendentals, Cengage Learning, 2020.
Learning Objectives:
1. Understand key concepts in transcendental functions, integration, and series.
2. Perform advanced calculus computations.
3. Use logical reasoning to justify mathematical results.
4. Apply calculus to solve theoretical and practical problems.
Course Outline:
Week 1:
Lecture 1-5: Transcendental Functions
Lecture 1: Exponential Functions
o Define exponential functions and their properties
o Explore limits and continuity
Lecture 2: Logarithmic Functions
o Define logarithms as inverses of exponentials
o Differentiate and integrate logarithmic functions
Lecture 3: Inverse Functions
o Define inverse functions formally
o Derivatives of inverse functions
Lecture 4: Inverse Trigonometric Functions
o Define inverse trigonometric functions
o Compute derivatives and integrals
Lecture 5: Applications of Transcendental Functions
o Model growth and decay
o Solve related rate problems
Assignment 1
More specific requirements (e.g., formatting requirements, deadlines, etc.) will be provided in the course.
Week 2:
Lecture 6-10: Techniques of Integration I
Lecture 6: Integration by Parts
o Derive the integration by parts formula
o Evaluate integrals using this method
Lecture 7: Trigonometric Integrals
o Evaluate integrals involving powers of sine and cosine
o Use trigonometric identities
Lecture 8: Trigonometric Substitution
o Apply substitution to simplify integrals
o Solve geometric integrals
Lecture 9: Partial Fractions I
o Decompose rational functions
o Integrate simple rational expressions
Lecture 10: Partial Fractions II
o Handle repeated and irreducible factors
o Apply partial fraction decomposition
Lecture 11-15: Integration and Applications
Lecture 11: Improper Integrals
o Define improper integrals
o Determine convergence and divergence
Lecture 12: Applications of Integration
o Compute area, volume, and arc length
o Apply integrals to physical problems
Lecture 13: Numerical Integration
o Use trapezoidal and Simpson’s rules
o Estimate errors
Lecture 14: Differential Equations I
o Solve separable differential equations
o Work with initial value problems
Lecture 15: Differential Equations II
o Model exponential growth and decay
o Apply differential equations
Midterm Exam: multiple choice and problem-solving questions
(Contains topics in Lecture 1-15)
Week 3:
Lecture 16-20: Parametric Equations and Polar Coordinates
Lecture 16: Parametric Equations
o Define parametric curves
o Eliminate parameters
Lecture 17: Calculus with Parametric Curves
o Compute derivatives and arc length
o Analyze motion
Lecture 18: Polar Coordinates
o Convert between polar and Cartesian coordinates
o Graph polar equations
Lecture 19: Calculus in Polar Coordinates
o Compute area and arc length
o Solve applications
Lecture 20: Polar Curves
o Analyze symmetry and graphing techniques
o Study applications
Lecture 21-25: Sequences
Lecture 21: Sequences and Limits
o Define sequences
o Determine convergence
Lecture 22: Monotonic Sequences
o Study boundedness and monotonicity
o Apply convergence theorems
Lecture 23: Divergence
o Analyze divergent sequences
o Study infinite limits
Lecture 24: Squeeze Theorem
o Apply the squeeze theorem to sequences
o Prove convergence
Lecture 25: Recursive Sequences
o Define recursive sequences
o Analyze behavior
Assignment 2
More specific requirements (e.g., formatting requirements, deadlines, etc.) will be provided in the course.
Week 4:
Lecture 26-30: Series
Lecture 26: Infinite Series
o Define infinite series
o Compute partial sums
Lecture 27: Basic Convergence Tests
o Study geometric series and p-series
o Determine convergence
Lecture 28: Comparison Tests
o Apply comparison and limit comparison tests
o Analyze series
Lecture 29: Ratio and Root Tests
o Apply ratio and root tests
o Determine convergence
Lecture 30: Alternating Series
o Study alternating series test
o Estimate errors
Lecture 31-35: Power Series
Lecture 31: Power Series
o Define power series
o Find radius of convergence
Lecture 32: Taylor Series
o Construct Taylor and Maclaurin series
o Approximate functions
Lecture 33: Applications of Series
o Use series for approximation
o Estimate errors
Lecture 34: Functions as Series
o Represent functions as power series
o Manipulate series
Lecture 35: Advanced Applications
o Apply series to differential equations
o Model problems
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 based on the topics covered in lectures, providing opportunities to apply the concepts and techniques learned in class. The problems will focus on key topics such as integration methods, sequences and series, and applications of calculus. Students are expected to present clear and well-organized solutions with logical reasoning. All work should include step-by-step explanations and proper mathematical notation. Where appropriate, graphs or diagrams may be used to support the solutions. Assignments must be submitted as written documents, and late submissions will incur a penalty unless prior arrangements are made.
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.
