University of the Philippines Manila
College of Arts and Sciences
Department of Physical Sciences and Mathematics
Mathematical and Computing Sciences Unit
Math 84 Course Syllabus
Second Semester, A.Y. 2025-2026
I. Course Catalogue Description
Course Number
Math 84
Course Title
Essentials of Analysis II
Course Description Techniques of integration; improper integrals; parametric equations; polar
coordinates; lines in space, planes, cylindrical surfaces, surfaces of revolution, and
quadric surfaces; vectors and vector-valued functions
Prerequisite
Math 83 or equivalent
Course Credit
5 units
Number of hours
4 hours (lecture), 3 hours (laboratory) [per week]
II. Course Content
0. Course Introduction
1. Techniques of integration and improper integrals
1.1. Review of formulas of integration and integration by substitution
1.2. Integration by parts
1.3. Trigonometric integrals
1.4. Integration by trigonometric substitution
1.5. Integration of rational functions by partial fractions
1.6. Integration by tangent half–angle substitution
1.7. Improper integrals
1.8. DiXerential equations
2. Parametric curves, polar coordinates, and surfaces
2.1. Conic sections
2.2. Parametric curves and the calculus of parametric curves
2.3. Polar coordinates and graphs of equations in polar coordinates
2.4. Conic sections in polar coordinates
2.5. Tangent lines to, areas enclosed by, and arc length of polar curves
2.6. The three–dimensional Cartesian coordinate system
2.7. Cylindrical surfaces
2.8. Surfaces of revolution
2.9. Quadric surfaces
3. Vectors, lines and planes in space, and vector–valued functions
3.1. Vectors in plane and in space
3.2. Magnitude and direction angles of vectors
3.3. Vector operations; dot and cross products of vectors
3.4. Scalar and vector projections; applications of the dot and the cross product of vectors
3.5. Lines and planes in space
3.6. Vector–valued functions
3.7. Calculus of vector–valued functions
3.8. Moving trihedral, arc length, and parametrization using arc length
3.9. Curvature
3.10. Motion in space; normal and tangential components of acceleration
4. DiXerential calculus of functions of several variables
4.1. Functions of several variables, level curves, and level surfaces
4.2. Limits and continuity of functions of several variables
Prepared by: John Justin C. Mesias, M.Sc.
4.3. Partial derivatives
4.4. Higher-order derivatives, the total diXerential, and tangent plane approximation
4.5. The Chain Rule and implicit diXerentiation
III. Course References
1. Anton, Bivens, Davis. Calculus: Early Transcendentals, 12th ed. Wiley, 2021.
2. Leithold. The Calculus 7. HarperCollins College Div, 1995.
3. Stewart. Calculus, Metric Edition, 9th ed. Brooks/Cole, 2020.
4. UP Diliman Institute of Mathematics. Math 22 Module. 2018
5. UP Diliman Institute of Mathematics YouTube Channe
Lecture: youtube.com/playlist?list=PL2kta8PKtfXTTRXytN5Az14dyfEJzlexo&si=Tod54zgeIKIMifXE
Discussion:
youtube.com/playlist?list=PL2kta8PKtfXShBnSy4R6Ark2ueQP349zL&si=ECpnYA7wZy8Ag9uh
Math 108 (Logic): https://youtube.com/playlist?list=PL2kta8PKtfXRhfYlhNTCA1HfjGE4Z1kHF
IV. Course Requirements and Grading Scale
Raw
[93, 100]
[90, 93)
[87, 90)
[84, 97)
[80, 84)
[75, 80)
Grade
1.00
1.25
1.50
1.75
2.00
2.25
Raw
[70, 75)
[65, 70)
[60, 65)
[55, 60)
[0, 55)
Grade
2.50
2.75
3.00
4.00
5.00
Lecture Exams
Laboratory Exams
Lecture Activities
Laboratory Activities
Final Exam
TOTAL
25%
25%
10%
15%
25%
100%
V. Class Policies
1. The deadline for dropping the course is on April 8 (Wed) A student is considered to have oXicially
dropped the course only upon the submission of the instructor’s copy of the dropping slip. The
deadline for filing Leave of Absence is on April 23 (Thu).
2. The Reading/Wellness Break is on March 31 – April 1.
3. Any form of academic dishonesty is subject to a grade of 5.0.
4. A student is allowed to take a make–up exam provided that they can present a valid excuse for
missing the regular exam.
5. All students are required to take the final exam. No one is exempted.
6. No special projects are to be given to compensate for either low or failing grades.
7. No cancellation of the exams with the lowest marks.
8. All students are required to get a minimum of 50% in each of the components of the course. If one
component goes below 50%, despite of a passing overall standing, the student automatically gets a
4.00.
Prepared by: John Justin C. Mesias, M.Sc.