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Njit Matlab Assignment 14

Math 112: Calculus II
Fall 2017 Course Syllabus

NJIT Academic Integrity Code: All Students should be aware that the Department of Mathematical Sciences takes the University Code on Academic Integrity at NJIT very seriously and enforces it strictly. This means that there must not be any forms of plagiarism, i.e., copying of homework, class projects, or lab assignments, or any form of cheating in quizzes and exams. Under the University Code on Academic Integrity, students are obligated to report any such activities to the Instructor.

Course Information

Course Description: Topics include integration, applications of integration, series, exponential and logarithmic functions, transcendental functions, polar coordinates, and conic sections.

Number of Credits: 4

Prerequisites: MATH 111 with a grade of C or better or MATH 132 with a grade of C or better.

Course-Section and Instructors

Math 112-001Professor V. Barreto-Aranda
Math 112-003Professor R. Kelly
Math 112-005Professor V. Barreto-Aranda
Math 112-009Professor R. Kelly
Math 112-011Professor J. Zaleski
Math 112-013Professor J. Porus
Math 112-017Professor J. Zaleski
Math 112-019Professor D. Schmidt
Math 112-023Professor N. Tsipenyuk
Math 112-029Professor N. Tsipenyuk
Math 112-101Professor P. Ward
Math 112-103Professor J. H . Ro

Office Hours for All Math Instructors: Fall 2017 Office Hours and Emails

Required Textbook:

TitleThomas' Calculus: Early Transcendentals
ISBN #978-0321981677
Notesw/ MyMathLab

University-wide Withdrawal Date:The last day to withdraw with a w is Monday, November 6, 2017. It will be strictly enforced.

Course Goals

Course Objectives

  • Students should (a) develop greater depth of understanding of integration and its importance in scientific and engineering applications, (b) learn about series, including their convergence properties and their use in representing functions, (c) gain experience in the use of approximation in studying mathematical and scientific problems and the importance of mathematically understanding and evaluating the accuracy of approximations, (d) learn new ways of mathematically representing curves and how to use calculus in these settings, and (e) learn alternative coordinate systems which are natural for many problems and learn how calculus can be applied in these systems.
  • Students should gain an appreciation for the importance of calculus in scientific, engineering, computer, and other applications.
  • Students should gain experience in the use of technology to facilitate visualization and problem solving.

Course Outcomes

  • Students have improved logical thinking and problem-solving skills.
  • Students have a greater understanding of the importance of calculus in science and technology.
  • Students are prepared for further study in mathematics as well as science, engineering, computing, and other areas.

Course Assessment: The assessment of objectives is achieved through homeworks, quizzes, and common examinations with common grading.


DMS Course Policies: All DMS students must familiarize themselves with, and adhere to, the Department of Mathematical Sciences Course Policies, in addition to official university-wide policies. DMS takes these policies very seriously and enforces them strictly.

Grading Policy: The final grade in this course will be determined as follows:

Quizzes, HW, and MATLAB15%
Common Midterm Exam I15%
Common Midterm Exam II20%
Common Midterm Exam III20%
Final Exam30%

Your final letter grade will be based on the following tentative curve.

A88 - 100C66 - 71
B+83 - 87D60 - 65
B77 - 82F0 - 59
C+72 - 76

Attendance Policy: Attendance at all classes will be recorded and is mandatory. Please make sure you read and fully understand the Math Department’s Attendance Policy. This policy will be strictly enforced. Students are expected to attend class. Each class is a learning experience that cannot be replicated through simply “getting the notes.”

Homework Policy: Homework is a requirement for this class.  Online homework will be completed with MyMathLab, which comes with a new copy of the textbook.  Access to it can also be purchased directly from the website.

MATLAB Assignments: MATLAB is a mathematical software program that is used throughout the science and engineering curricula. Two MATLAB assignments will be given during the semester; tutors are available to help students having difficulties in accordance with a posted schedule.

Quiz Policy: Quizzes will be given approximately once a week throughout the semester. They will be based on the lecture, homework and the in-class discussions. There will be 8-12 assessments given throughout the semester.

Exams: There will be three common midterm exams held during the semester and one comprehensive common final exam. Exams are held on the following days:

Common Midterm Exam ISeptember 27, 2017
Common Midterm Exam IIOctober 25, 2017
Common Midterm Exam IIINovember 29, 2017
Final Exam PeriodDecember 15 - 21, 2017

The time of the midterm exams is 4:15-5:40 pm for daytime students and 5:45-7:10 pm for evening students. The final exam will test your knowledge of all the course material taught in the entire course. Make sure you read and fully understand the Math Department's Examination Policy. This policy will be strictly enforced.

Makeup Exam Policy: To properly report your absence from a midterm or final exam, please review and follow the required steps under the DMS Examination Policy found here:

Mandatory Tutoring Policy: Based upon academic performance indicating a significant gap in understanding of the course material, students may receive a notice of being assigned to mandatory tutoring to assist in filling the gap. A student will have 2 points deducted from the course average for each instance in which the required tutoring is not completed by the stated deadline.

Cellular Phones: All cellular phones and other electronic devices must be switched off during all class times.

Additional Resources

Math Tutoring Center: Located in the Central King Building, Lower Level, Rm. G11 (See: Fall 2017 Hours)

Accommodation of Disabilities: Disability Support Services (DSS) offers long term and temporary accommodations for undergraduate, graduate and visiting students at NJIT.

If you are in need of accommodations due to a disability please contact Chantonette Lyles, Associate Director of Disability Support Services at 973-596-5417 or via email at lyles@njit.edu. The office is located in Fenster Hall Room 260. A Letter of Accommodation Eligibility from the Disability Support Services office authorizing your accommodations will be required.

For further information regarding self identification, the submission of medical documentation and additional support services provided please visit the Disability Support Services (DSS) website at:

Important Dates (See: Fall 2017 Academic Calendar, Registrar)

September 5, 2017TFirst Day of Classes
September 11, 2017MLast Day to Add/Drop Classes
November 6, 2017MLast Day to Withdraw
November 21, 2017TThursday Classes Meet
November 22, 2017WFriday Classes Meet
November 23 - 26, 2017R - SuThanksgiving Break - University Closed
December 13, 2017WLast Day of Classes
December 14, 2017RReading Day
December 15 - 21, 2017F - RFinal Exam Period

Course Outline

LectureSectionTopicAssignment in MyMathLabAssignment to Hand-in
16.1Volumes Using Cross Sections5, 9, 15, 17, 21, 28, 31, 3510, 36
26.1Volumes Using Cross Sections39, 43, 45, 47, 49, 51, 53, 5752
36.2Volumes Using Cylindrical Shells3, 5, 9, 11, 17, 19, 21, 25, 29, 3342, 47, 48
46.3Arc Length1, 2, 3, 4, 5, 7, 13, 2526
56.4Areas of Surfaces of Revolution9, 13, 15, 17, 19, 21, 2432
66.5Work3, 5, 6, 7, 9, 13, 15, 17, 188, 19
77.3Hyperbolic Functions1, 7, 9, 15, 17, 21, 23, 43, 45, 47, 49, 53, 55, 57, 8180
88.1/8.2Using Basic Integration Formulas; start Integration by PartsSection 8.1:  3, 5, 9, 10, 13, 15, 27, 33, 36, 3834, 43
98.2/8.3Finish Integration by Parts; start Trigonometric IntegralsSection 8.2:  3, 5, 11, 13, 23, 27, 29, 33, 35, 37, 39, 45, 47, 5528, 38, 46, 53
118.3/8.4Finish Trigonometric Integrals; start Trigonometric SubstitutionSection 8.3:  7, 9, 11, 17, 19, 21, 27, 31, 35, 37, 38, 39, 45, 65, 7163, 64, 68
128.4Trigonometric Substitution1, 5, 7, 11, 17, 19, 23, 29, 35, 37, 39, 41, 43, 5312, 20, 44, 57
138.5Integration of Rational Functions by Partial Fractions3, 7, 9, 11, 14, 16, 17, 1918
148.5Integration of Rational Functions by Partial Fractions23, 25, 27, 29, 33, 35, 39, 41, 45, 5530, 31, 38
158.7Numerical Integration3, 7, 13, 17, 21, 28Matlab Assignment
MATLAB #1 assigned:  DUE OCTOBER 30TH
168.8Improper Integrals1, 4, 6, 7, 9, 11, 13, 17, 21, 23, 25, 3116, 28
178.8Improper Integrals35, 39, 41, 47, 53, 55, 59, 61, 63, 67, 6954, 64, 71
1810.1Sequences3, 7, 9, 15, 17, 21, 23, 25, 31, 35, 37, 41, 45, 49, 5148, 50
1910.1/10.2Finish Sequences; start Infinite SeriesSection 10.1:  53, 61, 65, 67, 69, 79, 81, 87, 89, 9970, 74, 80
2010.2Infinite Series3, 5, 7, 13, 25, 29, 31, 37, 41, 43, 53, 55, 59, 61, 65, 69, 7163, 64
2110.3 Integral Test3, 6, 9, 11, 13, 19, 23, 25, 27, 29, 33, 35, 49, 5120, 34, 36
2310.4Comparison Tests1, 5, 18, 19, 21, 23, 254
2410.4Finish Comparison Tests; start Ratio and Root TestsSection 10.4:  28, 31, 32, 34, 37, 39, 41, 43, 47, 51, 5636, 40, 46
2510.5Ratio and Root Tests5, 7, 9, 18, 19, 21, 29, 31, 35, 42, 55, 57, 59, 6638, 56, 58
2610.6Alternating Series, Absolute vs. Conditional Convergence5, 7, 9, 10, 11, 13, 15, 19, 21, 23, 2512, 24
2710.6Alternating Series, Absolute vs. Conditional Convergence27, 34, 35, 37, 39, 41, 44, 47, 51, 5330, 42, 50
2810.7Power Series3, 5, 9, 11, 15, 19, 21, 23, 2722, 24
2910.7Power Series31, 37, 41, 43, 45, 53, 5432, 55
3010.8Taylor and Maclaurin Series3, 5, 8, 9, 11, 15, 18, 23, 29, 31, 3534
3110.9Convergence of Taylor Series1, 9, 10, 11, 13, 19, 20, 2516, 26
3210.9/10.10Finish Convergence of Taylor Series; start Binomial SeriesSection 10.9:  29, 35, 37, 39, 41, 43, 4931, 36, 48
3310.1Binomial Series and Applications of Taylor Series1, 3, 5, 13, 23, 25, 29, 31, 35, 39, 45, 49, 55, 6126, 40
3411.1Parametrizations of Plane Curves1, 3, 5, 7, 9, 1612
3611.1/11.2Finish  Parametrization of Plane Curves; start Calculus with Parametric CurvesSection 11.1  19, 21, 25, 27, 31, 33, 3932, 40
3711.2Calculus with Parametric Curves7, 9, 12, 13, 15, 21, 26, 28, 29, 31, 33, 3514, 47
3811.3Polar Coordinates1, 5, 7, 13, 17, 23, 27, 32, 37, 47, 51, 59, 60, 6138, 42
3911.4Graphing in Polar Coordinates1, 7, 9, 13, 17, 19, 25, 2718
4011.5Areas and Lengths in Polar Coordinates1, 7, 11, 13, 15, 1710
4111.5Areas and Lengths in Polar Coordinates21, 23, 27, 2824
42Review for Final

I am interested in modeling and analyzing various real world phenomena.  Some of the topics I have worked on and/or have an interest in working on are: Hydrodynamic Quantum Analogs, Logical circuit dynamics, Chaotic scattering, Cancer modeling, and Particle Accelerator Physics.  To find out more about my current research please refer to my research statement.

Cancer Drug Response:

According to the National Cancer Institute, almost 40% of men and women in the United States end up developing cancer in their lifetime, and total national expenditure on cancer is $125 Billion.  While we cannot hope to eradicate cancer in the near future, early detection and treatment can decrease death rates.  However, keeping a human being alive is not the main goal; it is to maintain their and their family's quality of life.  This can only be achieved through individualized treatment which increase efficacy and decrease toxicity.

Once again, borrowing statistics from the National Cancer Institute, over 60% of
cancer cases and over 70% of cancer related deaths occur in Africa, Asia, and South America, having the worst effect on the poorest populations.  Some of the easiest cancers to treat in the Western world, solid -- accessible tumors, are often fatal in poor nations.  In industrialized nations the answers to solid tumors is simple -- operate.  However, operation is generally quite complex, and costs a significant amount of money.  This can be remedied by the use of drug injections into solid tumors, which is cheap and does not require much skill.  We have developed mechanistic models for the fluidic interactions of the drug with the geometry and topography of the tumor coupled with statistical models of drug response for a population in order to achieve high predictive capabilities and show causality.  In oncological studies the data is represented by dose-response curves, which we also have from our model, however having
a model also gives us the ability to plot dose-response surfaces, which gives us a more fine-grained picture for the effects of the drug.

Hydrodynamic Pilot Wave Theory:

Source: John Bush

Source: Couder and Fort

For the past decade, scientists have been studying hydrodynamic systems, specifically walking droplets, that exhibit wave particle behavior and some quantum-like features.  These are generally modeled as quite complex integro-differential equations.  In recent years, mathematicians have developed discrete dynamical models for these walking droplets.

We analyzed and developed several models for walking in various geoemtries that verified and even extended what had been observed in the past.  Our analysis will aid in guiding experimentalists to develop new experiments that will improve our understanding of these systems.

Logical circuit dynamics:


Many biological systems including our brains and brains of other animals are capable arithmetic or more complicated logical operations.  In order to understand how biological systems become intelligent it is necessary to study logical circuits.

When one thinks of logical circuits, typically a simple-nondynamic electronic circuit comes to mind.  However, for biological systems the logical behavior isn't quite so linear.  Thought happens in a more nonlinear, perhaps chaotic, manner.  In more recent years electrical engineers have designed chaotic logical circuits, such as the RS flip-flop circuit.  We have modeled the chaotic dynamics of the threshold voltages of this circuit and analyzed these models.  We have also reproduced the chaotic RS flip-flop circuit and observed behavior similar to those of previous experiments.

Chaotic scattering:

Chaotic scattering has been studied from the early 70s and 80s in solitary wave collisions from the Phi-Four equation (called Kink-Antikink collisions).  These were mainly numerical studies that gave insight into the phenomena.  However, since the equation is so difficult to work with there has been very little analysis done.  In more recent years reduction techniques have been used to approximate the Phi-Four PDE with a system of ODEs and also as an iterated map.

We have gone further and developed a mechanical analog (a ball rolling on a special surface) of chaotic scattering in Kink-Antikink collisions.  This was done in order to conduct experiments.  In addition to experiments we have analyzed the system thoroughly, including the dissipation that comes from friction.  The experimental setup is shown bellow.


Distribution of metastases:

The proper prediction of how metastatic tumors are distributed can help save lives.

We used a model from Iwata et. al. (2000) and sought ways of simplifying it, improving upon it, and finding numerical solutions.  We saw that the Iwata model can be solved using an upwind scheme.  For the new models we focused on the effects of drugs on tumors, and simplifying the PDE into ODEs.  Drug affects was a major focus due to the lack of models taking drugs into account in the literature.  This is difficult to do, however, because drugs attack cells indiscriminately.  Probabilistically it has a bigger effect on larger tumors than smaller ones, but it seems as though a stochastic model is needed.

Alternate proof of Peixoto's theorem in 1-D:

Peixoto's theorem is one of the most important theorems in Dynamical Systems.  It was proved by Dr. Mauricio Matos Peixoto in 1962.  This proof is extremely involved - far too involved for most undergraduate students to follow.  We develop an alternate - pedagogical proof of the simpler 1-D case, with the goal of allowing senior undergraduate students to follow and understand the proof and consequently some of the ideas involved in the much bigger proof of the 2-D case.

Particle Accelerator Physics:

We numerically simulate the beam dynamics of the Energy Recovery Linear Particle Accelerator (ERL) design for Argonne National Lab's (ANL) Advanced Photon Source (APS).  The code BI, created by Ivan Bazarov, is benchmarked against our own code for simpler Accelerators.  Then the full ERL is simulated and the results analyzed.  We conclude that the ERL, if built, would theoretically be stable.  Therefore, it would be feasible to build it.

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