Math S115E — Calculus of Functions of One Variable II
Summer 2019
Course Description
In Math 115, we take complex problems and break them up into simpler ones. For instance, we
approximate the area of a region by subdividing it into rectangles. By breaking up a problem
into smaller and smaller parts, we get better and better approximations of the true solution. In
fact, sometimes we can compute the LIMIT of these approximations, giving us the best possible
answer. Otherwise, we can understand the ERROR involved in our approximation.
Over the semester, we will develop techniques for solving problems in geometry, economics, biology and physics. These techniques include Riemann sums, integration, differential equations, Taylor series, and parametric and polar equations. We will also appreciate
the beautiful way these ideas fit together within the theory of calculus, the mathematics of
change.
Course Goals
Math 115 focuses on the relationship between net changes and rates of change. Over the
course of the semester, you will learn how to...
• solve integral and differential equation problems.
• approximate integrals and analyze the error in those approximations.
• determine convergence or divergence of infinite sequences and series.
• represent functions with Taylor series.
• find lengths, areas and volumes of geometric objects.
• model curves using parametric and polar equations.
Active Learning
We learn best by doing. During our class meetings over Zoom, we will ask you to work on
problems, respond to polls and discuss concepts with your peers. Most importantly WE ENCOURAGE MISTAKES. Each misstep leads to new discoveries.
Pre-class Explorations
Before each class, you will complete online explorations that will help orient you to the material
that will be discussed in class. You will have up to six attempts to submit your exploration
answers online, and your highest score will be recorded as a homework grade.
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Campuswire Discussions
We use a discussion platform called Campuswire to facilitate discussions as you work on the
pre-class explorations (and other questions that may arise throughout the course). We hope
to generate a running dialogue about the material in the course. Please do not post full solutions to exploration or problem set questions on the discussions; hints and explanations are
welcome.
Problem Sets
We realize how well we understand something when we try to explain it to someone else. Our
weekly problem sets provide an opportunity to EXPLAIN your understanding of the material in
Math 115. Problem sets are evaluated based on both understanding and clarity—a correct
numerical solution alone is not enough to receive full credit. We encourage you to form study
groups and attend office hours regularly. Problems will be graded on the following scale:
3 : Work is completely accurate, essentially perfect. Ideas are fully developed.
Work is neat and easy to read. Complete sentences are used where appropriate.
2 : Work is good, on the right track, but development of ideas is incomplete.
Work is hard to read or disorganized.
1 : Work is sketchy, with some correct ideas, but mostly on the wrong track.
Work is messy or illegible.
0 : Answer is completely incorrect.
Work is minimal or non-existent. No explanations are given.
Problem sets are posted on Canvas and are due each week. You will submit your problem
sets as a .pdf file online by taking a picture or scanning your written work. We recommend an
app called Evernote Scannable for capturing your problem sets.
Exams
We will have two midterm exams and a cumulative final exam.
• Midterm 1
Week 2
• Midterm 2
Week 4
• Final Exam
Week 5
Calculators
Calculators and electronic devices are NOT allowed on any test in this course. The
midterms and exams are designed to minimize computation and emphasize conceptual understanding.
You may use calculators and computers while working on the pre-class explorations and
problem sets, and some homework problems may require more extensive computations than
you can do by hand. Make sure to reflect on how and why you are using these devices so that
their computing power does not obscure your understanding of the concepts.
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Grading
Our grading policy rewards improvement over the course of the semester. Your grade will be
determined as follows:
Midterm Score: Take the greater score of
• 50% Midterm 1 + 50% Midterm 2
• 40% Midterm 1 + 60% Midterm 2
Course Score: Take the greater score of
• 45% Final Exam + 40% Midterm Score + 15% Homework
• 55% Final Exam + 30% Midterm Score + 15% Homework
At the end of the semester, numerical scores will be converted into letter grades according to
the following scale:
Grade
Percentage
A+/A/A100 - 90
B+/B/B89 - 80
C+/C/C79 - 65
D
64 - 50
F
< 50
Resources
Textbook: Calculus, Early Transcendentals, 8th edition by James Stewart. We will assign
problems from this text for homework. WARNING : these exercises differ in older editions
of the book. The Yale Bookstore prints a special version of the text that contains only the
chapters we use in Math 115.
Canvas: We have a Canvas site containing all course documents and assignments. peer
tutors, homework, etc. The section sites contain pre-class modules and information
posted by your instructor.
Zoom: Our class meetings will be held over Zoom, a video conference platform. Zoom
can be downloaded for free at the web address: https://zoom.us/
Office Hours: Your instructor has weekly office hours on Zoom. Office hours are a great
opportunity to ask questions and gain a deeper understanding of the material.
Students with Disabilities
Yale provides accommodations to students with documented disabilities. You may request a
disability-related accommodation or service by contacting Yale’s Resource Office on Disabilities. More information is provided on the office website, https://rod.yale.edu/
Inclusive Learning Environment
The University adheres to the philosophy that all community members should enjoy an environment free of any form of harassment, sexual misconduct, discrimination, or intimate partner violence. If you have been the victim of sexual misconduct, we encourage you to report
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this. If you report this to a faculty/staff member, they must notify our college’s Title IX coordinator about the basic facts of the incident (you may choose to request confidentiality from the
University). If you encounter sexual harassment, sexual misconduct, sexual assault, or discrimination based on race, color, religion, age, national origin, ancestry, sex, sexual orientation, gender identity, or disability please contact the Title IX Coordinator, Stephanie Spangler,
at stephanie.spangler@yale.edu 203.432.4446 or any of the University Title IX Coordinators,
who can be found at http://provost.yale.edu/title-ix/coordinators
Academic Honesty
At Yale, academic honesty is taken very seriously. Please take a moment to read the above
homework and exam policies in Math 115, so that you can be sure to follow them. In particular,
the use of calculators, notes, cell phones, or any other aid on our exams is forbidden. If
you have any questions about our policies, please feel free to ask your instructor; they will be
happy to answer them for you.
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Schedule
Section
5.1
5.2
5.3
5.4
5.5
7.1
7.2
7.3
7.5
9.1
9.3
6.1
7.7.1
7.7.2
7.8.1
7.8.2
11.1
11.2
11.3
11.4
11.5
11.6.1
11.6.2
11.7
11.8
11.9
11.10.1
11.10.2
11.11
6.2
6.3
10.1
10.2.1
10.2.2
10.3
10.4
Topic
Areas and Distances
The Definite Integral
The Fundamental Theorem
Indefinite Integrals
The Substitution Rule
Integration by Parts
Trig Integrals
Trig Substitutions
Strategies for integration
Differential Equations
Separation of Variables
Areas Between Curves
Integral Approximation
Approximation Error
Improper Integrals
The Comparison Test
Sequences
Series
The Integral Test
The Comparison Tests
Alternating Series
Absolute Convergence
The Ratio Test
Strategies for Testing Series Convergence
Power Series
New Power Series from Old
Taylor Series
Taylor Polynomials and Error
Taylor Polynomial Approximations
Volumes of revolution – washers
Volumes of revolution – shells
Parametric Functions
Tangents & Areas with Parametric Curves
Lengths with Parametric Curves
Polar Coordinates
Area with Polar Curves
Week
1
1
1
1
1
1
1
2
2
2
2
2
2
2
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5