# MATH-UA 122: Calculus II, Summer 2015

**Location:** SILV
414

**Time:** 5:45PM — 7:50PM on Mondays, Tuesdays, Wednesdays, and Thursdays

(Summer session 2: July 6 — August 13)

**Lecturer**: Mark Kim, markhkim *[at]* math *[dot]* nyu *[dot]* edu

Jump to: general information | course logistics: office hours / textbook / attendance / homework / exams / grading policy | course notes

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### General Information

This is Calculus II, the second part of the standard three-semester calculus sequence offered at the Courant Institute of New York University. There are five major topics of study in Calculus II.

**1. Techniques of Integration.**You might remember from high school
geometry that the area of a disk with radius is ,
but have you ever wondered *why*? The disk of radius centered at
the origin of the coordinate plane is precisely the region enclosed by
the graph of two functions and . Since the area of the disk is the sum of the
area “under” and the area “over” , the formula for the
area of the disk should be

The value of the above
integral
is . Note, however, that we do not yet know how to
integrate functions of the form . The first
goal of the course is to introduce advanced techniques of
integration–such as *integration by parts*, *integration by
substitution*, and *integration by partial fraction decomposition*–that
allow us to handle complicated functions.

**2. Mathematical Modeling.**With integration techniques in our toolbox,
we can finally shed some light to the mantra you might have heard
before: *calculus is the language of science and engineering.* Our world
is made up of matter, and the
study of the rules that govern the behaviors of matter is known as
mechanics. The study of the
objects around us can be adequately carried out via *Newtonian
mechanics*, whose
flagship equation is the one many of you are familiar with:

Newton’s second law of motion. .

In essence, the equation gives a precise account of what
*force* is in terms of
*mass*, which describes the amount
of matter, and
*acceleration*, which
describes the movement of matter. The acceleration of an object can
change over time, so it would be better to rewrite the above equation as
follows:

But what *is* acceleration? It is the rate of change of
*velocity*, which is the rate
of change of
*position*. In
other words, if we consider an object of invariable mass with
position at time , then the net force acting on the
object at time is given by the formula

This is a *differential
equation*, which
describes function with derivatives of another function: in
this case, . Not only could we compute the
force from the position of the object, we could also compute the
position of the object from the force acting on it: indeed, the
fundamental theorem of calculus implies that integrating twice yields the position of the object. Indeed, calculus is
essential in understanding the relationship among position, velocity,
acceleration, and force–these concepts can then be used to understand
the nature of moving objects.

What we have done here is *mathematical modeling*: we gave a
mathematical description of a phenomenon, and then we analyzed it.
Taking a cue from this, we note that the second goal of the course is to
illustrate the basic ideas of mathematical modeling, with particular
attention to the language of ordinary differential equations.

**3. Approximation.** You might remember *Newton’s
method* from Calculus
I, which is a method of approximating the zeros of a function. In many
applications of calculus, it is rather difficult to compute the solution
of a differential equation precisely. Sometimes the integral is too
difficult; sometimes we’re not even sure which integral we should be
computing! The third goal of the course is to study a number of
approximation methods, which allow us to obtain results that are not
quite correct but “good enough”.

**4. Sequences and Series.** Consider *Zeno’s dichotomy
paradox*:

That which is in locomotion must arrive at the half-way stage before it arrives at the goal.

In other words, if we want to travel a mile, we first have to get to the half-mile point, and then to the -mile point, and then to the -mile point, and so on. Since, Zeno argues, we have to pass through infinitely many points, we can never get to the desired destination. This is absurd, of course, as we know we are capable of traveling a mile. What is the problem here?

For simplicity, let us assume that it takes twenty minutes to travel a
mile. It then takes ten minutes to travel half a mile, five minutes to
travel quarter a mile, and so on. Zeno tacitly assumes that the
*infinite sum*

cannot be finite. In other words, Zeno asserts that we cannot get to the
destination *in finite time*.

But we already know this is not true. The above infinite sum must equal
20, as we already *know* it takes twenty minutes to travel a mile. How
do we make sense of an infinite sequence of numbers that sums to a
finite number?

It turns out that the concept of
*limit*
introduced in Calculus I plays a crucial role in understanding the
notions of infinite sequences and series. Therefore, the fourth goal of
the course is to study the concept of limit in the context of infinite
sequences and series, and to develop an array of tools for analyzing
sequences and series.

**5. Analytic Geometry of the Plane.** Finally, this course serves as a
preparation for Calculus
III, which
covers differentiation and integration theory of functions of several
variables. For this purpose, it is imperative that you leave this course
with a thorough understanding of the geometry of the coordinate plane.
The fifth goal of the course is to introduce methods of describing a
large class of curves in the coordinate plane, called *parametrization*,
and to survey differentiation and integration theory of parametrized
curves.

### Course Logistics

**Office Hours** will be held in WWH
1109 on Wednesdays from 3 pm to 4 pm. You may also email me to ask
questions or schedule an appointment.

The mathematics department offers free tutoring in WWH 524: see the weekly schedule here.

**The course textbook** is *Essential Calculus: Early Transcendentals*
(2e) by James Stewart.

**Attendance** is not mandatory, and there will not be any in-class
quizzes. If you must miss a class, please talk to me *in advance* to
find a way to submit your homework write-up.

**Homework** will be assigned every day.

- On Mondays, Tuesdays, and Wednesdays, I will assign short exercises that reinforce the material covered in the lecture. Weekday assignments will be due on the day after they are assigned.
- On Thursdays, I will assign longer, more challenging problems for you to mull over through the weekend. Thursday assignments will be due on the following Monday.

See the course notes for the details on each assignment. Given the rapid pace of the course, I unfortunately cannot allow late submissions of homework write-ups.

You are encouraged to collaborate on the assignments. Nevertheless, the final write-up must be in your own words. Any act of plagiarism will result in an immediate “F” for the course: see the academic integrity guidelines for details.

- Exam I, on July 16;
- Exam II, on July 30;
- Final Exam, on August 13.

For courses like this one, exams are necessarily cumulative–please make
an effort to keep up with the course. Make-up exams will be administered
on a case-by-case basis: if you must miss an exam, please talk to me *in
advance* to find out whether you qualify for a make-up exam.

**Your final grade** for the course will be determined as follows. You
will be given a numerical grade, out of 100, based on the following
criteria:

- Weekly assignments: 10%
- Weekend assignments: 15%
- Exam I: 25%
- Exam II: 25%
- Final Exam: 25%

The numerical grade will then be converted to a letter grade in accordance with the following scheme:

(The *minimum passing grade* for taking further courses in the
mathematics department is a **C**.)

### Course Notes

- Diagnostic Test
- Set Theory
**Week 1**

July 6 - Riemann sums; approximate integration; the fundamental theorem of calculus : §5.2, §5.4, §6.5

July 7 – Integration by substitution; integration by parts: §5.5, §6.1

July 8 – Trigonometric integrals and substitutions; partial fractions: §6.2, §6.3

July 9 – Partial fractions; improper integrals: §6.3, §6.6

Weekend Problem Set 1 / Solutions**Week 2**

July 13 - Areas between curves; volumes of solids of revolution: §7.1, §7.2

July 14 – Arc length; area of a surface of revolution: §7.4, §7.5

July 15 – Review

July 16 –**Exam I**will cover material from July 6 — July 14

Exam 1 / Solutions

Weekend Problem Set 2 / Solutions**Week 3**

July 20 - Differential equations, part 1

July 21 – Differential equations, part 2

July 22 – Differential equations, part 3

July 23 – Differential equations, part 4

Differential equations reference sheet for Exam 2

Weekend Problem Set 3 / Solutions**Week 4**

July 27 – Parametric curves: §9.1, 9.2

July 28 – Polar coordinates: §9.3, 9.4

July 29 – Review

July 30 –**Exam II**will cover material from July 20 — July 28

Exam 2 / Solutions

Weekend Problem Set 4 / Solutions**Week 5**

August 3 – Sequences: an introduction

August 4 – Sequences and series: §8.1, §8.2

August 5 – Convergence tests: §8.3, §8.4

August 6 – Power series: §8.5, §8.6

Notes on telescoping sums

Notes on absolute convergence

Notes on the Dirichlet Test

Notes on the Bessel function of order zero

Weekend Problem Set 5 / Solutions**Week 6**

August 10 – Taylor series: §8.7, 8.8

August 11 – Extra problem set / Solutions

August 12 – Review

August 13 –**Final Exam**will cover material from August 3 — August 1

Notes on the binomial series

Final Exam / Solutions