Here are the student evaluations.
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.
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.
- 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.
- 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.)
- 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