Florida Atlantic University • Spring 2022 • Markus Schmidmeier

# Calculus II

Welcome to Calculus II !

You are registered for MAC 2312-006, 4 credits, CRN 17210. We meet Wednesday and Friday, 2:00 - 3:50 p.m. in-person in Physical Sciences Room 109.

The origins of calculus go back at least 2500 years to the ancient Greeks, who found areas using the "method of exhaustion". Limits arise not only when finding areas of a region, but also when computing the slope of a tangent line to a curve, the velocity of a car, or the sum of an infinite series. In each case, one quantity is computed as the limit of other, easily calculated quantities. Sir Isaac Newton invented his version of calculus in order to explain the motion of the planets around the sun. Today calculus is used in calculating the orbits of satellites and spacecraft, in predicting population sizes, in estimating how fast coffee prices rise, in forecasting weather, in measuring the cardiac output of the heart, in calculating life insurance premiums, and in a great variety of other areas.

In the first part of this course Calculus II, we will review integrals, study integration techniques, and apply integration to a variety of problems from science, engineering, and mathematics. In the second part, we will study sequences of functions. We all can differentiate and integrate polynomials, wouldn't it be nice if all functions were as easy to handle as polynomials? We will see, that many functions are power series (which are limits of polynomials) and hence in a way just as easy and as boring to deal with as polynomials!

Prerequisite

Calculus I with a minimum grade of C.

Textbook and Topics

Gilbert Strang, Calculus Volume 2, OpenStax College, PDF.
There are paperback and hardcover copies available for around \$25 to \$35 (new).

We are going to cover the following chapters:

 Chapter 2 Applications of Integrals(3 weeks) We review integrals by using them to compute a variety of things, for example the area between two curves or the length of a curve segment. A nice application are computations of volumes of rotated objects like cones or spheres or coffee cups (without handle). In physics, integrals help to compute moments of inertia and centers of masses. We'll touch on modeling using exponential growth and decay. Chapter 3 Integration Techniques(3 weeks) So far, we have computed integrals using the Fundamental Theorem of Calulus or the Substitution Rule. We'll learn three more techniques: Integration by Parts (for products like f(x)=xex), Trigonometric Integrals (for products of sines and cosines), and Partial Fraction Decomposition (to integrate rational functions). Our techniques are now so powerful that we can even compute areas of some interesting regions which are not bounded. Chapter 5 Sequences and Series(3 weeks) This chapter is an intermission, in a way, as there is no calculus! We consider sequences of numbers, for example, 1, 2/3, 7/9, 20/27, 61/81, 182/243, ... Can you see how this sequence is constructed? Where does this sequence go? A more basic question is: Does it converge? (meaning: does it go anywhere?) We will see that this sequence actually is a series, in fact, the n-th term is just the sum of the first n powers of -1/3, starting with n=0. For series, we will study tests that decide whether the series converges. Chapter 6 Power Series(4 weeks) You all can differentiate and integrate polynomials since you took Calculus I. It is easy, but after a while it gets boring. Here we will approximate arbitrary functions by polynomials. The higher degrees we allow, the better the approximation. If we are lucky, the given arbitrary function arises as a series of monomials (functions of type anxn) --- this is the famous Taylor series! We will see when we can do operations on Taylor series like: addition, multiplication, composition, differentiation and, of course, integration. Which is what this course is about!

Objectives

• Use integrals to compute various quantities like area, volume, mass...
• Recognize that there are ``easy'' integrals (which Calculus I students can do) ``mid level'' integrals (which we learn to do) and ``hard'' integrals (which nobody can do)
• Develop methods to discuss infinite sequences of numbers and of functions
• Approximate functions by sequences of polynomials and apply techniques to work with such sequences
• Communicate about calculus problems using computations, sketches and proper mathematical language.

Tutorials

The Math Learning Center (MLC) provides free academic support.

Credit

Homework:  I will assign homework problems every week. The problems will not be graded, but some may come up on a quiz. Please bookmark the link:

Homework problems

Quizzes:  We will have a quiz every Friday of about 20-25 minutes each; the ten best quizzes count for 50 % or 60 % of the grade. No calculators can be used during the quiz.

Presentation:  Depending on enrollment, I hope that every student can give one presentation of at most 10 minutes about a textbook problem. The presentation will count for 10 % of the grade. The last day for giving the presentation is Friday, April 15.

Final Exam:  The final exam is scheduled for Saturday, April 30, 10:30 a.m. - 1:00 p.m. in NU 113. It is comprehensive and counts for 40 % of your grade. Please bring a picture id (Owl card or drivers licence)!

Further Information

For the Disability Policy, the Make-Up Policy, the Code of Academic Integrity, Religious Accommodation, my Grading Scale and Financial Assistance Opportunities please visit Infos for all my courses.

Contact Me

Office hours:  MW 10:00 - 11:30 in SE 272.

Home page:  Markus Schmidmeier

Phone:  561-297-0275

E-mail:  markus@math.fau.edu.

Last modified:  by Markus Schmidmeier