Markus Schmidmeier
Mathematical Sciences
Florida Atlantic University

# Calculus - Analytic Geometry II

Fall 2016

Welcome to my course Calculus II (CRN: 11684, MAC 2312-002, 4 credits). We meet Monday, Wednesday, Thursday and Friday, 11:00 - 11:50 a.m. in SC 178.

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 --- of course --- 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 just as friendly to deal with as polynomials!

Prerequisite

Calculus I with a minimum grade of C.

Textbook and Topics

William Briggs, Lyle Cochran, Bernard Gillett: Calculus, Early Transcendentals, 2nd Edition

For extra reading, I recommend the Calculus Online Textbook, Calculus by Gilbert Strang, Massachusetts Institute of Technology, Wellesley-Cambridge Press. The book is MIT open courseware, available online under the Creative Commons Licence:
Calculus book

We are going to cover the following chapters:

 Chapter 6 Applications of Integration(4 weeks) We discuss a variety of applications of integration in mathematics (areas, volumes, and surface areas) and physics (work.) Chapter 7 Integration Techniques(3 weeks) Well known differentiation rules give rise to methods for integration: For example, the product rule yields the technique of integration by parts. While there are further methods (which we will study), we will also find integrals which are hard to compute --- and we will discuss how hard it can get. We also explore indefinite integrals, they let us study strange things like "Gabriel's horn" --- an amazing shape which has a finite volume yet an infinite surface area. Integration via Computer For definite integrals, we can use computers to approximate. We will explore how computer algebra (probably Sage) is used to define functions, compute integrals, and assist us in graphing. You can earn extra credit for using Sage or another computer algebra system to solve Calculus problems. Chapter 8 Sequences and Infinite Series(4 weeks) Some sums are easy to compute, for example 1/2 + 1/6 + 1/12 + 1/20. But note that there is a pattern: 1/(1*2) + 1/(2*3) + 1/(3*4) + 1/(4*5). What if we keep adding terms of the form 1/(n*(n+1)), where do we get? If we do get anywhere, then this is the sum of the series. Why should we bother? Well, infinite series have surprisingly many practical applications. P.S.: Can you find the sum of the above series? Chapter 9 Power Series(3 weeks) We approximate functions, like sine and cosine, by sequences of polynomials, the socalled Taylor polynomials. Whenever (even better, and almost true: wherever!) those polynomials converge against the given function, also the integrals converge. This allows us to compute integrals for functions like f(x)=sin(sin(x)) or f(x)=ex2, for which the antiderivative cannot be expressed in terms of functions known to us.

Objectives

• Use integrals to express and compute quantities like area, volume or work.
• Recognize that there are easy integrals, difficult integrals and integrals which we may not be able to solve.
• Understand the conceptual problem of computing an ``infinite sum''.
• Approximate functions by polynomials, and work with Taylor series.
• practise graphing and use calculus tools in computer algebra systems like maple.

Tutoring

There is free math tutoring available in the Math Learning Center in GS 211. See MLC for opening hours, appointments for one-to-one tutoring, and online tutoring.

Credit

Homework:  I will assign homework problems every week. The problems will not be graded, but may be related to problems on the weekly quiz:   Homework problems.

Quizzes:  We will have a quiz every Friday of about 20 minutes each; the eleven best quizzes count for 60 % of the grade. No calculators can be used during the quiz. Note that we will have a quiz on Thursday, November 10 (since Friday, November 11 is Veterans' Day). The last quiz is on Friday, December 2. Thus, there are 13 quizzes in total, and you can drop the weakest two grades.

Extra Credit:  You can obtain extra credit counting towards your quiz grade for assignments done on a computer algebra system, for example Maple.

Make-Up Class:  In order to make up for class time canceled due to Hurricane Matthew, we will have an extra class meeting on Tuesday, December 6, 11 a.m. - 12 noon.

Final Exam:  The final exam is on December 11 (Sunday), 4:00 - 6:30 p.m. in SO 250. It is comprehensive and will count 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:  MWRF, 12 noon - 1 p.m. in SE 272.