Approved by University Studies sub-committee 2/21/07. A2C2 action pending.

Math 165-02 -- Calculus II -- Fall 2006

This is a University Studies course that satisfies the Basic Skills in Mathematics requirement.

For details of the learning outcomes for this course please see below.

Course Meets: Mon/Tues/Thurs/Fri, 8:00 – 8:50 AM, in GI 326

Prerequisites:Successful completion (grade of “C” or better) in Math 160 (Calculus I) or equivalent

Instructor:Dr. Steve LeonhardiPhone:457 - 2359 (can leave a message)

Office:Gildemeister 203E-mail:

Office Hours:Monday/Friday: 9:00-9:50 AM and 2:00-3:50 PMWed.: 8:00-10:50 AM

(subject toTuesday/Thursday: 9:30-9:50 AM

revision)(or by appointment)

Text:James Stewart, Calculus: Early Transcendentals, 5th Edition.

Technology: A graphing calculator is strongly recommended. We may also use a computer algebra system such as Mathematica.

Course Objectives: This course builds upon your understanding of the major concepts of Calculus: limits, derivatives, definite integrals, and antiderivatives. In this course we will explore further applications of definite integrals, expand our techniques of integrating and adding up “infinite sums” (series), use differential equations to model exponential growth and decay, and use parametric equations and polar coordinates.

You are expected to develop your skills in

1) calculating and computing solutions to questions about functions;

2) understanding and explaining the concepts underlying the techniques that we use;

3) solving open-ended problems by creatively applying and synthesizing these techniques; and

4) expressing your reasoning and interpreting your solutions clearly, both verbally and in writing.

Tentative Grading Plan (any changes will be announced in class):

Gateway Tests(probably 2 Tests): These will test basic computational skills. You must pass each of these at a predetermined level (roughly 70-80%). You will have three opportunities to pass each of the gateway exams. If you pass all Gateway Tests, your grade will be based on the points and scale given below. If you fail any of the Gateway Tests three times, you will receive a grade of “F” for the course.

Midterm exams (4 @ 100 pts. each):400 points90% is sufficient for an A

Quizzes/Homework/Projects (scaled as needed): 200 points80% is sufficient for a B

Final Exam:200 points70% is sufficient for a C

------60% is sufficient for a D

TOTAL:800 points

Quiz/Exam Policy: No early exams and no make-up quizzes or exams will be given except in case of extenuating circumstances and at my discretion. Late homework or take-home quizzes will receive half-credit if turned in by the beginning of the next class period and before solutions are given. Cheating of any kind will result in failure and will be reported to school authorities.

Homework: I will assign many “suggested” problems that you should do on your own and then check your answers in the back of the book. I will also assign a number of “turn in” problems to be graded. You are responsible for knowing how to do all assigned problems. You should ask questions in class or at office hours about those problems which you do not understand.

Class Attendance: Essential for success. You are responsible for knowing what is covered and assigned in class regardless of whether or not you are present. Attendance and participation may be considered in determining your final grade.

Cooperative Learning: A significant portion of your grade will be based on group work, some of which will be done in class and some of which will need to be done outside of class. Cooperative learning will help you learn the material, will help you learn to present your solutions clearly, and will prepare you for working effectively as a team member after you graduate, which is a skill that employers value very highly.

The Student-Teacher “Contract”

It is my job to point the way toward what you should be learning, to assist you as much as possible in learning it, and to evaluate the results of your efforts. My underlying assumption is that you are reading the text carefully and actively (see below) and attempting all of the assigned homework problems. I will attempt to structure class time so that what we do in class supplements your work outside of class rather than serving as a substitute. I will spend some time lecturing on material in the text, but I will devote substantial time to answering questions, going over examples not in the text, and group work.

Learning the material in this course is primarily your responsibility. You will get as much out of this course as you put into it. A general rule of thumb is that you should study 2 hours outside of class for every hour spent in class. That means you should plan on at least 8 hours of study for this course outside of class each week. Put it on your schedule! MATHEMATICS IS NOT A SPECTATOR SPORT!

The most important advice I can offer you is: USE YOUR RESOURCES!!! Attend class faithfully, ask questions and participate actively, read the text actively, work on the homework every day, make use of my office hours, and find other students in the class with whom to study.

How To Read A Mathematics Text (Two possible methods)

Method (1): Turn directly to the homework problems. For each problem assigned, page backwards through the appropriate section of the text until you find an example which looks as close as possible to the problem you are trying to do. Copy the steps in the example, hoping that you receive at least partial credit. Go on to the next homework problem.

Method (2): Start by skimming the section, making note of the key ideas and trying to guess what the graphs, theorems, and equations are supposed to mean. Then, on the second pass, read the section carefully and slowly, with a calculator and pencil and paper handy. When you come to a computation in which the authors have skipped a few steps, verify the result on your own. When you come to a theorem, try to restate the theorem in your own words. When you come to an example that has been worked out, cover up the solution and try to solve the problem on your own. When you finish or get stuck, then look at how the authors have proceeded, just enough to give yourself a hint on how to continue. Then continue as far as you are able, uncovering the solution for more hints as needed. Use the margins to take notes next to steps or passages in the text which you did not understand, and then ask about these in class or at office hours. You should also make notes (explanations to yourself) next to parts which confused you at first, but which you now understand.

Once you have finished reading the section (which will usually take at least an hour), then proceed to the homework problems. When you solve a problem, check your answer for reasonableness. Summarize your solution strategy. Summarize what you have learned from solving this problem.

Analysis: Method (1) is by far the most popular among students. It takes (or at least appears to take) the least time, and often allows you to do the first few suggested homework problems. The disadvantage of this method is that you will not really understand what you are doing, and you will not have any idea about how to do, or even get started on, more difficult problems which are not exactly like the examples.

Method (2) requires more time spent reading but usually requires less time spent on problems. You may even find that your total time required is less for this method! Most importantly, this method gives you a much better understanding of the relevant concepts, and gives you much greater ability to creatively and confidently solve new, related but unfamiliar problems. Please give Method (2) a try!

Math 165, Calculus II -- Fall 2006 -- Weekly Schedule (as of 8/16/06)

The following schedule is extremely tentative. Exam dates are not official until announced by me in class.

Dates / Topics / Sections in Text
Aug. 28-Sept. 1 / Introduction and overview. Areas between curves.
Volumes by the washer method and by cylindrical shells. / 6.1-6.3
Sept. 4-8
(No class Monday,
Sept. 4) / Work. Average value of a function.
Integration by parts. / 6.4-6.5,
7.1
Sept. 11-15 / Trigonometric integrals. Trig substitution.
Integration of rational functions by partial fractions.
Strategy for integration. / 7.2-7.5
Sept. 18-22 / Exam 1 probably this week.
Integration using tables and computer algebra systems.
Approximate integrals. / 7.6-7.7
Sept. 25-29 / Improper integrals.
Arc length. Area of a surface of revolution.
Applications to physics and engineering. / 7.8,
8.1-8.3
Oct. 2-6 / Modeling with differential equations. Direction fields and Euler’s method. Separable equations. / 9.1-9.3
Oct. 9-13
(No class Friday,
Oct. 13) / Exponential growth and decay.
Exam 2 probably this week. / 9.4
Oct. 16-20 / Curves defined by parametric equations.
Calculus with parametric cruves. Polar coordinates. / 10.1-10.3
Oct. 23-27 / Areas and lengths in polar coordinates.
Conic sections. Conic sections in polar coordinates. / 10.4-10.6
Oct. 30-Nov. 3 / Exam 3 probably this week.
Sequences. Series. / 11.1-11.2
Nov. 6-10
(No class Friday,
Nov. 10) / The integral test and estimates of sums.
The comparison tests.
Alternating series. / 11.3-11.5
Nov. 13-17 / Absolute convergence and the ratio and root tests.
Strategy for testing series.
Power series. / 11.6-11.8
Nov. 20-21
(No class Nov. 22-24) / Exam 4 probably this week.
Nov. 27-Dec. 1 / Representation of functions as power series.
Taylor and Maclaurin series.
The binomial series. / 11.9-11.11
Dec. 4-8 / Applications of Taylor polynomials. / 11.12

Final Exam: Monday, December 11th, 7:00 - 10:00 AM

This course can be used to satisfy the University Studies requirements for Basic Skills in Mathematics. Thiscourse includes requirements and learning activities that promote students’ abilities to...

a.use logical reasoning by studying mathematical patterns and relationships;

  • understand the relationship of derivatives and errors made when using numerical approximation of definite integrals
  • understand the reasoning behind the existence of limits in improper integrals
  • explain why and when an improper integral converges and why and when it diverges
  • be able to compare an improper integral with another and explain its convergence
  • be able to use apply geometry knowledge e.g. similar triangle, Pythagoras theorem and create a function to integrate for finding areas, volumes, arc lengths, density, center of mass, work and force
  • accurately apply and compare the convergence tests for infinite series and improper integrals, demonstrating an understanding of the limitations of the tests and the difference between the behavior of the integrand/summand and the integral/series
  • understand the concept of radius of convergence and use it correctly for the convergence of power series
  • understand the role of higher order derivatives near/at a point and find Taylor’s series/polynomial for functions
  • understand the role of higher order derivatives in finding the error bounds for Taylor polynomials
  • understand when and why we use Fourier polynomials instead of Taylor’s polynomials for a function
  • understand the relationship of trig functions, sine and cosine for Fourier polynomials
  • be able to use logical reasoning to sketch slope fields for a differential equation
  • given a function in 3-D be able to characterize the solid and vice versa
  • apply distance function accurately with vectors
  • understand the meaning of dot product, cross product of vectors and projection of a vector

b. use mathematical models to describe real-world phenomena and to solve real-world problems - as well as understand the limitations of models in making predictions and drawing conclusions;

  • accurately model pressure and work problems involving continuously changing quantities.
  • given a particular solid, accurately formulate an integral which will provide the surface area and volume of that solid.
  • accurately model situations relating to growth, decay, heating, cooling and mixing using differential equations and solve the resulting equations using separation of variables or an integrating factor.
  • accurately model situations relating to oscillations using second order differential equations and solve and interpret the solution meaning fully to the context of the problem
  • be able to apply contour diagrams and graphs in context to real-world problems
  • understand why and how the contour diagrams looks like for a linear functions (in two variables)
  • understand why and how level surfaces are used to represent a function (in three variables)
  • apply the concept of limit, continuity and differentiability to functions in several variables
  • accurately model real-world problems using vectors to find direction of movement, velocity etc.

c.organize data, communicate the essential features of the data, and interpret

the data in a meaningful way;

  • accurately sketch a graph using the data sets in two variables
  • accurately interpret the behavior of a function representing a physical phenomenon using the given data set in more than one variable
  • use data to find average and instantaneous rate of change of a function (in more than one variable) and/or the rate of increasing or decreasing of a function (in more than one variable) in the direction of a particular variable
  • use data to find the limiting value of a function (in two or three variables) when (x,y) approaches (a,b) or (x,y,z) approaches (a, b, c)
  • use data to find upper and lower estimates for a certain quantity for e.g. given a data relating speed (mph) and corresponding fuel efficiency (mpg) find the lower and upper estimates of the quantity of fuel used
  • apply tables to sketch contour diagrams

d.do a critical analysis of scientific and other research;

  • Do assigned projects and group work with appropriate research and analysis of mathematical concepts

e.extract correct information from tables and common graphical displays, such

as line graphs, scatter plots, histograms, and frequency tables;

  • given the graph of a function, determine what can be said about all of the co-efficients in general, and the first three coefficients in particular, and the interval of convergence in a Taylor Series for that function about a point x=a.
  • given the graph of a simple wave-form, determine what can be said about the co-efficients of the Fourier Expansion of that wave-form.
  • given the graph of a function f(x,y), determine the nature of various cross-sections (directly or by a matching exercise) and of any critical points.

f.use appropriate technology to describe and solve quantitative problems.

  • demonstrate proficiency in using the TI-89 to solve messy algebraic equations and compute integrals and derivatives that arise from real-world problems with real data.
  • use a spreadsheet and various numerical methods to estimate the value of the integral of an unknown function whose values we know at a finite number of points.