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:
Prerequisites:
Instructor:
Office:
Mon/Tues/Thurs/Fri,
8:00 – 8:50 AM, in GI 326
Successful completion (grade of “C” or better) in Math 160 (Calculus I) or equivalent
Dr. Steve Leonhardi
Phone: 457 - 2359 (can leave a message)
Gildemeister 203
E-mail: sleonhardi@winona.edu
Office Hours:
(subject to
revision)
Monday/Friday: 9:00-9:50 AM and 2:00-3:50 PM
Tuesday/Thursday: 9:30-9:50 AM
(or by appointment)
Text:
Technology:
James Stewart, Calculus: Early Transcendentals, 5th Edition.
A graphing calculator is strongly recommended. We may also use a computer algebra
system such as Mathematica.
Wed.: 8:00-10:50 AM
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):
Quizzes/Homework/Projects (scaled as needed):
Final Exam:
TOTAL:
400 points
200 points
200 points
-------------800 points
90% is sufficient for an A
80% is sufficient for a B
70% is sufficient for a C
60% is sufficient for a D
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 halfcredit 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)
Sept. 11-15
Work. Average value of a function.
Integration by parts.
6.4-6.5,
7.1
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)
Oct. 16-20
Exponential growth and decay.
Exam 2 probably this week.
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.
9.4
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.
Exam 4 probably this week.
11.6-11.8
Representation of functions as power series.
Taylor and Maclaurin series.
The binomial series.
Applications of Taylor polynomials.
11.9-11.11
Nov. 20-21
(No class Nov. 22-24)
Nov. 27-Dec. 1
Dec. 4-8
Final Exam: Monday, December 11th,
7:00 - 10:00 AM
11.12
This course can be used to satisfy the University Studies requirements for Basic Skills in
Mathematics. This course includes requirements and learning activities that promote students’
abilities to...
a.
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b.
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c.
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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
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.
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
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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
do a critical analysis of scientific and other research;
Do assigned projects and group work with appropriate research and analysis of mathematical
concepts
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.
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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.