Prefix, Number and Name of Course: MAT 161 Calculus I
Credit Hours: 4
In Class Instructional Hours: 4 Labs: 0 Field Work: 0
Catalog Description: Graphic, symbolic, and numeric
representation and analysis of functions; limits; continuity; derivatives and
antiderivatives of algebraic, trigonometric, exponential, and logarithmic
functions; applications of the derivative and antiderivative. Appropriate for
math majors and students in partner disciplines requiring understanding of
fundamental principles of calculus with emphasis on deductive reasoning and
proof. Credit issued for either MAT 126 or MAT 161 (or equivalents), but not
for both.
Prerequisite: MAT 124 with a
minimum grade of C, or equivalent.
Corequisite: MAT 163
Reasons for Revision: This revision is based on current best
practices in the teaching of calculus, current undergraduate mathematics program guidelines, and the appropriate
use of technology. The department
proposes to revise both the course content and increase the credit hours earned
through successful completion of the course; the fourth credit hour will allow
us to incorporate a highly desired problem-solving session officially into this
course. Moreover, the idea of a 4-credit calculus course is not new:
Traditionally calculus courses have involved a recitation meeting or a
problem-solving session for which students receive credit, and we are currently
one of only two SUNY schools that attempt to teach our calculus courses for
mathematics majors in a three-hour format. [The rest of the SUNY schools teach
calculus courses as 4-credit courses that meet four hours per week.]
The
problem-solving session (fourth credit hour) is designed to provide
students with a student-centered learning environment and is directly tied to
Student Learning Outcomes #8 and #9, which are based on the new
recommendations from the Mathematical Association of America's (MAA) Committee
on the Undergraduate Program in Mathematics (CUPM). During the problem solving sessions students work in small groups on
challenging problems designed to give them hands-on
experience working multi-step problems that require them to make clear
sense of both the concepts and computational algorithms. The problems are
chosen from department-written pedagogical guides that direct instructors to
use this hour specifically for students to work on the problems with guidance
from the instructor. Ideally students and the instructor get instant feedback
during these sessions; by working with the students and observing their work,
the instructor can tell when certain central concepts and important algorithms
need additional instruction time. By working in small groups with the guidance
of the instructor, students can identify both their strengths and weaknesses
and get timely help where necessary. The department expectation is that
instructors use student participation and achievement in the problem solving
sessions as a formal part of the student's semester grade. The department has
experimented with this kind of problem-solving session through funding provided
by the STEP grant for the past 5 semesters. Early indications are that our
students benefit from these problem solving sessions and the department wants
to insure that they continue to benefit from this type of instruction long
after the STEP grant's funding runs out.
The revised
content reflects current scholarship in calculus reform practices and is driven
partly by computing possibilities and partly by different uses our students
will make of calculus. Thus, the revised course is characterized by studying
critically important concepts from multiple representations; more and deeper
modeling problems (where students build, not just use, calculus-based models);
more writing and verbal presentations of student reasoning; more open-ended,
investigative activities; and more challenging applications. Furthermore, to
address the needs of partner disciplines, trigonometric, exponential, and
logarithmic functions are now included in the first semester of study rather
than the second.
|
Student Learning Outcomes: Students will: |
Content
Reference |
Assessment: |
|
1. explain the concept of rate of change and its fundamental relationship to real-world phenomena. |
II-V |
1. Group work and classroom activities, individual assignments, quizzes, exams, projects |
|
2. analyze and compute limits graphically, symbolically, and numerically and use limit theory to solve problems and determine continuity and differentiability of functions. |
I, II |
2. Group work and classroom activities, individual assignments, quizzes, exams, projects |
|
3. explain the concept of derivative as instantaneous rate of change graphically, numerically, and symbolically, and its relationship to average rate of change. |
II-IV, VI |
3. Group work and classroom activities, individual assignments, quizzes, exams, projects |
|
4. compute derivatives of algebraic, trigonometric, exponential, and logarithmic functions using appropriate techniques of differentiation. |
II-VI |
4. Group work and classroom activities, individual assignments, quizzes, exams, projects |
|
5. analyze applied differentiation problems from related disciplines and describe results using appropriate mathematical language and notations. |
IV |
5. Group work and classroom activities, individual assignments, quizzes, exams, projects |
|
6. solve selected differential equations and related initial value problems with and without the use of technology.
|
V, VI |
6. Group work and classroom activities, individual assignments, quizzes, exams, projects |
|
7. use technology to solve problems and as a tool to provide insight into significant concepts of calculus. |
I-VI |
7. Group work and classroom activities, individual assignments, quizzes, exams, projects |
|
8. use deductive reasoning and proof as tools to generate mathematical knowledge and provide insight into significant concepts of calculus. |
I-VI |
8. Group work and classroom activities, individual assignments, quizzes, exams, projects |
|
9. solve problems from related disciplines individually and in small groups within a Socratic environment during weekly problem solving sessions. |
I-V |
9. Group work and classroom activities, individual assignments |
Course Content: Note:
Proof and deductive reasoning are an integral part of the course that should be
woven into the development of content knowledge and conceptual understanding at
every opportunity. Furthermore, concepts should be explored and developed
following the "rule of 4"– graphic, numeric, symbolic and verbal
representations.
I.
Limits and continuity
A. Functions
B.
Concept of limit (graphic,
numeric, symbolic and verbal representations)
i.
Formal definition of limit
ii. One-sided
limits
iii.
Limits that do not exist
C.
Limit theorems and
computational techniques (graphic, numeric, symbolic and verbal representations)
D. Continuity
i.
Definition
ii.
Intermediate Value Theorem
iii.
Extreme Value Theorem
E.
Limits at infinity and global
shape
F.
Indeterminate forms
II.
Derivatives
A. Introduction to the derivative (graphic, numeric,
symbolic and verbal representations)
i. Formal
definition of derivative
ii.
Tangent and secant lines;
best linear approximation
iii.
Average and instantaneous
rates of change
iv.
Derivative as an
instantaneous rate of change
v.
Connection between
differentiability and continuity
vi.
Mean Value Theorem and
Rolle's Theorem
B.
Techniques of differentiation
(graphic, numeric, symbolic and verbal representations)
i.
Power rule
ii.
Sum, product and quotient
rules
iii.
Chain rule
iv.
Implicit differentiation
III. Transcendental functions
A. Trigonometric functions
i.
Limits
ii.
Derivatives
B.
Exponential and logarithmic
functions
i.
Inverse functions
ii.
Definition
iii.
Limits
iv.
Derivatives
C.
Inverse trigonometric
functions (Arcsine, Arctangent)
i.
Properties
ii.
Limits
iii.
Derivatives
IV. Applications of the derivative
A. Applications to the behavior of functions
i.
Critical values, intervals of
increase and decrease, first derivative test
ii.
Concavity, point of
inflection, second derivative test
B.
Applications from related
disciplines including but not limited to:
i.
Relative and absolute extreme
values
ii.
Optimization
iii.
Related rates (optional)
C.
L'H™pital's Rule*
V.
Anti-derivatives and
differential equations
A. Graphic,
numeric, symbolic and verbal representations
B.
Initial value problems
C.
Slope fields *
D. Selected classes of differential equations *
E.
Exponential growth and decay
*
*Included
in MAT 161 or 162 depending on placement in adopted text.
Resources:
Classic
Scholarship in the Field:
Adams, C., Thompson, A., and Hass, J., How to Ace Calculus. W. H. Freeman, New York, 1998.
Apostol,
T., Calculus Volumes, I, II. Blaisdell
Pub. Co., Massachusetts, 1967.
Courant,
R. and John, F., Calculus and Analysis.
Interscience Publishers, New York, 1965.
Fraga, R., Calculus Problems for a New Century. The Mathematical Association of America (MAA), 1993.
Ganter, S., Changing Calculus A Report on Evaluation Efforts and
National Impact from 1988-1998. The Mathematical Association of America (MAA), 2001.
Kline,
M., Calculus: An Intuitive and Physical Approach. John Wiley & Sons, New York, 1977.
Roberts, W., Calculus,
the Dynamics of Change. The
Mathematical Association of America (MAA), 1995.
Steen, L. A., (Ed.) Calculus for a New
Century: A Pump, Not a Filter. Papers
Presented at a Colloquium, The Mathematical
Association of America (MAA) Notes Number 8,
1987.
Thomas,
G., Calculus and Analytic Geometry.
Addison-Wesley, New York, 1952.
Current
Scholarship in the Field:
Bressoud,
D., Launchings from the CUPM Curriculum Guide: Keeping the Gates Open. The Mathematical Association of America (MAA), 2006.
Buck, C., Advanced Calculus. Waveland Press, Illinois, 2003.
Hughes-Hallett,
D., Gleason A., McCallum W., et al., Calculus. 4th ed. Wiley, New York, 2005.
Kaplan, W., Advanced Calculus. 5th edition. Addison-Wesley, New York, 2002.
Krantz, S. G., Calculus DeMystified. McGraw-Hill, New York, 2003.
Larson, H., Calculus: Early Transcendental Functions. 4th edition. Houghton-Mifflin, 2007.
Lee, B., Forgotten Calculus. Hauppauge, NY, Barron's Educational Series, 2002.
Ostebee, A. and Zorn, P., Calculus from Graphical, Numerical, and Symbolic Points of View. 2nd edition, Houghton-Mifflin, 2002.
Stewart, J., Calculus, 6th edition. Brooks-Cole, 2007.
Undergraduate programs and Courses in the Mathematical Sciences: CUPM Curriculum Guide 2004. The Mathematical Association of America (MAA), 2005.
Periodicals:
American Mathematics Monthly
College Mathematics Journal
Math Horizons
Mathematics Magazine
Electronic or Audiovisual
Resources:
Buffalo State
Calculus Revision: http://www.bsccalculus.info
Calculus & Mathematica: http://cm.math.uiuc.edu/
Calculus on the Web: http://www.math.temple.edu/~cow/
The Calculus Page: http://www.calculus.org/
Carnegie Mellon's Open Learning Initiative (OLI):
http://www.cmu.edu/oli/courses/enter_calculus.html#aboutCalculus
Math Forum at Drexel: http://mathforum.org/library/topics/svcalc/
Project Calc: http://www.math.duke.edu/education/calculustext/
SUNY Stony Brook: Resources from their calculus
reform efforts:
http://www.math.sunysb.edu/~tony/calc/index.html
Tools for Enriching Calculus
Video CD-ROM, (iLrn Homework, and vMentor), Brooks-Cole, 2002.
Visual Calculus: http://archives.math.utk.edu/visual.calculus/