COURSE DESCRIPTION
Students study topics that include functions, models, and average rate of change, limits instantaneous rates
of change, the derivative, differentiation techniques applications of the derivative, and a brief introduction
to integration. This course may be used to fulfill the mathematics core curriculum requirement, if it
corresponds to the student's mathematics placement.
RATIONALE FOR THE EDUCATIONAL EXPERIENCE.
Mathematics is of primary importance in our increasingly technical world. The possibilities for its application are abundant.
Certain concepts inherent to mathematics -- for example, respect for cause and effect,
logic, honesty in presentation of material and critical thinking -- are also at the heart of Western science
and have been critical to the development of Western philosophy. Mathematics continues to be one of the
greatest cultural and intellectual achievements of humankind. This course is designed to provide students
with the opportunity to develop an understanding and appreciation of this on-going achievement.
GENERAL COURSE GOALS
To have students:
- Expand their understanding of the nature of mathematics.
- Learn to use mathematics to gain insight into areas of application, for example in the physical and life sciences, and economics.
- Appreciate the importance of mathematics in the development and advancement of human civilization.
- Learn the appropriate use of technology including how to use it and when to be skeptical of its results.
- Develop critical thinking and a basic understanding of logic essential for lifelong learning.
- Learn to behave in a professional manner (showing up to class on time, not disrupting the class, being respectful, preparing for class discussions and examinations, etc.)
- Ultimately, the course is designed to provide tools that can help in the search for truth.
If you look for truth, you may find comfort in the end;
if you look for comfort you will not get either comfort or truth only soft soap and wishful thinking to begin,
and in the end, despair. - C. S. Lewis.
CORE STUDENT LEARNING OUTCOMES
This course addresses the following two core student learning outcomes:
- 2.1. Demonstrate knowledge of the truth as it is pursued in the disciplines of mathematics and the sciences; the social sciences; the humanities; and the fine arts.
- 3.1. Utilize the liberal arts skills to analyze and evaluate significant texts and investigate mathematical and scientific processes.
SPECIFIC COURSE OBJECTIVES
For students:
- To understand the concepts of limit, continuity, average rate
of change, instantaneous rate of change.
- To understand and appreciate the difference between
average rate of change and instantaneous rate of change.
- To be able to apply calculus to real-life applications.
- To approach problems algebraically, geometrically, and numerically.
LEARNING OUTCOMES
By successfully completing this course, students should be able to:
- Define function; recognize polynomial, rational, exponential, logarithmic functions, and radical functions; find domains of functions; sketch the graphs of functions.
- Explain the properties of linear functions and their graphs, and interpret slope as a rate of change.
- Explain properties of quadratic functions, including their graphs and vertices (as maximum or minimum of quadratic functions).
- Calculate limits of functions with the graphing calculator.
- Calculate limits of constants, polynomials, exponential functions, and simple rational functions without the aid of calculators.
- Find vertical and horizontal asymptotes.
- Define of continuity, and find open intervals where certain types of functions are continuous, especially for polynomial, rational, exponential, logarithmic functions, and radical functions.
- Explain the difference between average and instantaneous rates of change.
- State the definition of derivative, and to use the definition to calculate derivatives of very simple functions (linear and quadratic).
- Explain the role of derivative as an instantaneous rate of change and use the derivative to find instantaneous rates of change, when appropriate. (The students should be able to determine when it is appropriate.)
- Differentiate sums, differences, products, quotients, and compositions of polynomial, rational, exponential, and logarithmic functions.
- Explain where functions are nondifferentiable (sharper corner, vertical tangent lines, and discontinuities).
- Apply differential calculus to solve related rates and optimization problems, and to sketch graphs of functions.
- Define critical numbers
- Define relative (i.e., local) extrema and absolute (i.e., global) extrema.
- Explain and apply the first derivative test for relative extrema.
- Explain and apply the second derivative test for relative extrema.
- Use the second derivative to determine a function's concavity and to find its inflections points.
- Explain the connection between a functions concavity and its slope.
- Explain the extreme value theorem and its role in optimizing continuous functions of closed intervals.
- Apply the second derivative test for absolute extrema, when appropriate.
- Define and apply relative rates
- Define, find, and interpret elasticity of demand, given a demand function.
- Reverse the differentiation process to find antiderivatives.
- Explain and apply the power rule for integration.
- Integrate sums and scalar multiples of functions.
- Integrate 1/x and exp(kx), where k denotes a real constant.
- Explain the difference between a specific antiderivative and the general antiderivative of a function.
- Define left, right, and midpoint Riemann sums and explain their connection with "area under a curve."
- Compute left and right Riemann sums.
- Explain and apply the Fundamental Theorem of calculus to evaluate definite integrals.
- Find a function given its derivative and a value at a point in its domain.
- Find the total accumulation at a given rate.
- Find average value of a function.
- Find the area between curves.
SUPPORT OF THE UNIVERSITY MISSION
This course supports the mission of the university as follows: "...through teaching ... prepares men and women for
responsible lives by imparting and expanding knowledge, developing skills, and cultivating enduring values.
... students develop their abilities for thinking clearly and creatively,
enhance their capacity for sound judgment, and prepare for the challenge of learning throughout their lives."
PREREQUISITES
High school Algebra II.
WHERE AND WHEN
- Section 01: Course meeting time/place: T, Th 8:00 A.M. - 9:15 A.M. Location: McAuley 104
- Section 02: Course meeting time/place: T, Th 9:30 A.M. - 10:45 A.M. Location: McAuley 104
REQUIRED TEXTBOOK AND MATERIALS
- Textbook with WebAssign: Brief Applied Calculus 7th Edition
by Geoffrey C. Berresford and Andrew M. Rockett
- Calculator: TI-83Plus or TI-84Plus graphing calculator.
Note: When calculators will be allowed on examinations, only the following models will be permitted:
TI-83, TI-83Plus, TI-84Plus. (Note that no TI-Nspire calculator will be allowed, even if it has a TI-84Plus style keypad!)
TEACHING METHODOLOGY
Lecture, question and answer, readings, homework exercises.
GENERAL REMARKS AND ADVICE
This web site is designed to, among other things, make it clear what is expected from you and what you can expect from the
course and from me. This course will be challenging; it is fast-paced, it requires a great deal of meticulous attention to detail as well as imagination and creativity, and just about everything in it depends
on your understanding of everything else in the course that preceded it. Nevertheless, if you work hard,
do not allow yourself to fall behind, and seek help when you need it, you should be successful in this course.
Reading assignments and homework problems for nearly each class meeting are posted on the course web site.
It is your responsibility to check the web site frequently (i.e., at least once a day) for the homework.
It is extremely important that you complete the reading assignments and try the homework problems before the following class meeting.
Last modified: 1/16/2020
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