Texas A&M University-Corpus Christi
Dept. of Mathematical and Statistics
Math 2312 Pre-Calculus
Fall 2008
I. Course Information
Meeting Time and Place: TR 2:00-3:15 PM BH 223
Instructor: Dr. Jose Giraldo
E-MAIL:
Office Address: CI 317
Phone: (361) 825-5827
Office hours: TR 12:30-1:45 pm or by appointment
II. Course Description
Topics Covered:
Data Analysis method; media-median line; re-expression, goodness of fit. Functions of models; graphing, transformation. Polynomials, finding zeros. Graphing rational functions. Exponential functions, graphing, compound interest. Logarithmic functions, graphing, solving exponential and logarithmic functions, growth and decay models. Logarithmic scales light and sound. Data analysis continued using Least Square Method. More modeling problems. Trigonometric functions, graphing, transformations. Symmetry, angles and radian measure and solving trigonometric equations.
Characteristics of the Course
Basic principles: the same as for our national reformed calculus. The rule of four: Every topic should be studied geometrically, numerically, and algebraically, and communicated back to the instructor in a literate fashion.
The way of Archimedes: Formal definitions and procedures evolve from the investigation of practical problems. Less emphasis on manual algebraic manipulation, more on concepts. Emphasis on cooperative learning, in-class and with homework groups; de-emphasis on lecturing.
III. Prerequisites: MATH 1314 (College Algebra) or placement into MATH 2312.
IV. Text and Other Supplies Required
Textbook: Contemporary Pre-Calculus through applications, by the North Carolina School of Mathematics, 2nd edition, published by Brooks/Cole.
Calculator: TI-83 plus or TI-89 is the most commonly used graphing calculator at this university. The calculator will serve as a tool for understanding and solving problems encountered in this course.
V. Student Learning Outcomes
This course is designed to prepare the student to use the tools of college mathematics in a variety of applications in physical and life sciences. In particular students will be given the chance to
Analyze and synthesize information concerning real-life data.
· Organize data, communicate the essential features of the data, and interpret the data in a meaningful way;
· Extract correct information from tables and common graphical displays, such as line graphs, scatter plots, histograms, and frequency tables;
· Express the relationships illustrated in graphical displays and tables clearly and correctly in words and/or use appropriate technology to describe and analyze quantitative problems (median-median line, least square line, etc)
Analyze and synthesize information concerning attributes of functions, relations, and their graphs.
· Understands when a relation is a function.
· Identifies the mathematical domain and range of functions and relations and determines reasonable domains for given situations.
· Understands that a function represents a dependence of one quantity on another and can be represented in a variety of ways (e.g., concrete models, tables, graphs, diagrams, verbal descriptions, symbols).
· Identifies and analyzes even and odd functions, one-to-one functions, inverse functions, and their graphs.
· Applies basic transformations to a parent function, f, and describes the effects on the graph of y = f(x).
· Performs operations (e.g., sum, difference, composition) on functions, finds inverse relations, and describes results symbolically and graphically.
Analyze and synthesize information concerning linear and quadratic functions, analyzes their algebraic and graphical properties, and uses them to model and solve problems.
· Understands the concept of slope as a rate of change and interprets the meaning of slope and intercept in a variety of situations.
· Writes equations of lines given various characteristics (e.g., two points, a point and slope, slope and y-intercept).
· Applies techniques of linear and matrix algebra to represent and solve problems involving linear systems.
· Analyzes the zeros (real and complex) of quadratic functions.
· Makes connections between the and the representations of a quadratic function and its graph.
· Solves problems involving quadratic functions using a variety of methods (e.g., factoring, completing the square, using the quadratic formula, using a graphing calculator).
· Models and solves problems involving linear and quadratic equations and inequalities using a variety of methods, including technology.
Analyze and synthesize information concerning polynomial, rational, radical, absolute value, and piecewise functions, analyzes their algebraic and graphical properties, and uses them to model and solve problems.
· Recognizes and translates among various representations (e.g., written, tabular, graphical, algebraic) of polynomial, rational, radical, absolute value, and piecewise functions.
· Describes restrictions on the domains and ranges of polynomial, rational, radical, absolute value, and piecewise functions.
· Makes and uses connections among the significant points (e.g., zeros, local extrema, points where a function is not continuous or not differentiable) of a function, the graph of the function, and the function's symbolic representation.
· Analyzes functions in terms of vertical, horizontal, and slant asymptotes.
· Analyzes and applies the relationship between inverse variation and rational functions.
· Solves equations and inequalities using a variety of methods (e.g., tables, algebraic methods, graphs, use of a graphing calculator), and evaluates the reasonableness of solutions.
· Models situations using polynomial, rational, radical, absolute value, and piecewise functions and solves problems using a variety of methods, including technology.
Analyze and synthesize information concerning exponential and logarithmic functions, analyzes their algebraic and graphical properties, and uses them to model and solve problems.
· Recognizes and translates among various representations (e.g., written, numerical, tabular, graphical, algebraic) of exponential and logarithmic functions.
· Recognizes and uses connections among significant characteristics (e.g., intercepts, asymptotes) of a function involving exponential or logarithmic expressions, the graph of the function, and the function's symbolic representation.
· Understands the relationship between exponential and logarithmic functions and uses the laws and properties of exponents and logarithms to simplify expressions and solve problems.
· Uses a variety of representations and techniques (e.g., numerical methods, tables, graphs, analytic techniques, graphing calculators) to solve equations, inequalities, and systems involving exponential and logarithmic functions.
· Models and solves problems involving exponential growth and decay.
· Uses logarithmic scales (e.g., Richter, decibel) to describe phenomena and solve problems.
· Uses exponential and logarithmic functions to model and solve problems involving the mathematics of finance (e.g., compound interest).
· Uses the exponential function to model situations and solve problems in which the rate of change of a quantity is proportional to the current amount of the quantity [i.e., f '(x) = k f(x)].
Analyze and synthesize information concerning trigonometric and circular functions, analyzes their algebraic and graphical properties, and uses them to model and solve problems.
· Analyzes the relationships among the unit circle in the coordinate plane, circular functions, and the trigonometric functions.
· Recognizes and translates among various representations (e.g., written, numerical, tabular, graphical, algebraic) of trigonometric functions and their inverses.
· Recognizes and uses connections among significant properties (e.g., zeros, axes of symmetry, local extrema) and characteristics (e.g., amplitude, frequency, phase shift) of a trigonometric function, the graph of the function, and the function's symbolic representation.
· Understands the relationships between trigonometric functions and their inverses and uses these relationships to solve problems.
· Models and solves a variety of problems (e.g., analyzing periodic phenomena) using trigonometric functions.
· Uses graphing calculators to analyze and solve problems involving trigonometric functions.
VI. Instructional methods and activities
Methods and activities for instruction include presentation of material and concepts from the text, problem solving techniques, use of the TI-83 or TI-89 Graphing Calculator, Geometer’s Sketchpad, class discussions and extensive work on class webpage.
VII. Evaluation, Grade Assignment, and Other Issues
Homework Assignments
Individual: Typically 4-5 exercises daily, but non-repetitive and nontrivial.
Group: Emphasis on applications and concepts. Writing skills and clarity of exposition stressed.
Typical Group Homework Problem
The following table represents the measured amount g, in grams, of a certain chemical compound that can be dissolved in one liter of water at various temperatures T, measured in degrees Celsius.
T / 10 / 20 / 30 / 40 / 50 / 60 / 70 / 80 / 90g / 1.70 / 1.23 / 0.86 / 0.65 / 0.47 / 0.34 / 0.25 / 0.18 / 0.13
1. In this relationship, which variable would it be natural to view as the independent variable and which as the dependent variable? Explain. (An aside: In practice, we would probably not add the command to "Explain." The students learn early in the course that this is a natural part of answering a question such as this.)
2. Using median-median line regression, find a linear expression that models the number of grams of solute as a function of the temperature of the water.
3. Describe the real-world physical meaning of the slope of the line given by your linear model.
4. Describe the real-world physical meaning of each of the axis intercepts of the line given by your linear model.
5. What are the natural domain and range of the function defining your linear model, taking into account physical constraints?
6. Find an exponential model for the number of grams of solute as a function of the temperature of the water.
7. Why would you expect the sign of the exponent in the equation that defines this model to be as it is?
8. Let the above data set be called Data Set A. Suppose that another experimenter performs the same experiment and obtains other figures that will be called Data Set B. For low temperatures, the figures of Data Set B show that more of the compound can be dissolved in a liter of water than do the figures of Data Set A, but for high temperatures the opposite is true. If Data Set B were to be used instead of Data Set A to construct the exponential model, what would be the likely effect on each of the numbers appearing in the equation that defines the model?
9. Which of the two models do you believe is better? Explain
Homework Groups:
There are groups of 3-4 students. The groups will be used for class discussions/activities, and homework discussion.
Group homework is held to high standards of completeness, accuracy, and literacy.
Group member roles (which should change for each homework set).
Recorder: keeps the "minutes".
Scribe: writes up the final version of the homework.
Clarifier: verbally summarizes and clarifies discussions during the group meetings. Manager: gets the group together, often provides refreshments.
In class activities are carefully chosen to get the students to interact, test out ideas on each other, and reinforce individual understanding. Watch for general points of confusion; be prepared to interrupt and clarify. Summarize when the group is reaching some conclusion. Make sure all the members of the group are engaged in the discussions
Favorite Data Sets: http://courses.ncssm.edu/math/CPTA/data/datalong.htm
Grade Distribution:
Class work, Homework, and Quizzes / (50, 100, 100)Two major tests / (250, 300)
Final Exam / 200
VIII. Tentative Course Schedule
Date / TopicAugust 28 / Introduction to the Course
Sept. 02 / 1.1, 1.2 Some thoughts about Models and Mathematics and a Modeling Activity
Sept. 04 / 1.3-6 Using Scatter Plots to Analyze Data, The Median-Median Line, and How good is the fit?
09 / 1.7, 1.8 The Least Squares Line and Error Bounds and the Accuracy of Prediction
11 / 2.1, 2.2 Functions as Mathematical Models, and Characteristics of functions
16 / Quiz #1, 2.3, 2.4 A Toolkit of Functions and Finding the Domain of a Function
18 / 2.5, 2.8 Functions as Mathematical Models Revisited and Basic transformations of Functions
23 / 2.9, 2.10 Combinations of Transformations, and Composition of Functions
25 / 2.12-14 Composition as a Graphing Tool, Inverses, and Using Inverses to Straighten Curves
30 / Test # 1
Oct. 02 / 3.1, 3.2 Functions Defined Recursively, Loans and the Binary Search Process
07 / 3.3, 3.5 Geometric Growth Models, and Geometric Series: Summing Geometric Growth
09 / 3.7, 3.8 Compound Interest, and Graphing Exponential Functions
14 / 3.9, 3.10 Introduction to Logarithms, and Graphing Logarithmic Functions
16 / 3.11, 3.12 Solving Exponential and Logarithmic Equations, and Logarithmic Scales
21 / Quiz #2, 3.13 Data Analysis with Exponential and Power Equations
23 / 4.1, 4.2 The Tape Eraser Problem, and Radioactive Chains
28 / 4.4, 4.6 Choosing the Best Product and Developing a Mathematical Model
30 / 5.1, 5.2 The Curves of Trigonometry, and Graphing Transformation of Trigonometric Functions
Nov. 04 / Test # 2
06 / 5.4, 5.5 Sine and Cosine on the Unit Circle and Getting to Know the Unit Circle
11 / 5.6, 5.7 Angles and Radians, and Solving Trigonometric Equations
13 / 5.9, 5.10 Using Trigonometric Identities, and Inverse Trigonometric Functions
18 / 5.12, 5.13 Compositions with Inverse Trigonometric Functions and Solving Triangles with Trig.
20 / 6.1, 6.3 Introduction to Combinations of Functions, and Sums and Product of Functions
25 / Quiz #3, 6.6, 6.8 Introduction to Polynomial functions, and Polynomial Functions
27 / Thanksgiving Holiday
Dec. 02 / 6.9, 6.10 Rational Functions, and Application of Rational functions
Dec. 04 / Reflection on the course
09 / General Review
IX. Class Policies
o Use the resources you have available: your classmates, the professor, the STEP mentor, the Tutoring and Learning Center. All of this will lead to our main objective, which is YOUR LEARNING.
o The course requires a solid and continuous effort. Since this is a three-credit course, you are expected to devote for each hour of class between two and three hours outside the class working on the subject (Some people need more time than others. Each individual has a different way to learn. All of us are different)
o I do expect that you come to each class prepared to talk about any assigned work and readings. One of the best ways to learn any subject and specially mathematics is by talking to other people about it after you have read and attempted the problems. Listening to a solution without trying and struggling through it will not benefit you very much. Be aware that reading the solutions and be able to follow the explanation does not mean that you know how to do the problem and understand all what is involved in it.
o At the beginning of each class you have the opportunity to ask questions about the homework. Use that time wisely. Remember that making a serious attempt to solve a problem and later discuss your solution or to clarify doubts is key in the learning process.