Students with Disabilities

Students with documented disabilities may be eligible for assistance and various accommodations. Please check the Student Support Services link on your Blackboard page.

Note that the course videos as distributed on DVD's do not currently have transcripts, though a version can be provided in which videos are embedded within screen-readable documents. If you require this version of the videos, or transcripts, due to a documented disability, please notify the instructor immediately.

Understanding of the nature of the mathematical modeling process, its uses and its limitations.

Proficiency in mathematical modeling using linear, quadratic, exponential, logarithmic, power and polynomial functions as well as recurrence relations.

More specifically, the following objectives are to be achieved.  This list is not exhaustive, but covers at least 80% of the topics for which students will be responsible:

Conceptual Objectives

Describe the behavior of a given table of y vs. x value, or a graph of y vs. x, in terms of increasing and decreasing behavior, asymptotes, cyclical behavior, and the increasing, decreasing or cyclical behaviors of the rates of increase or decrease.

Relate {table, graph, intervals of increasing and of decreasing values, horizontal and vertical asymptotes, interval of cyclic behavior, intervals of increasing rate of increase or decrease, intervals of decreasing rate of increase or decrease}

Quickly hand-sketch the graph of the basic points of each basic function as listed below, as well as the points as transformed by horizontal shift h, vertical shift c, horizontal compression k and vertical stretch A for any specified set of values h, c, k and A,  based on these points sketch the graph of the transformed function, and give a standard-form equation defining the transformed function:

• y = x with basic points (0, 0) and (1, 1)
• y = x^2 with basic points (-1, 1), (0, 0) and (1, 1)
• y = 2^x with basic points (-1, 1/2), (0, 1) and (1, 2)
• y = x^p for p = -2, -1, 0, 1/2, 1, 2, 3 or 4 with basic points at x = -1, 0, 1/2, 1, 2

Describe the behavior of each basic function and its transformations, and explain how each behavior is related to the formula for the function.

Explain the relationships among the formula, the standard table (using the standard graph points for each), and the graph for each basic family of functions over its standard domain. These families include the fundamental power functions (p=-2, -1, .5, 1, 2, 3 and 4), linear functions, quadratic functions and exponential functions.

Explain the relative shapes of the graphs of the fundamental power functions, with attention to how these shapes change from one power to the next.

Explain how to use the quadratic formula to graph a given quadratic function.

Give examples of how behavior which can be modeled with each of the basic functions can arise in the real world.

List and explain in commonsense terms the fundamental laws of real numbers, and show how each can be useful in understanding the real world.

Explain how the sequence behavior of each of the basic functions is related to the recurrence relation from which the corresponding sequence might arise. For each function type describe a real-world situation in which the sequence behavior corresponds to expected real-world behavior, and how the recurrence relation is related to that real-world behavior.

Explain how the sequence behavior of each of the given functions is related to the difference equation from which the corresponding sequence might arise. For each function type describe a real-world situation in which the sequence behavior corresponds to expected real-world behavior, how the recurrence relation is related to that real-world behavior, and how the difference equation is related to that real-world behavior.

For each basic function f(x), describe how each of the parameters A, k, c and h affects the graph of y = A f(k(x-h))+c, and describe how to construct for any function f(x) the graph of y = A f(k(x-h))+c from the graph of y = f(x).

Explain geometrically why the family of four-parameter transformations A f(k(x-h)) + c can be achieved by varying only two parameters if f(x) is a linear function, while three parameters are required for quadratic, exponential and power functions (with specified power), and four parameters are required for arbitrary power (power not specified) and trigonometric functions.

Explain how to construct a table of fluid amount vs. time from a table of flow rate vs. time, and a table of flow rate vs. time from a graph of fluid amount vs. time. Explain how to generalize to the task of finding an amount chronicle from a rate chronicle and a rate chronicle from an amount chronicle.

Explain how to construct a graph of fluid amount vs. time from a graph of flow rate vs. time, and a graph of flow rate vs. time from a graph of fluid amount vs. time. Explain how to generalize to the task of finding an amount graph from a rate graph and a rate graph from an amount graph.

Explain how to use trapezoids or fundamental triangles to determine the average rate at which some function changes over a specified time interval, or to determine the approximate change in amount over a specified time interval due to a given rate function.

Explain why the rate chronicle obtained from an amount chronicle obtained from a rate chronicle should be close to that original rate chronicle, provided the time intervals are sufficiently small. Explain how the graph of f(x) can be used to estimate the solution(s) of the equation f(x) = 0, f(x) = c, and how the graphs of f(x) and g(x) can be used to estimate the solution(s) of the equation f(x) = g(x).

Describe the rate-of-change behavior associated with linear, quadratic, power or exponential functions.

Describe situations in which linear, quadratic, power or exponential functions arise naturally.

Explain the connection between the rate-of-change behavior of a linear, quadratic or exponential function and its formula or its graph.

Explain how to use fundamental triangles to determine the equation of a line in the plane through two given points or of a circle with given center and radius.

Give examples of real-world in which simultaneous equations in two or more variables arise.

Give examples of how systems of simultaneous equations permit us to fit a linear function to two data points, a power, quadratic or exponential function to three data points, or a polynomial function to four or more points.

Explain how the process of elimination permits us to solve systems of linear equations, and how this process is represented by the process of matrix reduction.

Explain how to optimize the value of a given function f(x,y) over a region of the plane defined by a set of simultaneous linear equations in x and y.

Explain why we can fit a linear function to any two data points and a quadratic, exponential or specified power function to any three.

Explain how any equation can be arranged in the form f(x) = 0, and explain what to look for on the graph of f(x) to determine the solutions.

Give examples of real-world situations in which equations of the type f(x) = 0, f(x) = c or f(x) = g(x) arise.

Explain how to use the graph of a function f(x) to determine its approximate maximum.

Explain how to construct a table or graph of the reciprocal of a function, given a table or graph of the function.

Explain how, given tables or graphs of two functions, to construct the graph of the sum, the product, the difference, the quotient or (either) composite of the two functions.

Describe real-world situations naturally modeled by a function of a function, and explicitly describe the nature of the functions as well as their composite.

Explain how to use the table or the graph of a function to determine whether an inverse function exists and, if so, to construct the table or graph of that inverse function.

Explain how the concept of inverse functions can be used to solve an equation of the form f(x) = c, where f(x) is a function of x, c is a constant and solution means obtaining a value of x which makes the equation true.

Explain how logarithms arise naturally in the attempt to solve exponential equations.

Explain how logarithms are used to model and represent rapid growth in a way that can be grasped intuitively.

Explain how to linearize exponential or power-function data, then use the inverse transformation on the linear regression equation to obtain a functional model.

Explain how the function which linearizes a set of data is related to the function that models the data.

Mechanics and Manipulations

Given a linear, quadratic, polynomial, exponential or logarithmic equation, or any equation involving the basic functions, use appropriate numerical, algebraic, computerized algebraic or graphical techniques to find the solution(s) of the equation.

Algebraically determine the simplest form of the inverse of a given function

Show algebraically why the family of four-parameter transformations A f(k(x-h)) + c can be achieved by varying only two parameters if f(x) is a linear function, while three parameters are required for quadratic, exponential and specified-power functions, and four parameters are required for arbitrary-power and trigonometric functions.

Given a function, a range of values and an increment, construct the corresponding sequence of function values and construct a table for the function.

Using either the sequence and increment or the table, as specified, construct a corresponding table of approximate rates of change. Using either the sequence and increment or the table, as specified, and assuming that the function represents a rate of change with respect to its variable, construct a corresponding table of amount changes cumulative to the initial value of the variable.

Use DERIVE to author, simplify, graph and solve, as appropriate, a given algebraic expression, equation or function.

Use a spreadsheet to compute approximate rates of change and changes in amounts for given amount or rate functions, respectively, given a range of values of the variable and an increment, and to graph the results.

Use DERIVE to construct the graph of a family of functions obtained by changing one or more of the parameters A, k, h and c of the function y = A f(k(x-c))+h, where f(x) is a given function.

Modeling and Problem Solving

Given a situation which can be modeled by one or more algebraically defined functions, give a specific explanatory interpretation of the graph, its slope and/or area (as appropriate to the situation) over a given range of values, and any asymptotes. Show how to represent these quantities algebraically.

Given a situation in which the rate at which a quantity changes depends on that quantity and/or on some independent variable, write the rate-of-change equation that models the situation and use DERIVE to determine the function that solves the equation.

Given a set of dependent variable vs. independent variable data describing a real-world situation, fit an appropriate function to the data. Pose and answer the standard questions related to the behavior of the model (standard questions include determining the value of the independent variable from the dependent, or the dependent from the independent, and rate of change or accumulated amount questions).

Given an unfamiliar problem, use documented graphical and other imaging techniques, algebraic reasoning, geometric reasoning and intuition, and verbal reasoning to attempt a solution.

After registering for the course you will get an email, sent to your VCCS email account, with instructions for Orientation and Startup.  This process will constitute appropriately the first week's assignments for your course (about the first half of the week during the shorter summer term), and will show you the basic navigation of the website including how to communicate, submit work, locate assignments and due dates, and more.

The units to be covered include:

The Mathematical Modeling process, Modeling by Quadratic Functions, Basic Function Families, Introduction to Rates, and use of DERIVE

Modeling by Linear Functions, more DERIVE, Modeling with Proportionality and Power Functions

Modeling by Exponential Functions, Inverse Functions and Logarithms, Linearize Data and Curve Fitting

Polynomials, Properties of Functions, Combining Functions

The material is roughly equivalent to that in Chapters 1-5 of the recommended text.

A more complete and detailed set of measurable objectives is included in the Assignment Page for the course, with which you will become acquainted through the Orientation and Startup.

The instructor is available via web forms (to which you will be introduced at the very beginning of the course), and will normally respond by the end of the day following your submission (and more typically on the same day) with answers to properly posed questions, feedback on your efforts, and other information. Exceptions may occur in the event of Internet problems or other technical events. 

Use of email:  Prior to registration and receipt of initial instructions students my use Email to communicate with the instructor.  However email is much less reliable than web forms, and after registration and receipt of initial instructions anything sent through email should first be sent using the appropriate form.

In the event of a college-wide emergency 

In the event of a College-wide emergency, course requirements, classes, deadlines, and grading schemes are subject to changes that may include alternative delivery methods, alternative methods of interaction with the instructor, class materials, and/or classmates, a revised attendance policy, and a revised semester calendar and/or grading scheme. 

In the case of a College-wide emergency, please refer to the following about changes in this course:

·      Course web page http://vhmthphy.vhcc.edu/ (click on your course)

·      Instructor’s email dsmith@vhcc.edu (however, you should use your access page for the most reliable responses)

For more general information about the emergency situation, please refer to:

·      Web site  - www.vhcc.edu

·      Telephone Number - 276-739-2400

·         Emergency Text Messaging or Phone System- Virginia Highlands Community College uses VHCC Alert to immediately contact you during a major crisis or emergency. VHCC Alert delivers important emergency alerts, notifications and updates to you on your E-mail account (work, home, other), cell phone, pager or smartphone/PDA (BlackBerry, Treo & other handhelds). VHCC Alert is a free service offered by VHCC. Your wireless carrier may charge you a fee to receive messages on your wireless device. VHCC will test the alert system each semester. Register online at alert.vhcc.edu or by sending a text message to 411911 keyword: VHCC 

In the event of an emergency just regarding this class, the instructor will contact all students via email, and may post information to your access site.  You should check both email and your access site.