Study Guide for Exams in a Course on Mathematical
Modeling
MATH 465 Test #1 -- Wednesday, October 22
Bring a calculator and graph paper to the exam session.
Describe the process we call mathematical modeling, and illustrate
that process with a diagram and an example. [Class notes or p. 40]
Distinguish between "modeling from data" and "theoretical
modeling."
Identify the objective, or aim, the authors of our textbook had in
mind when they set out to write the book. Describe the strategy have they
employed to accomplish their objective.
Explain the meaning of the terms "mathematical modeling"
and "mathematical model."
Discuss some strengths and limitations of mathematical modeling.
Given a real phenomenon and relevant data, demonstrate a variety of
strategies one can employ to find a mathematical model that might express
the relationship between the physical quantities involved. (manipulating
data, transforming data, graphing data) [Technique illustrated in Example
1 on pp. 2-3; ln-ln plot used in class for Tutorial problem 7 on p.9]
Given a real phenomenon and relevant data, determine when one or more
of the following kinds of functions might provide a reasonable "fit"
to the data. (linear or affine, power or polynomial, exponential) [Similar
to the previous item.]
Specify several criteria for determining how well a functional model
"fits" a set of data. Employ several of those criteria to determine
"goodness of fit." [Revisit item #3 on Assignment #3.]
Distinguish between "change," "rate of change,"
"average proportionate rate of change," and "instantaneous
rate of change."
Employ analysis of different types of "change" in formulating
mathematical models. [linear, exponential, logistic, polynomial]
Employ qualitative analysis of graphs of data to predict what kind
of functional relationship might be represented in the underlying data; then
fit an appropriate functional model to the data. [linear, exponential,
logistic, power, polynomial]
Use differences and ratios to infer a functional relationship in the
data. [linear, polynomial, logistic, power, polynomial]
Given a real situation or problem, compile a "features list."
Refine that list so it includes only the most "relevant" or "sensible"
features. [Tutorial Problem 2 on p. 31; Sections 2.1, 2.2]
Given a real situation or problem and a list of relevant features,
formulate a model in the form of a sentence (a word model). Translate the
word model into mathematical symbols. [Tutorial Problem 3 on p. 33;
Sections 2.3, 2.4; Exercises 1(a), 2(a) on p. 41]
Given a recurrence relation, or difference equation, determine qualitative
properties of the relation and its graph without finding either a numerical
solution or a general solution. [Exercises 1(b), 2(b) on p. 41]
Given a recurrence relation, or difference equation, find (a) numerical
solutions. (b) a general solution. [Section 2.4; Exercises 1(c), 2(c)
on p. 41]
Given a recurrence relation, or difference equation, formulate, investigate,
and solve (numerical or general solution) a differential equations model
for the same phenomenon. [Tutorial Problem 13 (exponential model only)
on p. 38]
Given a verbal description of population in a particular kind of environment,
formulate a mathematical model capturing the essential elements of the
situation, analyze and solve equations in your model, and make predictions
based on your model. [Take-home problem similar to Exercises 3 and 4
on pp. 41-42]
Given a system of difference, or differential, equations, (a) determine
qualitative properties of the relations, and (b) numerical and graphical
solutions. (Later - Test #2)
Given a mathematical model for a phenomenon, evaluate and critique
the model. (Later - Test #2)
Construct, manipulate parameters in, and interpret a simple Stella
model. (Later - Test #2)