Goals. The primary purpose of this course is to
prepare students with only
two years of high school algebra and a limited introduction to trigonometry for
our courses in calculus. Since the mathematics treated in this course is of wide
utility it also serves as a general education course suitable for students in
some
disciplines. It should be noted that it is not the preferred general education
course for most students. The material covered in this course should ideally be
part of the high school background of most students intending to take calculus
as part of their degree program. It is offered at the university as a service to
students without this background.
Knowledge and abilities. The central theme of this course is the concept
of a
(real-valued) function (of a real variable ), the modes by which such functions
can
be represented and expressed, the language in which functions are described and
analyzed, the use of functions as models for relationships between variable
quantities
in mathematics and numerous applications, and a survey of the standard
families of elementary functions: linear, polynomial esp. quadratic, rational,
algebraic, logarithmic/exponential, and trigonometric.
What do we mean by “know?” In the following the word “know” will be
used as a shorthand for a level of mastery that entails being able to
• State an accurate and complete version of the definition, theorem, formula,
technique or example in question, using conventional mathematical
language and notation
• Describe or explain the item to someone else using clear written or spoken
English, graph sketches, or numerical examples
Basic skills and problem solving. Mathematical proficiency is achieved
cumulatively.
Students in precalculus must build upon, solidfy, and refine skills
that should have been acquired in high school mathematics courses in algebra
and geometry, including basic geometric trigonometry. This process of review
and progression in basic algebraic and problem- solving skills occurs along with
the study of the main classes of elementary functions, e.g. polynomial
arithmetic
skills in the study of polynomial functions, properties of logarithms in
the study of exponential functions, modelling and application (i.e. “word
problems”)
throughout.
All students of mathematics, at every level, learn mathematics by solving
problems
that involve the concepts and techniques they are studying and their
applications.
To be effective, these problems must be appropriately varied and
challenging and levels of mastery expected are best conveyed by sample problem
material. Likewise, assessment of student learning is most effectively done
through problem-solving in a variety of formats.
The appropriate use of technology is part of the skill set
of this course. Graphing
calculators are required and used to enhance the range of problems that students
can solve and the mathematical phenomena they can investigate.
Note on specificity. To make outcomes and expectations as clear as
possible,
examples and sample questions have been included in the topic oriented outcome
descriptions below. These should not be interpreted as the only specific
problems, techniques and concepts that students are expected to master, or on
which they will be assessed. They are intended as significant landmarks and as
indicators of level of depth.
Functions. Students should know the informal definition of a function as
a rule
of assignment, to use standard notation for functions, make inferences about
functions presented exactly by formulas or less completely by tables, graphs or
verbally, while moving between such representations. They should know about
domain, range, increasing/decreasing, one-to-one, arithmetic on functions,
function
composition and decomposition, inverse.
Examples of abilities:
• Be able to de termine if a given graph is the graph of a function.
• Given the graph of a function f, determine f(3) or f -1(2) when appropriate,
where f is increasing, the number of solutions of f(x) = 0 in a given
interval and make inferences about the domain, range, and existence of
an inverse.
• Given a formula for f, determine an expression e.g. f(x2+1), for (f o f)(x),
for f -1(x) in simple cases .
• Given a table of values for f determine if the function is increasing, is
plausibly one-to-one,
Outcomes by topic
Linear functions. Students should
1. Know about slopes and equations of lines . They should be able to quickly
give a rough estimate of the slope of a graphed line and determine such slopes
exactly from the graph, or from tables of function values. They should know
the point-slope and slope-intercept forms and the ax + by = c form, be able to
convert between these and know the graphical significance of the coefficients in
each case . They should know that linear functions are characterized by constant
rate of change and be able to check if tabulated data could plausibly arise from
a linear function. Students should know the slope relations between parallel and
perpendicular lines.
2. Be able to solve linear equations and inequalities
without a calculator . They
should be able to construct simple linear models from tabulated data when
appropriate,
and make simple predictions by extrapolation. They should know
that linearity often provides a reasonable first approximation over small
domains.
They should be able to provide some commonplace examples of linear
functions and examples of non-linear functions.
Quadratic functions. Students should
1. Know the general form f(x) = ax2 + bx + c and the signficance of the
parameters in the parabolic graph of f. They should be able to complete the
square to express f as f(x) = a(x−k)2+h and use this to determine the vertex
of the parabola . They should be able to use this algebraic/graphic information
to solve simple max/min problems.
2. Be able to solve quadratic equations by factoring (in simple cases),
completing
the square, and the quadratic formula. They should know the significance
of the discriminant and the way in which complex numbers arise from the
quadratic formula. Students should also be able to solve quadratic inequalities.
They should be able to solve problems involving things like areas, velocity and
acceleration using the techniques studied.
Polynomial functions. Students should
1. Know what a polynomial is and be able to quickly determine whether a given
expression is nor is not a polynomial. They should know the meaning of degree
and leading term and the relation of degree to the maximum number of zeros
and turning points of its graph. They should know how to determine the the
global behavior (“end behavior”) of a polynomial function by examining the
leading term.
2. Know the relation between the factors and the roots of a polynomial and use
this relation to find zeros from factors and to construct a polynomial with
given
zeros.
3. Be able to determine polynomial zeros approximately using a graphing device.
4. Be able to add and multiply polynomials, including those in more than one
variable, using the distributive property (not the limited FOIL method), be able
to perform polynomial division and interpret the results of that division, be
able
to perform simple factoring.
