Outline: Test 2

Outline for Math 119-Test 2
Date: Wednesday, June 10, 2009

The format of test 2 will be open questions (ie: not mult. choice or t/f). There will be about 8-12 questions many of which will have parts. Partial credit may be awarded if the work shown warrants it. You will be required to show all of your work in order to receive full credit for a solution. You may use a calculator (not the TI 89 or TI92). You may not use any notes, books, etc.

Disclaimer: The fact that a certain specific type of problem is not named below does not preclude it from being covered on the test. This document is intended to give you an outline of the most important topics that have been covered.

1) You must know all terminology in all sections.
Expect some short answer questions that test your knowledge of vocabulary;
some of these may require short computations.
2) You will be expected to solve applied problems comparable to those assigned in each section.

Section 2.1: Quadratic Functions and Models

know what a quadratic function is and what a parabola is (ie: the graph of a quadratic function)
be able to graph any quadratic function algebraically; given a graph, be able to write the rule for the quadratic function
be able to write a quadratic function in standard form by completing the square
be able to algebraically compute the vertex (from standard form)
be able to solve word problems by setting up and solving a quadratic equations

Section 2.2: Polynomial Functions of Higher Degree

know what a polynomial function is
be able to use the properties of continuity and smoothness to decide if a graph is the graph of a polynomial function
be able to use the following properties to decide if a graph of a poly. function is complete:

end behavior (leading coefficient test), number of x-intercepts (zeros), number of local extrema

know how to compute the multiplicity of a zero and its relationship to the graph (ie: even/odd mult)
a polynomial of degree n has at most n distinct zeros
know very well the four equivalent statements on p. 143 (in box: "Real Zeros of Polynomial Functions") and how to get any one of those pieces from any other
be able to sketch the graph of a polynomial function using all of the information above, sign charts and plotting points

Section 2.3: Polynomial and Synthetic Division

be able to find the quotient and remainder for division of two polynomial functions using long division or synthetic division, as appropriate
be able to rewrite the polynomial function, f(x), using the division algorithm (p. 154)
know how to determine if a divisor is a factor of the polynomial (ie: remainder is zero)
be able to compute the remainder if the polynomial f(x) is divided by (x-c) without doing division (ie: f(c)=remainder)
know the relationship between remainders, roots and factors and how to use the remainder theorem and the factor theorem (p. 157)

Section 2.4: Complex Numbers

know what a complex number is
be able to perform all necessary computations with complex numbers
know how to deal with square roots of negative numbers
know when and how to introduce and manipulate complex conjugates
be able to solve quadratic equations over the set of complex numbers (ie: finding complex solutions to quadratic equations)

Section 2.5: Zeros of Polynomial Functions

see class handout for summary of properties you must be able to use in this section
be able to find the zeros for any polynomial f(x) over the complex number system, then use the factor theorem (section 2.3) to write f(x) as a product of linear factors or irreducible quadratic factors
given information about degree, roots, multiplicity, etc, be able to write the rule for a polynomial satisfying the given conditions (must know that given any complex root of polynomial with real coefficients, the conjugate is also a root)

Section 2.6: Rational Functions

know what a rational function is and how to compute the domain of a rational function
know how to find the horizontal and vertical asymptotes for rational functions (and write as eqn of a line)
know how to find the x and y intercepts of a rational function
know how to put all of these pieces together to sketch a complete graph of a rational function with the aid of your calculator
see class handout for detailed guidelines for the steps above

Section 2.7: Non-linear Inequalities

be able to solve polynomial and rational inequalities algebraically using sign charts
be able to use your calculator to solve these inequalities
be able to find the domain of a function using these techniques

Section 3.1: Exponential Functions and Their Graphs

know the definition of an exponential function with base a and how to evaluate function values (by hand and using calculator)
know the graph of an exponential function very well including domain, range, asymptotes, intercepts
be able to use your knowledge of this graph and transformations to answer questions about more general exponential functions and their graphs including domain, range, asymptotes, intercepts
Recommendation: Study Section 1.7, Transformations of functions
be able to use the one-to-one property to solve exponential equations
know that e is a number (about 2.71828182845....) and that it serves as a very common base (natural base) for the exponential function
Memorize the formulas for compounding interest and be able to use these to solve applications

Section 3.2: Logarithmic Functions and Their Graphs

know the definition of the logarithmic function with base a, notation for log base a, and how to evaluate function values (by hand and using calculator)
in particular, know the inverse relationship between log base a and exp fcn with base a
know the graph of a logarithmic function very well including domain, range, asymptotes, intercepts
be able to use your knowledge of this graph and transformations to answer questions about more general logarithmic functions and their graphs including domain, range, asymptotes, intercepts
know the shorthand notation for the common log (base 10) and the natural log (base e)
know the properties of logarithms (p. 230)
be able to use the one-to-one property to solve logarithmic equations

Section 3.3: Properties of Logarithms

know the change of base formula and be able to use your calculator to evaluate log functions with bases other than 10 or e
know the Properties of Logarithms (p. 240) and be able to use them to expand logs, write as a single log, and more generally, for simplifying expressions

Section 3.4: Exponential and Logarithmic Equations

be able to solve exponential and logarithmic equations using the various properties of logs and exp functions covered in the previous three sections
for exponential equations, be sure you isolate the "a^x" part before taking logs of both sides
for the log equations, be sure you "write as a single log" before you exponentiate both sides
where appropriate, you may be required to solve equations exactly (no calculator approximations)

(*): it is required that you check solutions to spot extraneous solns for log equations

Section 3.5: Exponential and Logarithmic Models

be able to use the exponential growth and decay models and the logistic model to solve applications

Additional Problems
Chapter 2 Review pp. 208-211: #1-21, 23-42, 47-88, 91-106, 111-135, 139-147
Chapter 3 Review pp. 271-274: #1-14, 23-30, 35-69, 71-94, 97-115, 119-131, 135, 143-147, 150, 153-155

Good Luck!