Review Topics for Final Exam
20 June, 2012
Format of the Exam
The exam will consist of approximately 40-50 questions from the entire course. Approximately 50% of these questions will be from Chapters 1-4 and 50% will be from Chapters
5-9.
There will be one problem that is designed to test your knowledge of the concepts from the
course, and one problem that is designed to test your critical thinking and problem-solving
skills.
The exam will take place from 10 a.m.-12:15 p.m. in ST 104.
§ Sets and Numbers
• Recognize and use proper set notation.
• Know what each of the following symbols means, and how to use them:
N, Z, Q, R, ∈, ⊆
• Determine whether one set is a subset of another.
• Determine whether an object is in a set.
§ Rules for Numbers
• Know the laws of associativity, commutativity, identities, and inverses (and be able to
name them) for addition and multiplication.
• Know the distributive law.
• Determine whether two expressions are equal, and identify which rule(s) is/are followed
or violated.
• Recognize and use interval notation.
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§ Solving Some Simple Equations
• Use the rules from § Rules for Numbers to solve equations.
§ Intro to Functions
• Determine whether a rule is a function.
• Know what the domain and target of a function are (know the definition and be able
to identify the domain/target of a given function).
• Evaluate a function.
§ Sequences
• Classify a sequence as arithmetic, geometric, or neither.
• Find the common difference (for an arithmetic sequence) or common ratio (for a geometric sequence).
• Find the nth term of an arithmetic or geometric sequence, or a sequence which is defined
by an equation.
§ Sums and Series
• Find the value of a generic sum (either with a small upper limit or that you can use a
trick to simplify).
• Find the sum of the first k terms of an arithmetic sequence.
• Find the sum of the first k natural numbers.
• Find the value of a geometric series with −1 < r < 1.
§ Counting I
• Find the number of ways to put n objects in order.
• Find the number of ways to choose & arrange k objects from a pool of n objects.
• Use one of the techniques above or the ”options multiply” technique to find the number
of different ways to do something. (See class examples and homework problems.)
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§ Counting II
• Find the number of ways to choose k objects from a pool of n objects without regard
to order.
• Know the binomial theorem.
• Know how to find the nth row of Pascal’s Triangle
• Expand a binomial (e.g. (2x−3y)7 ) using Pascal’s Triangle and the Binomial Theorem.
• Rewrite nk as a natural number in standard form.
§ More on Functions
• Find the composition of two functions.
• Find
the implied domain of any function we have studied so far (examples: g(x) =
√
4
1
x − 7, f (x) = x−2
, h(x) = 2x2 + 4) and use set notation to write it.
• Know what the domain, range, implied domain, and target are.
• Know how to solve inequalities.
§ Intro to Graphs
• Determine whether a graph is the graph of a function.
• Given the graph of a function:
– Evaluate the function (e.g. Find f (1).)
– Find the range of the function.
– Find the domain of the function.
– Find the x- and y-intercepts of the graph.
• Know the graphs of the basic types of functions:
– id(x)
– x2
– x3
–
–
1
x
1
x2
n
– x where n is even
3
– xn where n is odd
–
–
1
xn
1
xn
where n is even
where n is odd
– constant functions
§ Graph Transformations
• Be able to use, recognize and describe each of the 10 graph transformations from the
section: (d and c are real numbers with d > 0, and c > 1)
– f (x) + d
– f (x) − d
– cf (x)
–
1
f (x)
c
– −f (x)
– f (x + d)
– f (x − d)
– f (cx)
– f ( xc )
– f (−x)
§ Inverse Functions
• Determine whether two functions are inverses.
• Determine whether a function has an inverse.
• Know the definitions of one-to-one and onto, and how they are related to whether a
function has an inverse.
• Find the inverse of a function.
• If a function has an inverse, and you know its graph, graph the inverse function.
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§ nth roots
• Know the definitions of the functions
their targets, and their graphs.
√
n
when n is both even and odd, their domains,
• Find the implied domain of a function involving
• Graph a function involving
√
n
√
n
(either even or odd).
(and possibly some transformations).
§ Basics of Polynomials
• Know the definitions of the terms polynomial, degree, degree k coefficient, leading coefficient, leading term.
• Know how to add, subtract and multiply polynomials.
• Know the relationship between the degree of the product of two polynomials (p(x)q(x))
and the degrees of the polynomials (p(x) and q(x)).
• Find the leading term of a product of polynomials from the leading terms of the
polynomials.
§ Division
• Know how to use long division to find the quotient of two polynomials.
• Know how to write the answer to a division problem (e.g.
4
3
= 1 + 13 ).
• It may be useful to know how to use synthetic division, but you will not be required to
use it. If you do choose to use it, be sure you know which types of problems it works
for, how to set up the problem, how to perform the division, and how to translate your
answer into the language of polynomials.
§ Roots and Factors
• Know the definitions of the terms root, factor.
• Determine whether one polynomial is a factor of another polynomial.
• Determine whether a real number is a root of a polynomial.
• Given a root of a polynomial, write it as the product of a linear polynomial and a
higher degree polynomial. (This requires you to understand the relationship between
factors and quotients and the original polynomials.)
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• Find the roots of a polynomial based on its graph.
• Determine the degree of a product of polynomials.
• Know that a polynomial of degree n has at most n roots. (Important: It may have
fewer. It cannot have more.)
§ Constant and Linear Polynomials
• Know how to graph any constant or linear polynomial.
• Know how to find the roots of any constant or linear polynomial (if they exist).
• Know how to find the slope of a line that passes through two points.
§ Quadratic Polynomials
• Know how to complete the square for any quadratic polynomial.
• Know how to graph any quadratic polynomial in the ”completed square” formula.
• Know what the vertex of a parabola is.
• Know how to determine how many roots a quadratic polynomial has, and how to find
those roots.
§ Factoring Polynomials
• Know the statement of the Fundamental Theorem of Algebra.
• Know how to completely factor any constant, linear, or quadratic polynomial.
• Know how to factor a higher degree polynomial once you know one or more of its roots.
• Know how to find a root of a higher degree polynomial (by checking the factors of the
constant term.)
• Know how to find the roots, x-intercepts, and factors of a polynomial by looking at its
completely factored form.
• Know the relationship between roots and x-intercepts.
§ Graphing Polynomials
• Know how to graph any completely factored polynomial.
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§ Rational Functions
• Know how to graph any completely factored rational function.
• Know how to find the roots and vertical asymptotes of a rational function.
• Be able to find the implied domain of any rational function.
• Be able to calculate the end behavior of any completely factored rational function.
§ Exponential Functions
• Know the definition of the exponential function with base a, including its domain and
target, its graph, and the possible bases for exponential functions.
• Be able to use the rules for exponential functions to rewrite an expression involving
exponentials as a single exponential (when possible).
• Know the rules for exponential functions.
• Be familiar with the number e, both the description in the textbook of where it comes
from and its numerical approximation (2.71).
• Be able to use exponential functions to model situations where a certain quantity
increases or decreases exponentially (by a proportion of the existing quantity after
every year/day/month/million years/unit of time).
• Be able to solve some simple exponential equations using what you know about exponents.
§ Logarithms
• Know the definition of the logarithmic function with base a, including its domain and
target, its graph, and the possible bases for logarithmic functions.
• Be able to find the implied domain of any (transformation of a) logarithmic function
using what you know about the domain of the logarithmic function with base a.
• Know the rules for logarithms and be able to use them to break a logarithm down into
the simplest possible pieces, and to combine logarithms into a single logarithm.
• Know the change of base formula and be able to use it to rewrite logarithms in a
particular base.
• Know how to graph any (transformation of a) logarithmic function.
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• Find the exact value of a logarithm when it is an integer or a rational number, using
the rules for logarithms and/or the change of base formula.
• Find the nearest integers to a logarithm (in particular the largest integer less than the
logarithm, and the smallest integer greater than a logarithm).
• Solve simple logarithmic equations using the fact that loga (x) is the inverse function
of ax .
§ Exponential and Logarithmic Functions
• Solve logarithmic and exponential equations using the rules from §Logarithms and
§Exponential Functions.
§ Piecewise Defined Functions
• Be able to evaluate a piecewise defined function.
• Be able to graph a piecewise defined function.
• Know the definition of the absolute value function |x|, including its domain, range,
graph, and meaning in relation to distances.
• Know how to solve absolute value inequalities.
• Know the two rules for absolute values (multiplication, and the triangle inequality).
§ Linear Equations in Two Variables
• Know the definitions of linear equation in two variables, coefficients, variables, constants and solution set of the equation.
• Know the geometric interpretation of the solution set to a linear equation in two
variables.
• Be able to plot the solution set to a linear equation in two variables.
• Be able to identify or reject a potential solution to a system of linear equations in two
variables.
• Know what is means for a system to have infinitely many solutions, no solutions, or
a unique solution, and be able to determine which category a particular system falls
into by graphing the solution set.
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§ Substitution
• If a system of linear equations in two variables has a unique solution, know how to find
it using substitution.
§ Linear Equations in Three Variables
• Recognize a linear equation in three variables, and know the geometric interpretation
of the solution set.
• Be able to identify or reject a potential solution to a system of linear equations in three
variables.
• Know the different geometric possibilities for the solution set for a system of linear
equations in three variables.
§ Vectors and Matrices
• Know the definitions of vector, matrix, identity matrix.
• Know the relationship between R2 , R3 , and vectors.
• Know how to add matrices and vectors, multiply them by scalars, and multiply matrices
and vectors (if the dimensions are appropriate).
• Be able to check whether two matrices are inverses.
§ Determinants and Inverse Matrices
• Know how to find the determinant of a 2x2 or a 3x3 matrix.
• Know the relationship between determinants and inverses.
• Know how to find the inverse of a 2x2 matrix, if it exists.
§ Matrix Equations
• Know the relationship between matrix equations and systems of linear equations.
• Know how to use properties of matrices and inverses to solve systems of linear equations.
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Formulas and Expressions
You will need to know the following formulas and expressions, what they mean, and how to
use them on the exam:
• Formula for the nth term of an arithmetic sequence.
• Formula for the nth term of a geometric sequence.
• Formula for the sum of the first k terms of an arithmetic sequence.
• Formula for the value of a geometric series with −1 < r < 1.
• n!
n!
(n − k)!
n
n!
•
=
k
(n − k)!k!
n X
n i n−i
n
• (a + b) =
ab
i
i=0
•
• Discriminant of a quadratic polynomial
• Quadratic formula/equation
• Change of Base Formula for logarithms
• Formula for the inverse of a 2x2 matrix
• Formula for the determinant of a 2x2 and a 3x3 matrix
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