Exam 1 Rules and Definitions Shormann Precalculus

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364 Terms

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What people group are the oldest know mathematical documents from?

Babylonians (2500 BC to AD 260); not much is known of the pre-flood world.

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solute divided by solution x 100

How do you find the % of solute in a solution?

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commutative property for addition

if a+b=c, then b+a=c

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Difference of 2 squares rule:

a squared - b squared = (a+b)(a-b)

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polynomial in more than one unknown

one term, or a sum of individual terms of the form ax^n y^m z^p..., where a is a real #, x,y,z, etc. are unknown quantities, and n,m,p, etc. are whole #'s.

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"What 2 factors, when multiplied together equal 'c', and when added together equal 'b'?"

What question can you think about when factoring a quadratic polynomial?

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Equation of a line in General form

Ax + By + C = 0

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fundamental theorem of algebra

any polynomial of degree 'n' has 'n' complex roots; NOTE: real #'s are also complex #'s if the form a+bi. In other words, the imaginary part =0

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line

a width-less length; its location is represented on paper by using a pencil and straight edge.

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perpendicular bisector of a chord theorem

if any segment 'AB' passing through the center, 'A', is drawn perpendicular to any chord 'j', then it bisects the chord. The converse is also true: if any segment 'AB' passing through the center bisects any chord 'j', then it is perpendicular to the chord.

<p>if any segment 'AB' passing through the center, 'A', is drawn perpendicular to any chord 'j', then it bisects the chord. The converse is also true: if any segment 'AB' passing through the center bisects any chord 'j', then it is perpendicular to the chord.</p>
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linear function graph

Name the type of function shown
f(x) = x:

<p>Name the type of function shown<br>f(x) = x:</p>
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logarithmic function graph

name the type of function shown f(x)=log (sub b)times x for b>1

<p>name the type of function shown f(x)=log (sub b)times x for b&gt;1</p>
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quadrant

the 4 parts of a coordinate plane, as described below:

<p>the 4 parts of a coordinate plane, as described below:</p>
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inverse sine function

opposite/hypotenuse

<p>opposite/hypotenuse</p>
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Parallelogram Law

the sum, or resultant, of any two vectors equals the diagonal of a parallelogram whose sides equal the magnitudes and directions of the 2 vectors.

<p>the sum, or resultant, of any two vectors equals the diagonal of a parallelogram whose sides equal the magnitudes and directions of the 2 vectors.</p>
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Steps to convert from Rectangular to Polar Coordinates:

1. Get your bearings (draw the vector on a coordinate plane)
2. Draw a right triangle, marking the magnitude, M and direction, 'theta'.
3. Find M using the distance formula.
4. Find 'theta' using the inverse of tangent, recording its actual value relative to the positive x-axis.
5. Write answer in the form M angle (theta).

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disjoint sets

two sets that have no elements in common

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union

when referring to 2 sets, {A} & {B}, where all #'s belong to both sets. When both sets have an identical #, it is only included once when describing this.

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intersection

when referring to 2 sets, {A} & {B}, where only the #'s that the sets have in common is referred to.

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mass divided by volume

How do you calculate density?

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sas

2 triangles are similar if the lengths of 2 corresponding sides are proportional, and the angles between those sides are congruent: side, angle, side

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vertex

where 2 polygon sides, or 2 angle rays, meet

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cone

a solid with a circular base and a lateral surface that comes to a point

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sphere

a solid, which is the set of points a given distance from a given point called the center

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deductive reasoning

the process of applying rules. Mathematics is deductive. There are mathematical rules yet to be discovered.

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hypothesis

the premise or initial facts started with the "if" statement. In science courses, it is often referred to as an educated guess.

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A3

From Euclid's 5 axioms, or common notions:
If equals be subtracted from equals, the remainders are equal.
If a=b and c=d, then a-c=b-d

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horizontal line test

a graph on a coordinate plane is a one-to-one function if a horizontal line intersects the graph at only one point.

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slope

a ratio expressing the change in the dependent variable (y) with respect to the independent variable (x). Also referred to as "rise over run", "change in y over change in x", and "delta 'y'/delta 'x' where "delta (triangle shape)" = "change in".

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ratios of 45-45-90 right triangle

knowt flashcard image
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polar form

a way to represent a vector by its magnitude (M) and direction (degrees-"theta"). Often written using the notation M (angle) "theta"

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30-60-90 triangle

knowt flashcard image
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infinitesimally

a quantity whose value is decreasing without bound. The idea that a value can be close to, but not equal to, zero. super small change; instantaneous.

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inductive reasoning

the process of discovering rules.

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deductive reasoning

the process of applying rules

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numeral

a symbol or symbols used to express the idea of #

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abstract

dealing with the properties & ideas of things

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concrete

a word used to describe real objects; abstract ideas are always based on real things.

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digit

any of the Hindu-Arabic numerals 1-9, and 0

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natural #'s

counting #'s; 1,2,3...etc.

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whole #'s

counting #'s & zero

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integers

whole #'s & all negative #'s

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real #'s

any number used to describe a positive or negative #; includes all integers & all decimal & fractions.

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rational #

a # that can be written as a fraction of integers

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irrational #

a # that cannot be written as a fraction of integers; have non-repeating decimal patterns

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trichotomy axiom

for any 2 real #'s and b, exactly one of the following is true:
a<b a=b a>b

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square root

if x is greater than 0, then the square root of x is the unique positive real # such that the square of the square root of x = x.

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imaginary #'s

the result of taking the square root of a negative number. Normally the square root of -1 is factored out and exchanged for the symbol "i". i squared = -1

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complex #'s

numbers that have a real part & an imaginary part: ex. a+bi
note - all real #'s are complex with b=0

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subset

a special group of elements that is entirely contained within a larger set.

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logic

the art of reasoning well.

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idea

a number is an _______.

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mathematics

This is Dr. Shormann's definition of what term:
"_______ is the language of science and a God-given tool for measuring and classifying pattern and shape."

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{}

this represents a 'set' of things

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U

union of 2 sets

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upside-down U

intersection of 2 sets

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empty set; null set

An "o" with a diagonal line through it

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vertical line '|'

means "such that"

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A fancy, bold 'N' represents what in set notation?

"the set of natural #'s"

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A fancy, bold 'Z' represents what in set notation?

"the set of integers"

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A fancy, bold 'Q' represents what in set notation?

"the set of rational #'s"

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A fancy, bold 'R' represents what in set notation?

"the set of real numbers"

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universal set

The enclosing rectangle of 2 sets depicting a Venn diagram; maybe illustrating an intersection or union.

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Jeremiah 33:3

"Call unto me, and I will answer thee, and shew you great and mighty things which thou knowest not." (KJV)

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sexagesimal

the base 60 numeral system which we still use today (60 seconds in a minute, 60 min. in an hour, 360 degrees in a circle.) This originally comes from the ancient Babylonians.

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one

Any # divided by itself equals what?

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1/x

What is the reciprocal of x?

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Logarithm rule

knowt flashcard image
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ratio

the size of one thing relative to another.

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fraction

part of a whole; a numerical quantity that indicates division of one # by another.

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prime #

a # that is only divisible by itself & 1, such as 2,3,5,7 etc.

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numerator & denominator

What is the "top" and "bottom" # of a fraction respectively?

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logarithm

in any exponential relationship of the form N=b to the 'L' exponent; 'L' can be referred to as the _______ of N

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proportional

having a constant ratio

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analogy

resemblance in some particulars between things otherwise unlike; similar

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Logos

the original Greek manuscripts used this in place of 'Word'

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proportion

equal ratios

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circumference divided by the diameter of a circle

Where does the calculation for "pie" come from?

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Natural log is also called Log 'e'. What is it's value and where is this # used?

the 'e' is named for the scientist, Euler. Euler's # is approximately = to 2.718 and is used in science, specifically in biology to measure bacteria.

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Log 10 & Log 'e'

What are the 2 most commonly used logarithms?

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distance divided by time

How do you calculate rate?

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1000 grams

What does 1000mL of water weigh?

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commutative property of multiplication

if ab=c, then ba=c

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associative property for addition

(a+b)+c = a+(b+c)

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associative property for multiplication

(a times b)c = a(b times c)

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distributive property

a(b+c) = ab + ac

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additive identity

a+0=a ; any # plus zero equals that #

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multiplicative identity

a(1) = a ; any # times 1 equals that #

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power rule for exponents

if m and n are real #'s and x does not = zero, then

<p>if m and n are real #'s and x does not = zero, then</p>
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Logarithm Laws:

knowt flashcard image
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algebra

a generalization of arithmetic, where letters representing #'s are combined according to the rules of arithmetic, often to solve for a unknown value

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one

in algebra, you don't have to write the 1 (one)

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polynomial in one unknown:

one term, or a sum of individual terms of the form ax^n, where a is a real #, x is an unknown quantity, and n is a whole #.

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Hisab al-jabr

a book written by Islamic astronomer meaning "science of transposition & cancellation." This is where we get 'algebra' from. (sound it out)

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additive property of equality

if a, b, and c represent real #'s, and if a=b, then a+c=b+c also: c+a=c+b

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multiplicative property of equality

if a, b, and c represent real #'s, and if a=b, then ca=cb also: ac=bc
and: a/c = b/c

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parallel lines

"train tracks" have the same slope

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perpendicular lines

where 2 lines cross making 4 right angles; these lines have slopes that are negative reciprocals of each other

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slope

a ratio expressing the change in the dependent variable (y) with respect to the independent variable (x). Also referred to as "rise over run", "change in y over change in x"; also:

<p>a ratio expressing the change in the dependent variable (y) with respect to the independent variable (x). Also referred to as "rise over run", "change in y over change in x"; also:</p>
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delta

"change in"