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math 20 cumulative theory

set

a collection of objects, the objects in the set are called elements.

prime number

natural number whose only divisors are 1 and itself, 1 is not a prime number.

subset

all of the elements in one set are also contained in another set

union

the set of all objects in A or B or both

intersection

set of all objects contained in A and B.

complement

complement of A is the set of elements in the universe but not in A.

natural numbers

1,2,3,4... N

integers

0, natural numbers, negatives of the naturals. Z

rational number

can be expressed as a ratio of two integers, A/B where B cant be 0. Q

irrational number

cant be expressed as a ratio of two integers. ex pi, square root of prime numbers, R\Q

real numbers

all numbers on a number line. R. rationals take up 0% of R

theorem about rational numbers

every rational number can be written both as a ratio of integers and as a terminating or repeating decimal

density

A set has the density property if between any two numbers of the set there is another number that is also a member of the set. As a result, any two members of the set have infinitely many other members between them.

limit

a value that a sequence approaches as the number of terms approaches infinity

fundamental theorem of arithmetic

any natural number greater than 1 can be written as a unique product of primes. ex) 6=2x3

which roots can be negative

cube roots can, square roots cannot

product law

a²+a³ = a²+³

quotient law

a³ / a² = a³-²

power rule

(a²)³ = a²³

distributivity over products

(ab)³ = a³b³

distributivity over quotients

(a/b)³ = a³/b³

power of zero

a⁰ = 1

radical

n√x is a symbol used to denote the nth root of some number x. there can be a coefficient, n is the index, x is the radicand. when there is no coefficient it is 1.

entire radical

radical where the coefficient must be 1 or -1

mixed radical

radical where the coeffiecent cannot be 1 or -1

when is a radical simplified

when the radicand does not contain any factors that are perfect powers of n

cartesian product

cartesian product of two sets A and B is denoted by AxB, is the set of all possible ordered pairs where the first element comes from A and the second from B. (a,b)

cartesian plane

contians all possible pairings of the real numbers, if the cartesian plane represents RxR=R²

relation

between two sets A and B is a subset of AxB, a set of ordered pairs

domain

(a,b) first element

range

(a,b) second element

function

a relation is a function if each element in the domain is sent to only one element in the range. f(n)

a line segment contains..

infinitely many points

parallel lines

never meet, always have the same slope, no solution, different y intercept

perpendicular lines

meet at a right angle, slopes are negative reciprocals of each other, multiplying them gives -1

slope intercept form

y=mx+b, m is slope b is y intercept

point slope form

y-y1=m(x-x1)
x1 y1 is a point on the line, m is slope

general form

Ax+By+C=0. a is a whole number and b and c are integers

vertical line test

draw line through each point in domain. if it intersects the graph at ony one point, it is a function

continuous data

can take any value, no gaps in the graph. denoted by intervals.

discrete data

can only take on certain values, gaps in the graph

closed interval

A closed interval includes all real numbers between two given endpoints, and it also includes those two endpoints. [a,b], closed circles

open interval

An open interval contains all real numbers between two distinct real numbers, but excludes the two numbers themselves. (a,b). not the same as ordered pair, open circles

undefined

some functions are undefined at certain values. this is a discontinuity at zero, causes a hole in the graph, represented by an open circle

slope

rise/run, vertical change/horizontal change

horizontal line

slope of zero and equation y=#

vertical line

undefined slope and equation x=#

solve

means to find the value of x that satisfies the equation

how do you solve a linear equation with two variables

need to have two equations, a system of linear equations

three ways to find solution of system

graph lines, elimination, substitution

intersecting lines

one solution, different slopes

coincident lines

infinite solutions, same slope and same y intercept, equations are related by multiplying or dividing by same number

rearrange for radius of cylinder

r²=V/πh

rearrange for radius of cone

r²=3V/πh

rerrange for radius of sphere given SA

r²=SA/4π

rearrange for radius of sphere from volume

r³=3V/4π

surface area of hemisphere

3πr² circular base on surface of hemisphere, so the surface area will be half the surface area of sphere, plus the area of circular base

foiling

multiply each term of the first binomial with each term of the second binomial, also called expanding

perfect square

number multiplied by iteself, can also be expanded/foiled

perfect square trinomial

result after binomial squared is expanded

difference of squares

when two binomials are multiplied together that are the same except with the opposite sign between them, middle terms cancel, so you are left with the first term of each binomial squared minus the second term of each binomial squared

what is factoring

reverse process of expanding

what can a trinomial be factored as (a=1)

two binomials.
ax²+bx+c where a is equal to one

(x+_)(x+_)

with first blank being add to get b, second being mulitply two get c

how to factor a trinomial (a not equal to 1)

ax²+bx+c
(_x+_)(_x+_)

with first and third blanks being factors of a, second and fourth blanks being factors of c, so that the outside and inside products combine to give the middle term bx

trigonometry

study of triangles. examines relationship between sides and angles. use capital letters to label angles, lowerscase to label sides, written opposite from their respective angles

how to tell the sides

hyp= longest, opposite from right angle
opp= opposite from reference angle

adj= adjacent to reference and right angle

how can trig ratios be expressed

reduced fractions or decimals (4 places)

solve a triangle

determine measures of all unknown angles and lengths of all unkwon sides.

angle of inclination

acute angle it makes with horizontal

angle of elevation

angle between horizontal and line of sight. object above observer

angle of depression

angle between horizontal and line of sight. object below observer

pneumonic for metric units

king henry doesnt usually drink chocolate milk
kilo hecto deka unit(m) deci centi milli

each step is ten times as much as step on other side

1kilo=10hecto=100deka=1000m, etc

right cone

circular base and curved surface. base of cone is a circle, so the perimeter is the circumference of the circle (2πr), so surface area becomes πrs+πr²

volume of a prism

Ah

how does volume of right pyramid compare to prism

1/3

how does volume of cone compare to cylinder

1/3

volume of hemisphere

V=2/3πr³ half the volume of a sphere

composite objects

two or more distinct objects, to find volume, add volumes of distinct objects. to find surface area, you have to remove the bases and then add the surface area.
ex) hemi (no base) + cylinder (-1 circle)

2πr² + (πr²+2πrh)

3πr²+2πrh

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