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