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Math II > Midterms > Flashcards

Midterms Flashcards

1
Q

two-valued logic

A

every statement is either True or False

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2
Q

truth table

A

used to determine the truth or falsity of a complicated statement based on the truth or falsity of its simple components

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3
Q

conjunction

A

“and”; true when both statements are true; ^

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4
Q

disjunction

A

“or”; true when @ least 1 statement is true, V

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5
Q

negation

A

“not”, ~

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6
Q

inclusive “or”

A

doing 1/other/both

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7
Q

when is p → q not true?

A

when p is true and q is false

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8
Q

tautology

A

rule of logic

a formula which is “always true” - all the end results are true

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9
Q

p ⇔ q

A

p iff q

both p & q r equivalent. true if p & q r both true/both false

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10
Q

contradiction

A

opposite of a tautology, a formula which is “always false”

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11
Q

what r p &q called in p ⇒ q?

A

p = hypothesis

q = conclusion

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12
Q

p ⇒q

A

if p, then q

p implies q

p if q

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13
Q

4 ways to rewrite a statement

A

1) if p, then q
2) Every p has q.
3) The fact that p, implies that q
4) p iff/if/only if q

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14
Q

converse

A

q ⇒ p

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15
Q

inverse

A

~p ⇒ ~q

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16
Q

contrapositive

A

~q ⇒ ~p

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17
Q

Direct Argument

A

p ⇒ q

p

…q

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18
Q

premise

A

a statement that is assumed to be true

a given statement in an argument. the resulting statement is called the conclusion

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19
Q

Indirect Argument

A

p ⇒ q

~q

… ~p

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20
Q

Chain Rule

A

p ⇒ q

q ⇒ r

…p ⇒ r

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21
Q

Or Rule

A

p V q

~p

…q

p V q

~q

…p

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22
Q

good definition

A

built from a true conditional with a true converse

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23
Q

invalid argument

A

argument that doesn’t use rules of logic

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24
Q

4 rules of biconditionals

A

p ⇔ q

p

… q

p ⇔ q

q

… p

p ⇔ q

~p

… ~q

p ⇔ q

~q

… ~p

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25
Q

Venn diagram placement for conditionals/implications

A
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26
Q

two-column proof

A

a proof written in 2 columns

statements r listed in 1 column & justifications r listed in the other column

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27
Q

paragraph proof

A

a proof whose statements & justifications r written in paragraph form

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28
Q

flow proof

A

’s written over the arrows refer to a #-ed list of the justifications 4 the statements

a proof written as a diagram using arrows to show the connections b/w statements

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29
Q

postulate

A

a statement assumed to be true w/out proof

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30
Q

Addition Property of Equality

A

If the same # is added to equal #’s, the sums r equal

a = b → a + c = b + c

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31
Q

Subtraction Property of Equality

A

If the same # is subtracted from equal #’s the diff’s r equal

a = b → a - c = b - c

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32
Q

multiplication property of equality

A

If equal #’s r multiplied by the same #, the products r equal

a = b → ac = bc

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33
Q

division property of equality

A

if equal #’s r divided by the same nonzero #, the quotients r equal

a = b and c =/ 0 → a/c = b/c

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34
Q

reflexive prop of equality

A

a # is equal to itself

a = a

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35
Q

substitution property

A

if values r equal, 1 value may be substituted 4 the other

a = b → a may be substituted for b

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36
Q

distributive prop

A

An expression of the form a(b + c) is equivalent to ab + ac

a(b + c) = ab + ac

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37
Q

square root

A

one of 2 equal factors of a #

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38
Q

straight angle postulate

A

If the sides of an angle form a straight line, then the angle is a straight angle with a measure of 180

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39
Q

angle or segment addition postulate

or

whole and parts postulate

A

for any segment/angle, the measure of the whole is equal to the sum of the measures of its non-overlapping parts

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40
Q

Supplements of Angles Theorem

A

If 2 angles r supplementary to the same angle, then they r equal in measure

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41
Q

complements of angles theorem

A

If 2 angles r complements of the same angle, then they r equal in measure

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42
Q

vertical angle theorem

A

All vertical angles r equal in measure

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43
Q

corresponding angles postulate

A

if 2 parallel lines r intersected by a transversal, then corresponding angles r equal in measure

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44
Q

alternate interior angles theorem

A

if 2 parallel lines r intersected by a transversal, then alternate inteiror angles r equal in measure

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45
Q

what kinds of angles r these?

A

angles 3 & 6 r alternate interior angles

angles 1 & 8 r alternate exterior angles

angles 3 & 5 r cointerior angles

angles 2 & 6 r corresponding angles

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46
Q

co-interior angles theorem

A

If 2 parallel lines r intersected by a transversal, then co-interior angles r supplementary

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47
Q

what kinds of angles r shown below?

A

Angles A & D r consecutive angles

Angles A & C r opposite angles

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48
Q

Consecutive Angles th

A

If a quadrilateral is a parallelogram, then consecutive angles r supplementary

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49
Q

opposite angle theorem

A

If a quadrilateral is a parallelgoram, then opposite angles r equal in measure

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50
Q

Venn diagram placement for biconditionals

A
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51
Q

hypothesis

A

the if part of an if-then statement

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52
Q

conclusion

A

the then part of an in-then statement

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53
Q

implication/conditional

A

a statement with an if part & a then part

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54
Q

biconditional

A

the conjunction of a true conditional & its true converse, usually written using the phrase if and only if (iff)

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55
Q

converse of corresponding angles postulate

A

If 2 lines r intersected by a transversal, & corresponding angles r ocngruent, then the lines r parallel

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56
Q

converse of cointerior angle th

A

if 2 lines r cut by a transversal & cointerior angles r supplementary, then the lines r parallel

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57
Q

converse of alt interior angle theorem

A

if 2 lines r cut by a transversal & alternate interior angles r congruent, then the lines r parallel

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58
Q

perpendicular & parallel th

A

if 2 lines r perp 2 the same transversal, then they r parallel

59
Q

perp & transversal theorem

A

if a transversal is perp to one of 2 parallel lines, then it is also perp to the other line

60
Q

trapezoid

A

quadrilateral w/ only 1 pair of parallel sides

61
Q

triangle sum theorem

A

the sum of the measures of the angles of a triangle is 180

62
Q

quadrilateral sum theorem

A

the sum of the measures of the angles of a quadrilateral is 360

63
Q

quadrilateral is parallelogram theorem

A

if both pairs of opp angles of a quadrilateral are equal, then the quadrilateral is a parallelogram

64
Q

exterior angle theorem

A

an exterior angle of a triangle is equal in measure to the sum of its 2 remote interior angles

65
Q

definition of similar

A

2 triangles are similar iff their vertices can be matched up so that corresponding angles r equal & corresponding sides r equal in proportion

66
Q

definition of congruent

A

2 triangles r congruent iff their vertices can be matched up so that the corresponding parts (angles & sides) of the triangles r equal in measure

67
Q

triangle similarity postulate (AA Post.)

A

If 2 angles of a triangle r equal to 2 angles of another triangle, then the 2 triangles r similar

68
Q

how to find any angle of a triangle?

A

A = 1/2ab sin C

69
Q

trig

A

SOH CAH TOA

70
Q

overlapping similar triangles

A

if a line is drawn from a pt on 1 side of a triangle parallel to another side then it forms a triangle similar to the original triangle

71
Q

ASA theorem

A

if 2 angles & the included side of 1 triangle r equal to the corresponding angles & side of another triangle, then the triangles r congruent

72
Q

AAS theorem

A

if 2 triangles & a non-included side of 1 triangle r equal to corresponding angles & side of another triangle, then the triangles r equal

73
Q

ASS

A

IZ NOT GOOD

74
Q

SAS Post

A

if 2 sides & the included angle of 2 triangles r equal to the corresponding sides & angle of another triangle, then the triangles r congruent

75
Q

angle bisector

A

a ray, line, or segment that divides an angle into two equal parts

76
Q

segment bisector

A

a line, ray, / segment that divides a segment into 2 equal parts

77
Q

perpendicular bisector

A

a line, ray, / segment that bisects the segment & is perpendicular to it

78
Q

Hypotenuse-Leg Theorem

A

2 right triangles r equal if the hypotenuse & leg of 1 triangle r equal to the hypotenuse & leg of the other triangle

79
Q

isosceles triangle

A

a triangle w/ 2 sides = in measure

80
Q

equilateral triangle

A

a triangle in which all the sides r equal in measure

81
Q

equiangular triangle

A

a triangle in which all the angles r equal in measure

82
Q

Isosceles Triangle Theorem

A

if 2 sides of a triangle r = in measure, then the angles opposite those sides are = in measure

83
Q

converse of isosceles triangle th

A

if 2 angles of a triangle r equal in measure, then the sides opp those angles r = in measure

84
Q

equilateral triangle th

A

if a triangle is equilateral, then it’s also equiangular

85
Q

converse of equil triangle th

A

if a triangle is equiangular, then it’s also equilateral

86
Q

perpendicular bisector th

A

if a pt is the same distance from both ends of a segment, then it lies on the perp bisector of the segment

87
Q

similar right triangle th

A

if the altitude is drawn to the hypotenuse of a right triangle, then the 2 triangles formed r similar to each other & the original triangle

88
Q

geometric mean

A

if a, b, and x r pos #’s, and a/x = x/b, then x is the geometric mean b/w a and b. the GM is ALWAYS the pos root

89
Q

geometric mean th

A

if the altitude is drawn to the hyp of a right triangle, then the measure of the altitude is the geometric mean b/w the measures of the parts of the hypotenuse

90
Q

30-60-90

A

opp 30 - x

hypotenuse - 2x

opp 60 - x[3

91
Q

45-45-90

A

opp 45’s - x

hypotenuse - x[2

92
Q

pythagorean theorem

A

in any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs

a2 + b2 = c2

93
Q

polynomial

A

an expression tht can be written as a monomial/sum of monomials w/ whole # exponents

94
Q

degree of a polynomial

A

the largest exponent

95
Q

what can a polynomial NOT be?

A

it can’t have negative/fractional exponents

can’t have variables in denominators

BE SURE TO SOLVE FIRST SO THAT IT’S JUST AN EXPRESSION NOT AN EQUATION

96
Q

polynomial equation

A

an equation tht can be written w/ a polynomial as 1 side & 0 as the other side

97
Q

standard form of polynomials

A

from left to right, exponents go from largest to smallest

98
Q

rational number

A

can be written as quotient of 2 integers

99
Q

rational expression

A

an expression that can be written as the quotient of 2 polynomials

100
Q

rational equation

A

an equation w/ only rational expressions on both sides

101
Q

product of powers rule

A

am • an = am+n

102
Q

quotient of powers rule

A

am/an = am-n

a does not equal 0

103
Q

how to factor polynomials

A
  1. factor out any GCF’s
  2. factor normally (ax2 + bx + c)
104
Q

ratio of the volume of 2 spheres

A

equal to the ratio of the cubes of their diameters

105
Q

power of a power rule

A

(am)n = amn

106
Q

power of a product rule

A

(ab)n = anbn

107
Q

power of a quotient rule

A

(a/b)n = an/bn

provided tht b doesn’t equal 0

108
Q

how to solve rational equations

A
  1. find LCD of all 3 & make the denominators all the same & cancel them out OR find LCD of left side denominators, make them the same, & then cross multiply
  2. simplify
109
Q

extraneous solution

A

when a solution isn’t a solution of the original equation

test it

110
Q

point of no return

A

the pt where it would take as much time to return to the starting point as to continue on to the destination

use t = d/r

d/r = d/r

111
Q

cubic function

A

a polynomial function of degree 3

112
Q

zero of a function

A

value of x that makes y = 0

same as x-intercept

113
Q

double zero

A

cubic function w/ a squared factor

114
Q

triple zero

A

a cubic function w/ a cubed factor

115
Q

y = (x - 2)3

A

intersects x-axis once

116
Q

y = (x - 2)2(x + 1)

A

intersects x-axis twice

(x+1) part goes thru at 1

(x-2)2 part touches at -2

117
Q

y = (x - 2)(x + 1)(x + 4)

A

intersects x-axis three times

118
Q

steps to solve a cubic equation w/out graphing

A
  1. put y on left side & set it to 0
  2. find the GCF of the right side if possible & factor it out
  3. if ax2 + bx + c format, use quadratic formula
119
Q

how to solve cubic equations w/ calculator

A
  1. input equation
  2. find x-intercept (where y = 0)
120
Q

square root of negative number

A

becomes i • pos. square root

121
Q

parametric equation

A

equations where 2 variables r expressed in terms of a 3rd variable

122
Q

parameter

A

3rd variable of a parametric equation

123
Q

how to graph a parametric equation

A
  1. make a table of values (time, x, y)
  2. graph using x & y

OR

  1. change to PAR mode
  2. input equations
124
Q

how to write an equation for y in terms of x

A
  1. solve for t with either of the equations (preferably x)
  2. substitute into the other equation not used in step 1, t
  3. simplify
125
Q

3D coordinate plane

A
126
Q

how to find a coordinate on a coordinate plane

A
127
Q

3D midpt formula

A

(x1 + x2/2, y1 + y2/2, z1 + z2/2)

128
Q

length of a diagonal of a rectangular prism

A

w/ edges of length x, y, and z & a diagonal length of d

d = [x2 + y2 + z2]

129
Q

3D distance formula

A

d = [(x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2]

130
Q

equations of circles & spheres centered @ the origin

A

circle ⇒ x2 + y2 = r2

sphere ⇒x2 + y2 + z2 = r2

131
Q

how to graph circle x2 + y2 = 36

A
  1. since it’s in this form, u kno the center is in (0,0) and that r2 = 36 → r = 6.
  2. sketch the circle w/ center at Origin & (0,6) at top, (6,0) at right, (0, -6) at bottom, and (-6,0) at left
  3. use a graphing calc to graph 2 semicircles: x2 + y2 = 36 → y2 = 36 - x2 → y = pos/neg [36 - x2]
           \*to make circle circular, use Zoom Square
132
Q

how to find the equation of a circle/sphere

A
  1. find center
  2. decide if it is at O or other & use the right equation
  3. find length of radius
  4. plug in radius
133
Q

equations of circles not centered at origin

A

center (h,k) with radius r

(x - h)2 + (y - k)2 = r2

134
Q

coterminal angles

A

angles in standard position (angles with the initial side on the positive x-axis) that have a common terminal side.

For example 30°, –330°

135
Q

unit circle quadrant formulas

A

Quadrant I - All +, leave alone, (+,+)

Quadrant II - sin +, 180 - a, (-,+)

Quadrant III - tan +, a - 180, (-,-)

Quadrant IV - cos +, 360 - a, (+,-)

136
Q

“family” unit circles

A

over 2 family

0: 0, (1,0)
90: pi/2, (0,1)
180: pi, (-1,0)
270: 3pi/2, (0, -1)
360: 2pi, (1,0)

over 6 family

30: pi/6, ([3]/2, 1/2)
150: 5pi/6, (-[3]/2, 1/2)
210: 7pi/6, (-[3]/2, -1/2)
330: 11pi/6, ([3]/2, -1/2)

over 4 family

45: pi/4, ([2]/2, [2]/2)
135: 3pi/4, (-[2]/2, [2]/2)
225: 5pi/4, (-[2]/2, -[2]/2)
315: 7pi/4, ([2]/2, -[2]/2)

over 3 family

60: pi/3, (1/2, [3]/2)
120: 2pi/3, (-1/2, [3]/2)
240: 4pi/3, (-1/2, -[3]/2)
300: 5pi/3, (1/2, -[3]/2)

137
Q

sin, cos, tan quick sheet

A
138
Q

quadrantal angles

A
139
Q

Converting from Degrees to Radians & Radians to Degrees

A

Degree → Radians = Degree x pi/180

Radians → Degrees = Radian x 180/pi

140
Q

how to find the sin cos & tan of t

A
  1. Convert to Degrees: 7pi/4 • 180/pi = 315
  2. 315 is in Quadrant IV
  3. sin315 = -sin(360 - 315) = -sin(45) cos315 = +cos(45) tan315 = -tan(45)
  4. -sin(45) = -[2]/2, cos(45) = [2]/2, -tan(45) = -1
141
Q

degrees, radians, revolutions chart

A

how to find revolution:

2pi / radian = x

1/x = revolution

142
Q

converting to/from DoM’S” & Decimal Form

A

1’ = one minute = (1/60)(1o)

1” = one second = (1/60)(1’) = (1/3600)(1o)

Ex: convert 152o15’29” to decimal degree form

152 + (15/60) + (29/3600) = 152 + .25 + .0081 = 152.26o

Ex: convert 153.66o to DMS form

153o

.66•60 = 39.6

153o39’

.6 (from 39.6) • 60 = 36

153o39’36”

143
Q

how to calculate sin cos tan for degrees & radians

A

be sure to change mode b/w Degree & Radians

Math II (17 decks)

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