Praxis 2: Mathematics Content Knowledge 0061 Flashcards
The objective of this deck is to study for the math teaching certification test Praxis 2: Mathematics Content Knowledge, number 0061 or 5061. This was made specifically for the creator to study, so topics are geared toward what the creator needed to review. (227 cards)
1
Q
Set notation for natural numbers
A
N
2
Q
Set notation for whole numbers
A
W
3
Q
Set notation for integers
A
Z
4
Q
Set notation for rational numbers
A
Q
5
Q
Set notation for irrational numbers
A
S
6
Q
Reflexive property
A
a=a
7
Q
Symmetric property
A
if a=b, then b=a
8
Q
Transitive property
A
if a=b and b=c, then a=c
9
Q
What is an equivalence relation?
A
reflexive, symmetric, and transitive properties all hold
10
Q
form of complex numbers
A
a + bi
11
Q
i
A
root of -1
12
Q
i^2
A
-1
13
Q
i^3
A
-i
14
Q
i^4
A
1
15
Q
quadratic formula
A
(-b +- root b^2 - 4ac) / (2a)
16
Q
simplifying a/b + c/d
A
(ad + bc) / (bd)
17
Q
A dependent equation in a system of equations means…
A
The equations are the same line.
18
Q
formula for discriminant
A
b^2 - 4ac
19
Q
number of roots based on the discriminant
A
D0, roots are real and unequal
20
Q
method for completing the square
A
ax^2 + bx + c = 0
ax^2 + bx + (0.5z)^2 = -c + (0.5z)^2
where z = -b/a
21
Q
definition of extraneous roots
A
a solution to a quadratic equation that is found to be false when checked
22
Q
altitude of a polygon
A
a line perpendicular to a side
23
Q
median of a polygon
A
a line that splits both sides into two equal segments (bisects)
24
Q
bases of a trapezoid
A
the parallel sides
25
isoceles trapezoid
the nonbases are equal
26
scalene triangle
no equal sides
27
isosceles triangle
at least 2 equal sides
| vertex angle, 2 base angles
28
polygon # sides and # angles
sides = #angles
29
equilateral polygon definition
all sides equal
30
equiangular polygon definition
all angles equal
31
regular polygon
all sides equal and all angles equal
32
bisector definition
divides a segment into two congruent segments
33
slope-intercept form
```
y = mx + b
m= slope=rise/run
b= y-intercept
```
34
point slope form
y - yo = m(x - xo)
35
concavity of quadratic graphs
a>0, opens up
| a<0, opens down
36
vertex of quadratic graphs
( -b/2a , (4ac - b^2)/4a )
37
axis line of quadratic graphs
x = -b/2a
38
form of homogeneous equation
ax^2 + bxy + cy^2 + dx + ey = 0
39
sense definition
two inequalities have the same sense if their inequalities point the same direction
40
scalar vs vector
```
scalar = magnitude only
vector = magnitude and direction
```
41
free vector definition
can be translated without change in magnitude or direction
42
unit vector
length one
43
zero vector
magnitude zero
44
vector i
magnitude 1 along x-axis
45
vector j
magnitude 1 along y-axis
46
vector addition
A + B = (a1 + b1)i + (a2 + b2)j
47
vector multiplication by a scalar
cA = cai + cbj
or
cA = c(ai + bj)
48
formula for magnitude of a vector
lsl = root a1^2 + a2^2
49
scalar (DoT) product
A * B = a1b1 + a2b2
50
angles of vectors formula
cosZ = (a1b1 + a2b2) / lABl
51
projection of A on B formula
lAlcosZ
or
(A*B) / lBl
52
variation: directly proportional formula
x/y = k
53
variation: inversely proportional formula
xy = k
54
variation: jointly proportional formula
x = kyz
55
reflex angle measure
between 180 and 360
56
converse statement
A iff B = B iff A
57
formula for number of sides on a regular polygon
360/n
| where n = exterior angle measure
58
apothem of a polygon definition
segment from the center of a polygon to the side at a perpendicular angle
59
radius of a polygon definition
segment from the center of a polygon a vertex
60
Area of a regular polygon formula
A = 0.5aP
| half x apothem x perimeter
61
concurrent definition
If there exists one point common to 3+ lines
| i.e. they all intersect at a common point
62
Area of a triangle formula
A = 0.5bh
63
Area of a parallelogram formula
A = bh
64
Area of a trapezoid formula
A = 0.5h(b1 + b2)
| height x mean length of the bases
65
Area of a circle formula
```
A = pr^2
A = pi x radius squared
```
66
Area of a rhombus formula
A = 0.5(d1 x d2)
67
secant definition
a segment that intersects a circle at 2 points
68
tangent definition
a segment that intersects a circle at exactly one point
69
Surface Area of a sphere formula
SA = 4pr^2
| 4 x pi x radius squared
70
Volume of a cylinder formula
V = phr^2
| pi x radius squared x height
71
Volume of a sphere formula
V = 4/3pr^3
| 4/3 x pi x radius cubed
72
Volume of a pyramid formula
V = 1/3Bh
| 1/3 x area of base x height
73
midpoint formula
( [x1 + x2]/2 , [y1 + y2]/2 )
74
equation of a circle
(x - h)^2 + (y-k)^2 = r^2
| where the center is (h,k) and radius is r
75
equation of an ellipse
x^2/a^2 + y^2/b^2 = 1
76
eccentricity of an ellipse formula
e = c/a
| where c are the foci
77
axis of a parabola
the line running perpendicular to the vertex and focus
78
directrix of a parabola
line running perpendicular to the axis of the parabola and equidistance from the vertex as the focus
79
equation of a parabola
y^2 + Dx + Ey + F = 0 (opens right/left)
| x^2 + Dx + Ey + F = 0 (opens up/dpwn)
80
finding the vertex of a parabola formula
```
(y-k)^2 = c(x-h)^2
vertex at (h, k)
```
81
finding the vertex of a quadratic
find y min for
| y = ax^2 + bx + c
82
finding max x for a quadratic
max x = -b/2a
83
equation of a hyperbola
[(x-h)^2]/a^2 - [{y-k)^2]/b^2 = 1 (top/bottom)
| [(y-k)^2]/a^2 - [(x-h)^2]/b^2 = 1 (left/right)
84
sinA =
opp/hyp
85
cosA =
adj/hyp
86
tanA =
opp/adj
87
cscA =
hyp/opp
88
secA =
hyp/adj
89
cotA =
adj/opp
90
trig radians
```
360 = 2pi (full period)
180 = pi
90 = 1/2pi
45 = 1/4pi
```
91
sin^2x + cos^2x =
1
92
tanx =
sinx/cosx
93
cotx =
1/tanx
or
cosx/sinx
94
cscx =
1/sinx
95
secx =
1/cosx
96
1 + tan^2x =
sec^2x
97
1 + cot^2x =
csc^2x
98
sin(A+B) =
sinAcosB + cosAsinB
99
cos(A+B) =
cosAcosB + sinAsinB
100
tan(A+B) =
[tanA + tanB] / [1 + tanAtanB]
101
cot(A+B) =
[cotAcotB + 1] / [cotB + cotA]
102
sin2A =
| double angle formula
2sinAcosA
103
cos2A =
| double angle formula
2cos^2A - 1
or
cos^2A - sin^2A
104
tan2A =
| double angle formula
[2tanA] / [1-tan^2A]
105
sin(A/2) =
| half angle formula
[root 1+cosA] / 2
106
cos(A/2) =
| half angle formula
[root 1 + cosA] / 2
107
tan(A/2) =
| half angle formula
[sinA] / [1 + cosA]
108
cot(A/2) =
| half angle formula
[sinA] / [1 - cosA]
109
sinA + sinB =
2sin(A+B / 2)cos(A-B / 2)
110
sinA - sinB =
2cos(A+B / 2)sin(A-B / 2)
111
cosA + cosB =
2cos(A+B / 2)cos(A-B / 2)
112
cosA - cosB =
-2sin(A+B / 2)sin(A-B / 2)
113
sinAsinB =
1/2[cos(A-B) - cos(A+B)]
114
cosAcosB =
1/2[cos(A+B) + cos(A-B)]
115
sinAcosB =
1/2[sin(A+B) + sin(A-B)]
116
cosAsinB =
1/2[sin(A+B) + sin(B-A)]
117
graphing trig functions change in amplitude
y = Asin(Bx + C) + D
| A changes amplitude
118
graphing trig functions change in period
y = Asin(Bx + C) + D
| 2pi/B changes period
119
graphing trig functions change in phase shift
y = Asin(Bx + C) + D
BC moves left
D moves center line
120
function definition
one y for each x
121
dependent variable is
y
122
independent variable is
x
123
composite function definition
fog = f(g(x))
124
finding the inverse of a function method
switch x and y
solve for y
*the solution must be a function*
125
meaning of a log function logbX = Y
b to the Y power = X
126
logb1 =
0
127
logbb =
1
128
logbb^x =
x
129
logM^p =
PlogM
130
logMN =
logM + logN
131
logM/N
logM - logN
132
one-to-one function definition
exactly one y for every x
133
lim f^n =
(lim f)^n
134
lim cf =
c(lim f)
135
lim cx^n =
c(lim x^n)
136
lim sinx/x =
| x->0
1
137
lim (1 + 1/n)^n =
| x->inf
e
138
lim a^x =
| x->inf
inf.
139
lim (1 - cosx / x) =
| x->0
0
140
lim (1+n)^1/n =
| x->0
e
141
lim 1/a^x =
| x->inf
0
142
lim logaX =
| x->inf
inf.
143
lim logaX =
| x->0
-inf.
144
common nonexistent limits
| x->0
1/x^2
and
lxl/x
145
continuous limit definition
if a limit can be found from both sides
146
Intermediate Value Thm
If f is cont. on [a,b] and f(a) not = f(b),
| then f takes on every value between f(a) and f(b) in the interval.
147
derivative formal formula
```
lim f(a+h) - f(a) / h
h->0
```
148
derivative uses
rate of change
| slope of tangent line to the curve at point x
149
derivative product rule
(fg)' = fg' + f'g
150
derivative quotient rule
(f/g)' = gf' - g'f / g^2
151
implicit function definition
one variable is not directly expressed in terms of the other
152
method for differentiation of implicit funcitons
-diff leaving dy/dx with any ys
-solve for dy/dx
(use chain product rule for any combined variable terms)
153
L'Hopital's Rule
if lim = 0/0 or inf/inf
| diff both numerator and denominator and we obtain the same limit
154
Rolle's Thm
```
f is cont. on [a,b]. f' exists at each point in (a,b)
If f(a) = f(b) = 0 then there is at least one point (xo) in (a,b) s.t. f'(xo) = 0
```
155
derivative sinx =
cosx
156
derivative cosx =
-sinx
157
derivative tanx =
sec^2x
158
derivative secx =
tanxsecx
159
derivative inverse sin
1 / [root 1 - u^2]
160
derivative inverse cos
-1 / [root 1 - u^2]
161
derivative inverse tan
1 / 1 + u^2
162
derivative inverse sec
1 / lul[root u^2 - 1]
163
derivative inverse cot
-1 / 1+u^2
164
derivative inverse csc
-1 / lul[root u^2 - 1]
165
derivative e^x
e^x
166
derivative ln y
1/y
167
derivative a^x
e^(xlna)
168
normal definition
a line normal to a curve at a point has a slope perpendicular to the tangent line
1/-m
169
finding a normal line method
- find derivative to get m
| - use 1/-m
170
finding min and max values method
- slove for the variable to be optimized
- differentiate
- set equal to zero and solve to find critical values
- test values in 2nd derivative, +=min, -=max
171
finding concavity method
test f'' at a point in the interval
+=concave up (convex)
-=concave down
(at f''=0 are points of inflection, concavity may change)
172
Mean Value Thm
f is cont. on [a,b] and has a derivative at every point in the interval, then there exists c in the interval s.t.
f'(c) = f(b) - f(a) / b-a
173
using Mean Value Thm
- plug in a and b
- set equal to f'
- solve for xo
or
1/b-a[integration from a to b of f(x)dx]
174
rectangular (x,y) to polar coordinates (r,8)
```
r^2 = x^2 + y^2
tan8 = y/x
```
175
polar (r,8) to rectangular (x,y) coordinates
```
x = rcos8
y = rsin8
```
176
Law of Cosines
c^2 = a^2 + b^2 - 2ab(cosc)
177
Law of Sines
sinA/a = sinB/b = sinC/c
178
modulus m = k modn
m and k have the same remainder when divided by n
179
method for proof by induction
```
assume P(n)
prove P(1)
prove P(n+1)
therefore P(n) true
```
180
converse statement
if a then b = if b then a
181
contrapositive statement
if a then b = if not b then not a
182
inverse statement
if a then b = if not a then not b
183
conjunction
a and b
184
disjunction
a or b
185
negation
not a
186
conditional
if a then b
187
biconditional
a iff b
188
deductive reasoning
valid iff it is impossible that all premises be true while the conclusion is false
189
invertible matrices (A^-1)^-1 =
A
190
invertible matrices (kA)^-1 =
(1/k)(A)^-1
191
inverting a matrix formula
(1/ad-bc) [d -b]
| [-c a]
192
Multiplying matrices
must be mxn and nxp
| mult A row by B column and add each result
193
Bayes Theorem
P(YlX) = (YX)/YX+Y'X
194
augmented matrix definition
turning a system of linear equations into a matrix
195
inconsistent system means
no solutions
196
consistent system means
at least one solution
197
sample size required
```
n = ms^2/A^2
A = 2s/root n
```
198
linear regression definition
regress one variable to the other
| makes scatterplot into linear
199
linear regression formula
Y = a + bX
b=EXY - n X_Y_/EX^2 - nX_^2
a=EY - bEX/n
200
correlation coefficient
r = E(x-x_)(y-y_)/root(E(x-x_)^2(E(y-y_)^2)
positive cor, r>0
negative cor, r<0
no cor, r=0
201
standard deviation formula
root[(E(x-x_)^2)/n]
202
variance
s^2 = E(x-x_)^2/n-1
203
On a symmetrical curve the measures of central tendancy ...
mean=mode=median
204
n total objects, combinations of r objects
nCr
| n!/r!(n-r)!
205
permutations of groups of like objects
n!/r1!r2!...rk!
206
n total objects, r objects permutated
nPr
| n!/(n-r)!
207
combinations without replacement
n!
208
Fundamental Principle of Counting
| how many combinations?
n1 x n2 x ... x nk
209
marginal probability definition
sums of joint probability
210
joint probability definition
combined probability of 2 events
211
dependent probability
| one affects the other
P(x and y) = P(x)P(ylx)
212
independent probability
| have no effect on each other
P(x and y) = P(x)P(y)
213
non-mutually exclusive probability
| x or y, can be simultaneously
P(x or y) = P(x) + P(y) - P(x and y)
214
mutually exclusive probability
| cannot happen simultaneously
P(x or y) = P(x) + P(y)
215
integrate 1/x =
ln x
216
integrate 1/x^2 + a^2 =
1/a tan^-1 x/a
217
integrate a/root(a^2 - x^2) =
sin^-1 x/a
218
integrate sinax =
1/a cos ax
219
integrate cosax =
1/a sin ax
220
integrate e^ax =
1/a e^ax
221
integration by parts formula
int udv = uv - int vdu
222
Fundamental Theorem of Calculus
int from a to b of f(x)dx = F(b) - F(a)
223
int kf =
k int f
224
int f + g =
int f + int g
225
use for definite integral
find area under a curve
226
rate of change formula
f(d) - f(c) / d - c
227
acceleration -> velocity -> position
a(t) = v'(t) = s''(t)
