Mtel Math 47 51

Front 1

prime factors

Back 1

1 and number

Front 2

GCF

Back 2

greatest common factor

if nothing else, 1

Front 3

LCM

Back 3

least common multiple

Front 4

Reflexive axiom of equality

Back 4

a=a

Front 5

Symmetric axiom of equality

Back 5

if a=b, then b=a

Front 6

Transitive axiom of equality

Back 6

a=b, b=c, then a=c

Front 7

Closure operations

Back 7

a, b a+b, axb

Front 8

Associative operations

Back 8

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

Front 9

Identity operation (+/x)

Back 9

+ 0, x 1

Front 10

Inverse operation (+/x)

Back 10

-a, 1/a

Front 11

Commutative Axiom of +/x

Back 11

a+b=b+a

Front 12

Distributive operation of multiplication over addition

Back 12

a(b+c)= ab+ac

Front 13

Substitution property

Back 13

if a=b, then can use either

Front 14

Transitive vs Substitution

Back 14

T showing equality between two previously unrelated things

S replacing parts known to be equal.

Front 15

Complex number

Back 15

-i, sqrt-1

Complex vs Real (rational + irrational)

Front 16

Real numbers

Back 16

Rational + Irrational

Rational: Integers, Whole, Naturals

Irrational: no fraction or repeating dec.

Front 17

Rational

Back 17

any number st a/b,
fractions of whole numbers, positive and negative,
no zero in denominator,
repeating decimal

Front 18

Irrational

Back 18

no fraction or repeating dec

Front 19

Integers

Back 19

Naturals (no fractions)
Whole (zero)
Integers (-)

Front 20

Natural

Back 20

counting numbers

Front 21

Whole

Back 21

zero

Front 22

integers

Back 22

negative

Front 23

index
radical sign
radicand

Back 23

n
sqrt radicand^m = base^m/n

Front 24

sqrt a x sqrt b

Back 24

sqrt ab

Front 25

a^p x a^q

Back 25

a^p+q

Front 26

a^p/a^q

Back 26

a^p-q

Front 27

(a^p)^q

Back 27

a^pq

Front 28

(ab)^p

Back 28

a^p b^p

Front 29

(a/b)^p

Back 29

a^p/b^p

Front 30

Arithmetic Series

Back 30

a(n) = a + (n-1)d

Front 31

Arithmetic sequence = sum

Back 31

S(n) = .5 n (a(1) + a(n))
n > 0 only
find difference, recursive, linear function, exponential function y=ab^x
(i.e. a(n)=10n=10,20,30, 40....

Front 32

Geometric series

Back 32

a(n) = a(1) r^n-1

n>0

a(1) = 2x5^(1-1) = 2 x 1

Front 33

Geometric Sequence = sum

Back 33

S(n) = a(1) ((1-r^n)/1-r)

n>0

Front 34

to solve xyz

Back 34

must have three equations

Front 35

first degree equation

slope intercept form

from points on a line

Back 35

ax + by = c

y = mx +b , m=change y/change x

Y-ya = m(X-xa)

Front 36

Inequality word problems

Back 36

find examples in c8

Front 37

composition function

Back 37

(fog)x or f(g(x))

their domain requires both functions be defined (nonneg squares and no zero in denominator)

ie sqrt x+2 x>2

Front 38

solve for xyz

Back 38

homogenous system has one trivial solution

if # variables greater or equal to number of equations, then non-trivial solutions. Use matrices and substitution.

Front 39

Factor

x^2 + 10x - 24 = 0

Back 39

(x+12) (x-2) = 0
x= -12, 2

Front 40

Complete the square

Back 40

move the nonvariable number to the other side and figuring out what number will complete factor st it is squared, add to both sides, take sq root.

Front 41

Quadratic Formula

Back 41

-b +/- sqrt (b^2 - 4ac)
---------------------------------
2a
Discriminant tells roots (0=one real, >0 2 unequal, <0 complex)

Front 42

Quadratic from roots

Back 42

add roots = sum; multiply = product
x^2 + (opp of the sum)x + product = 0
eliminate fractions with common denom

Front 43

roots of quadratic equation

Back 43

x intercepts

Front 44

composite factors

Back 44

greater than two primes

Front 45

divisibility for 3s and 4s

Back 45

sum multiples of three
last 2 digits 4
last 3 digits 8

Front 46

scientific notation operations

Back 46

add/subtract: maintain same exponent
multiply add exponents
120 add exp to left (1.20 x 10^2)

divide subtract exponents
.012 subtract exp to right (1.20 x10^-2)

Front 47

what counts as significant digits?

Back 47

nonzero number
zeros between nonzeros
final zeros (right of the decimal place)

Front 48

mapping vs function

Back 48

vertical line test

Front 49

angle of exterior linear pair

Back 49

sum of opposite internal angles

Front 50

angle of central angle

Back 50

length of the minor arc

Front 51

inscribed angle

Back 51

half the intercepted arc measure

Front 52

vertical angles (where two chords intersect)

Back 52

half the sum of intercepted arcs

Front 53

tangent chord angle

Back 53

half the intercepted arc

Front 54

exterior vertice of a circle

Back 54

half the difference between the two arcs it inscribes

Front 55

relationship between segments of intersecting chords x and y

Back 55

x1x2 = y1y2

Front 56

relationship between intersecting tangent chords x and y

Back 56

x = y

Front 57

relationship between intersecting tangent and segments of secant chord x (tan) and y (sec)

Back 57

x^2 = y1 y2

Front 58

Perimeter of a regular polygon of n sides

Back 58

n x side length

Front 59

Area of a regular polygon of n sides

Back 59

half the sum of apothem x perimeter

to find apothem: if hexagon, 30 60 90 rules (radius=side length; a =halfside x sqrt3)

Front 60

apothem of a polygon

Back 60

use angle relationships, if 345 triangle or if 30 60 90: radius = side length.
apothem = half radius x sqrt 3

(radius)^2 = (half radius)^2 + (halfradius x sqrt 3)^2

Front 61

Finding the domain of a function

Back 61

anything not negative in denominator and any square not negative

undefined and complex

Front 62

To find max values of coins

Back 62

remember 5n +10d= ____
graph, use x and y intercepts for linear equalities

Front 63

30 60 90


45 45 90

Back 63

x, x sqrt3, 2x


x, x, x sqrt2

Front 64

median

Back 64

count data points, divide number in half. (average if even data set)

Front 65

mean

Back 65

sum data/number of data points
One problem with using the mean, is that it often does not depict the typical outcome. Outlier will skew the mean strongly affecting the outcome. The median is the middle score. If we have an even number of events we take the average of the two middles. Median better for typical value. It is often used for income and home prices.

Front 66

mode

Back 66

most frequent data point

Front 67

range

Back 67

high to low data points

Front 68

percentiles

Back 68

data/100

one hundred equal parts of data

Front 69

stanines

Back 69

linked to standard deviation: values to probability (y axis); steep curve is tight cluster around mean; low slope large standard deviation.

mean at the middle (highest part) of the bell curve, negative and positive standard deviations from the mean x axis

Front 70

quartiles

Back 70

using medians to divide the data set into four subsets.
Quartiles are the medians of each of the four subsets (Q1-4)

Front 71

variance

standard deviation

Back 71

Calculate the mean, x.
Write a table that subtracts the mean from each observed value.
Square each of the differences.
Add this column.
Divide by n -1 where n is the number of items in the sample This is the variance.
Take the sqrt(variance) = standard deviation from the mean


To get the standard deviation we take the square root of the variance.

Front 72

Why is standard deviation useful?

Back 72

The mean may either be closely spaced or spread out; it may be a good approximation of the data or a dangerously poor one. Examples: insulin level after a drug trial.

Front 73

radians to angles
30
45
60
90

Back 73

30 = pi/6
45 = pi/4
60 = pi/3
90 = pi/2

Front 74

sinx

cscx

Back 74

OPP/HYP = SINX

HYP/OPP = CSCX

Front 75

cosx

secx

Back 75

ADJ/HYP = COSX

HYP/ADJ = SECX

Front 76

tanx

cotx

Back 76

OPP/ADJ = TANX

ADJ/OPP = COTX

Front 77

direct variance in functions

Back 77

y= cx or y=cx^2

proportional equation

Front 78

indirect variance in functions

Back 78

xy = c or y=c/x or y= c/x^2

inverse variation

example: speed vs driving time

Front 79

Absolute value equation

Back 79

y=m(x-h)+k
(h,k) max or min point
+ or - m = slopes

solve for both equations (0, 1, 2)
resubstitute finding null sets

Front 80

multiply an inequality by -1

Back 80

reverses the inequality

Front 81

factoring cubes x^3 + y^3

acronym

equation

Back 81

SDP== same different plus

x^3 + y^3 =(x+y) (x^2-xy+y^2)

x^3 -- y^3 =(x--y) (x^2+xy+y^2)

Front 82

matrices

Back 82

easy

Front 83

synthetic division

(x^4-3x+5)/ (x-4)

Back 83

x-4---> 4
PC polynomial coefficients 1 0 0 -3 5

bring down first PC (1)
multiply by 4
replace in second column (under 0)
add column
repeat multiplication/replacement

answer
x^3+4x^2+16x+61+(249/(x-4))

Front 84

geometric postulates of congruency

Back 84

SSS

SAS (included angle)

ASA (included side)

NO ASS!

Front 85

postulates

Back 85

accepted as true without proof

Front 86

geometric theorems derived from ASA

Back 86

AAS (two angles and noninclusive side)

HL if hypotenuse and leg of two right triangles are congruent, congruent.

Front 87

Theorem

Back 87

derived from postulate

Front 88

Undefined terms

Back 88

point, line, plane

Front 89

defined geometric terms

Back 89

ray-- one direction from end point

Front 90

Property of addition

Back 90

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

Front 91

Property of Subtraction

Back 91

if a=b and c=d, then a-c=b-d

Front 92

Property of Multiplication

Back 92

if a=b then ac=bc

Front 93

Property of division

Back 93

if c does not equal zero and a=b,
then a/c=b/c

Front 94

Reflexive Property

Back 94

a=a

what use is reflexive?

Front 95

Symmetric property

Back 95

If a=b, then b=a

Front 96

Transitive property

Back 96

If a=b, b=c, then a=c

Front 97

Distributive Property of mult over addition

Back 97

a(b+c) = ab+ac

Front 98

Substitution property

Back 98

If a=b, b may be substituted for a

Front 99

distance between two points

Back 99

abs value (a-b or b-a)

Front 100

Induction

Back 100

examples patterns
sometimes true

Front 101

Deduction

Back 101

from postulates->conclusion
always true

Front 102

Internal angles of a polygon

Back 102

((n-2)180)/n

Front 103

External angles of a polygon

Back 103

360/n

Front 104

Isosceles Angle Theorem

Back 104

If two sides (or two angles) are equal, the angles (or sides) are equal.

Front 105

Parallel line theorem

Three examples

Back 105

corresponding angles congruent

alternate interior angles congruent

alternate outside angles congruent

Front 106

Area of a Circle

Volume of a Sphere

Surface area of a sphere

Back 106

pi r^2

4/3 pi r^3

4 pi r^2

Front 107

Area Trapezoid

Back 107

h/2 (b1+b2)

Front 108

Area square

Area triangle

Area parallelogram

Back 108

Asquare = lw

Atriangle = 1/2 base height

Aparallelogram = base height

Front 109

Volume cylinder

Surface Area

Back 109

V cylinder = pi r^2 h


SA cyl = 2 pi r h + 2 pi r^2

Front 110

Volume cone

Surface area

Back 110

Vcone = pi r^2 height/ 3

SA cone = pi r sqrt(r^2 + h^2) + pi r^2

Front 111

slant height of cone

Back 111

sprt(r^2 + h^2)

application of pythagoras theorem

Front 112

Describe this parabola

y=a(x-h)^2 +k

y=-3x^2 + 6x-1

Back 112

vertex (h,k)
sym about the y axis x^2
+a indicates opens upward

Complete the square
y=-3(x^2-2x+1) -1 +3
y=-3(x-1)^2 +2

Front 113

Describe this parabola

x=-a(y+k)^2 -h

Back 113

vertex (-h,-k)
sym about the x axis (y^2)
-a indicates opens downward

complete the square

Front 114

Describe this ellipse

(x-h)^2 //a^2 + (y-k)^2//b^2 =1

Back 114

complete the square for y and x polynomials separately

center (h,k)
x and y intercepts by solving for x=0 and y=0
length of major axis is 2a, minor axis is 2b
c^2=a^2-b^2
foci of two circles that form the ellipse are either (h+/-c,0) or (0,h+/-c)

Front 115

Describe an ellipse where b>a


a>b

Back 115

symmetrical about the y axis


symmetrical about the x axis

Front 116

Describe this hyperbola

(x-h)^2//a^2 - (y-k)^2//b^2 = 1

Back 116

-y^2 about y axis
and equation for asymptote is
y= +/-b/a(x-h) +k

-x^2 about x axis
and equation for asymptote is
y=+/-a/b(x-h)+k

foci (h+/c, k) or (h,k+/-c) where c^2=a^2+b^2 (not pythag)

vertices are (h+/-a, k) or (h, K+/-a)

Front 117

Describe this circle

x^2 + y^2 = 9

Back 117

if (x-h) or (y-k), the center is on (h,k) (ignore the negative sign)

here, center is (0,0) and the x and y intercepts are at sqrt9=+/-3

If

Front 118

objective (No of queens/Total cards)

Back 118

P(x) =
Total number (outcomes)/
Total Possible

4/52

P(x) is always between 0 and 1, and the sum of all probabilities in a set is equal to 1

Front 119

P(heads) + P(tails)

Back 119

1/2 + 1/2 = 1

Front 120

mutually exclusive events

P(spade) + P(clubs)

Back 120

13/52 + 13/52 =26/52 =0.5

x or y

Front 121

non-mutually exclusive

P(queen) + P(spade)

Back 121

4/52 + 13/52 - 1/52 = 16/32

x + y - P(x+y)

Front 122

Independent Events

3 heads on 3 tosses

Back 122

1/2 x 1/2 x 1/2 = 1/8

P(x) x P(y)

Front 123

Dependent Events

6 red, 4 green, 5 purple
P(red) 1st draw and
P(purple) on second draw

Back 123

P(red1)= 6/15
P(purple2)= 5/14

P(both) = 6/15 x 5/14 = 30/210 = 1/7

Front 124

Define Odds

Back 124

favorable outcomes//
unfavorable outcomes

If odds against, then unfavorable//fav

Front 125

Odds a head will turn up if three coins are tossed

Back 125

plot out the combinations, count favorable/unfavorable

Front 126

Combinations Problem


How many possible combinations of heads from five tosses?

Back 126

Pascal's Triangle: (1s are row zeros) fifth line summarizes total combinations of possible heads from 5 tosses (1,5, 10, 10, 5,1)

The full formula is =(n!/((n-r)!*r!))

Front 127

"how many different ways can you choose two objects from a set of three objects?"

Back 127

Using Pascals Triangle
1
11
121
13(3)1 = 3 ways. (place 2)

If 5 objects,
14641
15(10) 10 5 1 = 10 ways.

Front 128

permutations

10 objects, 4 selections without repeating, order is not important (ie. CACA is not the same as ACAC)

Back 128

10!/(10-4)! = 5040


More permutations than combinations

Front 129

Combinations

10 objects, 4 selections without repeating, order is important (ie CACA equals AACC and ACAC ...)

Back 129

10!/[4!(10-4)!] = Permutation/4!

essentially dividing permutation by the 4 possible selection factorial

Front 130

unit circle

pi/6=30
find sinx, cosx, tanx

Back 130

sinx = 1/2
cosx=sqrt3/2
tanx=1/sqrt3

Front 131

unit circle

pi/2 =90
find sinx, cosx, tanx

Back 131

sinx=1/1
cosx=0/1
tanx=1/0 or =/- infinity

Front 132

Trig Identities

sin^2x + _____ = 1

Back 132

cos^2x

sin^2x +cos^2x = 1

Front 133

Trig Identities

1 + _____ = sec^2x

Back 133

tan^2x

1 + tan^2x = sec^1x

divide basic identity by cos^2x; solve.

Front 134

Trig Identities

1 + _____ = csc^2x

Back 134

cot^2x

1 + cot^2x = csc^x

divide basic identity by sin^2x

Front 135

Law of Sines

Triangle with side a opposite Angle A

Back 135

sinA/a = sinB/b = sinC/c

The Law of Sines is used for solving triangles SSA (two sides and an angle opposite one of them) and AAS (or ASA) (two angles and any side).

Front 136

Law of Cosines

assuming SAS (aCb) and want opposite c

Back 136

c^2 = a^2 +b^2 - 2ab cosC

very useful for finding third side length if you know other two and its opposite angle measure.

Front 137

Derivatives of trig functions
d/dx(cosx)

d/dx (sinx)

Back 137

-sinx

cosx

Front 138

d/dx (tanx)

d/dx (cotx)

Back 138

quotient rule:
d/dx(sinx/cosx) =
cosx (cosx) - sinx (-sinx)/cosx^2=
cosx^2 -(-sinx^2)/cosx^2=
1/cosx^2= sec^2

d/dx(cotx) = -cscx^2

Front 139

Product Rule

d/dx (f(x) x g(x))

Back 139

= f d/dx(g(x)) + d/dx(f(x) g(x)

Front 140

Quotient Rule
d/dx(f(x)/g(x))

Back 140

g(x) d/dx(f(x)) - f(x) d/dx(g(x))// g(x)^2

Front 141

exp derivatives:
d/dx (e^x)

Back 141

d/dx (e^x) = e^x * d/dx(x) = 1 * e^x

The derivative of the exponential function is the exponential function

Front 142

Three rules of logarithms
ln x*y
ln x/y
ln x^n
ln1

Back 142

ln x*y = ln x + ln y

ln x/y = lnx - lny

ln x^n = n*lnx

ln1 = 0
x intercept is zero; translation if ln(x-2) to +2 thus x intercept is at 3 and vertical asymptote is at +2

Front 143

log base 3 of 1/9

Back 143

log base 3 (1/3^2)= log base 3 (3^-2) =

-2 *log base 3 (3) = -2 *1 = -2

Front 144

b^n = x

e.g. 8^2 = 64

Back 144

log base b (x) = n

2 = log base 8 (64)

Front 145

log base b (b^x) =__

What exponent on the right will raise the base (b) to produce b^x

Back 145

log base b (b^x) = x
or b^x = (b^x)

x -- on the right -- is the exponent to which the base b must be raised to produce bx.

Front 146

Exponential functions

y=b^x

y=b^0=

Back 146

y intercept is +1 thus b^0 = 1
why?

Front 147

b^(logbaseb(x)) = _______

log base b (b^x) = _________

Back 147

The inverse of any exponential function is a logarithmic function. For, in any base b:

b^(logbaseb(x)) = x
(logbx is the exponent to which b must be raised to produce x.)

log base b (b^x) = x

composite inverse functions =x

Front 148

f(g(x)) = ln e^x = ___

g(f(x)) = e^ln x =__________.

Back 148

x

lnx or log base e (x) is the inverse function of e^x: e^x y intercept 1 and lnx x intercept 1 about y=x

Front 149

Solve this equation for x :

5^(x + 1) = 625

Back 149

Solution. To "release" x + 1 from the exponent, take the inverse function -- the logarithm with base 5 -- of both sides.
logbase5(5^x + 1) = logbase5(625)

x + 1 = logbase5(625)

x + 1 = 4

x = 3.

Front 150

Solve for x:
log2x + log2(x + 2) = 3.

remember the domain of log base 2 x

Back 150

log2[x(x + 2)] = 3.

If we now let each side be the exponent with base 2, then

x(x + 2) = 23 = 8.

x² + 2x − 8 = 0

(x − 2)(x + 4) = 0

x = 2 or −4.

See Skill in Algebra, Lesson 37.

We must reject the solution x = − 4, however, because the negative number −4 is not in the domain of log2x.

Front 151

Chain Rule

d/dx(x^3)

d/dx(2/(x + x^2)3=d/dx 2(x+x^2)^-3

Back 151

3x^2

-6(2x+1)//(x+x^2)^4

Front 152

Limits

lim(2x+1) = _____
->2

lim (x^2-9)/(x+1) = ____
jx->3

Back 152

=5 substitution

=0 if substitution results in inf or zero, use L'Hospital's rule: take the derivative of the equation (no rules) and plug in the limit again.

Front 153

lne = _____

d/dx b^x =_________
d/dx 4^2x+5=_______
d/dx 2x(4^x)=________

Back 153

ln e = 1

b^xlnb
2ln4(4^2x+5) =d/dx(b^u) =b^u*lnb du/dx
(The derivative of e raised to a quantity is e raised to that quantity, times the derivative of that quantity.)
2(4^x) + 2x(4^x)ln4 (use this with product rule)

Front 154

d/dx log base b (x) = 1/x*ln(b)

d/dx ln x = d/dx log base e (x) = 1/x

Back 154

The derivative of the natural logarithm of a quantity is the reciprocal of that quantity, times the derivative of that quantity.

Front 155

d/dx sin u =

d/dx cos u =

Back 155

cosx * du/dx

-sin x * du/dx

Front 156

d/dx e^x=__________

d/dx e^2x+4 =_______

Back 156

e^x

2e^2x+4
The derivative of e raised to a quantity is e raised to that quantity, times the derivative of that quantity.

Front 157

d/dx lnx = ________

Back 157

1/x

Front 158

Derivative

Back 158

instantaneous rate of change

equation which describes the slope of the tangent at any point on a curve

plug in value to find the slope or rate

Front 159

Why is d/dx e^x = e^x

Back 159

e^x is the only function which grows at a rate of change equal to itself

Front 160

Quotient Rule

f(x) = g(x)//h(x)

Back 160

h(x) g'(x) - g(x) h'(x)//h(x)^2

straight bottom goes first and subtract it's differential

Front 161

Differentiate

y=2^3x+1 * ln (5x-11)

3ln2=_______

Back 161

Product rule
=2^3x+1*(1/5x-11)*5 + 3*ln2*2^3x+1 *ln(5x-11)

ln8 = 2^3

Front 162

The natural log gives you the time needed to reach a certain level of growth.

Back 162

* e^x lets us plug in time and get growth.
* ln(x) lets us plug in growth and get the time it would take.

Front 163

e represents the idea that all continually growing systems are scaled versions of a common rate.

Back 163

This is wild! e^x can mean two things:

* x is the number of times we multiply a growth rate: 100% growth for 3 years is e^3
* x is the growth rate itself: 300% growth for one year is e^3.

Well, since the crystals start growing immediately, we want continuous growth. Our rate is 100% every 24 hours, so after 10 days we get: 300 * e^(1 * 10) = 6.6 million kg of our magic gem.

Front 164

lne = ?

Back 164

1
* The math robot says: Because they are defined to be inverse functions, clearly ln(e) = 1
* The intuitive human: ln(e) is the amount of time it takes to get “e” units of growth (about 2.718). But e is the amount of growth after 1 unit of time, so ln(e) = 1.

Front 165

Rate of change of a bubble

R=dV/dt=0.2 cm^3/sec and radius r=0.5 cmgiven

differentiate sphere volume to get change with time: dV/dt V(sphere) =d/dt 4/3(pir^3)dr/dt

Back 165

V' = 4pir^2*r'
I want to know when r'=0.2 cm^3/sec

set given rate equal to V' and solve for r' (rate of change of radius) given the radius (0.2cm^3/sec//4pi(0.5)^2

Front 166

Cost of production minimize ave cost

c(x) = x^3....
average cost(x) = c/x

Back 166

differentiate average cost
C' = x^2....
set equal to zero

Front 167

Max profit
C = 150 + 40x
x=80-price or price = 80-x

Find maximum price/unit

Back 167

Profit = Revenue -Costs
R=price/unit*units = px
C is given
P = (80-x)x - (150+40x)
P' = -2x + 20
set equal to zero to find maximum
x=10

Front 168

Max profit truck rentals
costs 30 trucks at 20/day to run
or 5/day storage

Back 168

Profit=Revenue - Costs = N(R-C)
N (#trucks) rented =
30 -(R-20(cost))=50-Rent
P=(50-R)(R-C)
C=5/day
P=(50-R)(R-5) solve
differentiate and set equal to zero

Front 169

Race car: rate of change in distance bt it and spectator in final 100 feet if spectator is 200 ft from finish
A x
y h

0 spectator
car dx/dt is 176; 100ft out at x

Back 169

Pythagorean to find h as func x
differentiate to find changing h'dx/dt

dx/dt is known, speed of auto. Plug in and solve. (-79 ft/sec)

Front 170

y=cosx, x is given, dx/dt is given
find dy/dt in seconds at x

Back 170

differentiate y'=-sinx dx/dt
Plug in dx/dt value
confused about how to solve

Front 171

V shaped tank is
Vprism =1/2(xyL) (xy is fluid volume)
(wh is tank dims given)
V'-0.002 m3/s

Find rate of vertical (x) decrease with time

Back 171

similar triangles can use proportions to write in other terms: y=wx/h so it is in x

solve differentiate for V in terms of x and dx/dt

using given values of w and h, solve for dx/dt

Front 172

implicit differentiation

Back 172

used when functions are in two variables when you cannot solve y in terms of x: solve for dx/dy and plug in values for x and y to find slope (instantaneous rate of change)

Instance of chain rule

Front 173

Probability

Rolling sum greater than 7 or doubles

Back 173

Not mutually exclusive

Add together Pa + Pb - instances of doubles greater than seven

Front 174

Postulates - given

Back 174

SSS
SAS
ASA
Reflexive leads to symmetric (another) leads to transitive (a third through a second)

Front 175

Theorems - der from postulates

Back 175

AAS
HL
Isosceles Triangle
Parallel Lines (corr, alt in, alt out)
Vertical angles congruent

Front 176

Axioms -given

Back 176

Add, Multi, Div, Sub

Front 177

Why Reflexive?

Back 177

Thus, if perhaps two triangles share a side and you wish to prove those two triangles congruent using the SSS method, it is necessary to cite the reflexive property of segments to conclude that the shared side is equal in both triangles.

Front 178

Why Transitive?

Back 178

This holds true in geometry when dealing with segments, angles, and polygons as well. It is an important way to show equality.

Front 179

Other postulates

Back 179

midpoints, bisectors, defn of a line, supplementary 180 and complimentary 90 angles

Front 180

commutative

Back 180

add or mult or sub b from two equal lengths, get same amount.

vs transitive a=b, b=c, a=c. or a=c, b=d, a=b therefore c=d.

Front 181

Additive equality

Additive inverse

Additive identitiy

Back 181

AE= subtract four
AI= a+-a =0 cancels out 4
AIdentity= remove zero a+0=a

Front 182

minimum even divisor for 72 and 42

Least common multiple

Back 182

factor, cross off only one similar in a pair between the two numbers. Multiply. 504

Front 183

Least common denominator

Larry 10 min/ele
Moe 6 min/ele
Curly x min/ele
total together 3 min/ele

Back 183

to change to ele/min, invert.

Find least common Denominator, and reduce terms to solve for x

Front 184

Inequality word problem

YMCA raffle wants >-32 K
Cost of event 7250, P of ticket 25
How many need to sell?

Back 184

25t-720>=32K
t>=1570

Front 185

Bicycle Profits 3 speed vs 10 speed

Back 185

time cost equation for 480 minute work day

want profit per bike ,=300??

Max where x or y is zero.

Front 186

Inflection point

Back 186

second derivative where slope = zero

a local max or min if first derivative on either side of point at zero is pos and neg (to show concavity changing)
A) If on both sides of the point the f'slope is positive (or neg), then the inflection point is not a max or min.
B)if on both sides of the point the f''slope is pos or neg, then the inflection point could be a local max/min (x^4) local max

Front 187

Discern a functions max, mins concavity, and inflection points

Back 187

factor f' = 0
plug those numbers into original eq

take second derivative to find if concave up or down.
factor f'' set equal to zero to find inflection points

Front 188

Maximize area if given circumference

Back 188

solve for one variable, substitute into Area equation, take derivative, set equal to zero and solve for variables.

Front 189

Area given, find dimensions for greatest volume

Back 189

isolate terms in Area to make single variable in volume. take derivative and solve for zero.

Front 190

Surface area of a cylinder without a top, find height and radius for max V

Back 190

SA isolate terms on either side of equation; plug into Volume. Derivative, set to zero, check the radius possibilities.

Front 191

Minimize distance from a graph of f(x) = sqrtx to the point (4,0)

Back 191

Pythagorean theorem

Front 192

binomial theorem

complex
one real root
two real roots

Back 192

-b+/-sqrt(b^2-4ac)/2a

b^2-4ac < 0 complex
b^2-4ac = 0 one real
b^2-4ac > 0 two real

Front 193

Output maximum profit
50 apple tree = x
800 apples/tree
Each tree loses 10 apples for each additional tree. What is maximum number of trees i can add before losses/tree take over?

Back 193

Output Total =
xtrees*Output/tree = 4000
Output/tree =
800 apples-10 apples *(x-50)
=x( 800 apples -10 apples(x-50)
Reduce, differentiate, solve for zero
x=65

Front 194

Chain Rule

Back 194

Used when composite of two functions
not two separate functions (product or quotient)

Front 195

synthetic sub

Back 195

bring down first coefficient of the dividend (not the divisor), mult by (x-2) then 2, add, continue to end.

Root if zero. Can Find other roots by factoring easier polynomial.

Remainder/(x-2) ends new polynomial.

Front 196

Integrate:
sinxdx
cosxdx
tanxdx

Back 196

sinxdx = -cosx
recall d/dx -cosx=sinx

cosxdx = sinx
recall d/dx sinx= cosx

tanxdx = ln absvalue(secx)

Front 197

Integrate
sin^2xdx
cos^2dx

Back 197

sin^2x= x/2 - sin2x/4

cos^2x=x/2 + sin2x/4

Front 198

d/dx 2^(3x+1)

d/dx2^x

Back 198

Can't take derivative of base 2. Change to something we know how to derivate: e^x

d/dx 2^2x+1 = e^ln2^3x+1 =
d/dx e^ln2*3x+1= 3ln2* d^ln2*3x+1

e^x^y = x^x*y

In the same way:
d/dx 2^x = d/dx e^ln2*x =
ln2*e^lnx

Front 199

d/dxlnx = d/dx log base e x =______

Find d/dx log base b x= ____________

Back 199

1/x

Given b=e^lnb
and b^k = x
log base b (b^k)
Substitute and take ln:
ln( e^lnb*k)=lnx
lnbk=knx
k=lnx/lnb Differentiate
lnb is a constant
d/dx lnx = 1/x

this 1/lnbx

Front 200

Sequences are just progressions of numbers with a common difference separated by commas

Arithmetic sequence a(n) =

Back 200

a(n) = a(1) + (n-1)d

where d is the common difference (+/-) between arithmetic sequences

Front 201

Sequences just numbers, numbers
with a common ratio separated by commas

Geometric Sequence a(n) =

Back 201

a(n) = a(1) * r^n-1

where r is the common ratio between terms (*/divide)

Front 202

Arithmetic sequences

1 5 9 13 17

Back 202

a(n) = 1 + (n-1)4

common difference; only thing that changes is the starting point. Linear.

Front 203

Geometric Sequences

2 6 18 54

3 1 1/3 1/9 1/27

Back 203

a(n) = 2*3^(n-1)

a(n) = 3*(1/3)^(n-1)

Common ratio structured so that for n=1, the ratio is r^0 or 1, which leaves the first term, or n=1.

Front 204

Arithmetic Series progression of numbers with a common difference (+/-) separated by plus or minus

Back 204

The sum of an arithmetic series is found by multiplying the number of terms times the average of the first and last terms.

Front 205

Arithmetic Series

3 + 7 + 11 + 15 + ··· + 99 has a1 = 3 and d = 4. We solve 3 + (n – 1)·4 = 99 to get n = 25.

Back 205

Find a(n) using Arithmetic Sequence equation = a(1) + (n-1)d.

Multiply number of terms n, by the average of the first and last term.
Series S(25) = 25(3+99/2)

Front 206

Geometric Series progression of numbers with a common ratio between them and separated by a plus or minus

3 1 1/3 1/9 1/27 1/81

Back 206

Sum of Series S(n) = a(1)(1-r^n)/(1-r)

S(6) = 3(1-(1/3)^6)/(1-(1/3))

************************************

Front 207

Infinite series

convergent ratio

divergent ratio

Back 207

ratios less than 1 converge

ratios greater or equal to 1 diverge