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Math II > Finals > Flashcards

Finals Flashcards

1
Q

degrees → radian

A

degree • pi/180

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2
Q

radians → degree

A

radian • 180/pi

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3
Q

evaluate sin x, cos x, and tan x for a value x

A
  1. if necessary, convert x to degrees
  2. Check what quadrant it’s in
  3. Do whatever the quadrant says to do
  4. Use special/quadrantal angles tables to solve
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4
Q

special angles table

A
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5
Q

quadrantal angles table

A

y = sin x graph → 0 1 0 -1 0

y = cos x graph → 1 0 -1 0 1

tan x = sinx/cosx

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6
Q

state the number of revolutions for an angle

A
  1. convert to radians
  2. radians/2pi
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7
Q

convert to decimal degree form

A

Ex: 152o15’29”

152 + (15/60) + (20/3600) =

152.26o

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8
Q

decimal degree form

A

156.33o

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9
Q

DMS form

A

122o25’51”

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10
Q

convert to DMS form

A

Ex: 24.240

24o(.24*•60)’ (.4**•60)”

24o14’24”

*.24 is from 24.24

**.4 is from (.24•60 = 14.4)

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11
Q

1’ = ?

A

one minute = (1/60)(1o)

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12
Q

1” = ?

A

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

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13
Q

quadrant rules

A
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14
Q

terminal ray

A

the pipe cleaner

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15
Q

in which quadrant does the terminal side of each angle lie when it is in standard position?

A
  1. convert to degrees
  2. if negative, + 360; if over 360, - 360
  3. find which quadrant it’s in
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16
Q

find the exact value of sin/cos/tan x. no calculator.

A
  1. if radians, convert to degrees
  2. use quick charts
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17
Q

use a calc to approximate sin/cos/tan x to four decimal places

A
  • if degree, change calc to Degree mode
  • if radians, change calc to Radians mode
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18
Q

sketch w/out a calculator a sin/cos/tan curve

A

xmin = -2pi

xmax = 2pi

xscl = pi/2 (unless stated otherwise)

ymin = -5

ymax = 5

yscl = .5

sin, cos

  • if y = c + cos(x) → period & amplitude same; c = pos moves max/min up, c = neg moves max/min down
  • if y = a sin(x) → period same; move max to a
  • if y = sin(bx) → max/min/amp same; normal period/b
  • if y = sin(x + b) → if b = pos, move left; if b = neg, move right

**tan, cot, sec, csc - **no ampl/min/max

  • y = c + tan(x) → if c = pos, move up; c = neg, move down (easier to just move x-int’s)
  • y = a csc(x) → move min/max’s to a
    • y = cot(bx) → period/b
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19
Q

period

A
  • for sin, cos curves → the shortest distance along the x-axis over which the curve has one complete up-and-down cycle
  • for tan, distance b/w consecutive x-intercepts
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20
Q

amplitude

A

max - min

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21
Q

vertical asymptotes

A

lines that the graph approaches but doesn’t cross

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22
Q

periodic

A

repeating

Ex: tan function

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23
Q

csc, sec, cot

A

csc = 1/sin

sec = 1/cos

cot = 1/tan

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24
Q

what happens to y = csc(x) whenever y = sin(x) touches the x-axis?

A

vertical asymptote

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25
Q

why are y = sin(x) and y = csc(x) tangent whenver x is a multiple of x

A

they r reciprocals, so csc’s max is at sin’s min, and csc’s min is at sin’s max

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26
Q

sine function: y = sin(x)

A

“wave”

amplitude - 1

period - 2pi

frequency = 1 cycle in 2pi radians (1/2pi)

max - 1

min - (-1)

one cycle occurs between 0 and 2pi with x-int @ pi

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27
Q

cosine function: y = cos(x)

A

also “wave”

amplitude = 1

period = 2pi

period = 2pi

frequency = 1 cycle in 2pi radians (1/2pi)

max = 1

min = -1

one cycle occurs b/w 0 & 2pi w/ x-int’s @ pi/2 & 3pi/2

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28
Q

tangent function: y = tan(x)

A

amplitude = none, go on forever in vertical directions

period = pi

one cycle occurs b/w -pi/2 and pi/2 (x-int = 0)

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29
Q

cotangent function: y = cot(x)

A

amplitude = none

period = pi

one cycle occurs b/w 0 and pi (x-int pi/2)

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30
Q

relationship b/w tan & cot graphs

A

the x-int’s of y = tan(x) are the asymptotes of y = cot(x)

and vice versa

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31
Q

cosecant function: y = csc(x)

A

amplitude = none

period = 2pi

one cycle is between 0 & 2pi, with the center being @ pi/2

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32
Q

relationship b/w sin & csc graphs

A

the maximum values of y = sin(x) are min values of the pos sections of y = csc(x)

the min values of y = sin(x) are the max values of the neg sections of y = csc(x)

the x-int’s of y = sin(x) are the asymptotes for y = csc(x)

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33
Q

secant function: y = sec(x)

A

amplitude = none

period = 2pi

one cycle occurs between -pi/2 and pi/2, with the center being at 0

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34
Q

relationship b/w cos & sec graphs

A

the max values of y = cos(x) are the min values of the pos sections of y = sec(x)

the min values of y = cos(x) are the max values of the neg sections of y = sec(x)

the x-int’s of y = cos(x) are the asymptotes for y = sec(x)

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35
Q

algorithm

A

a set of step-by-step directions for a process

simple sequence of steps that u follow in order

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36
Q

loop

A

a group of steps that r repeated for a certain # of time or until some condition is met

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37
Q

solving algorithms in two ways

A
  • algebra - just solve
  • graph - for inequalities
    • graph both sides of equation separately
    • estimate x-coord of intersection
    • Ex: - 3x + 9 < 4
      • all values of x for which y = -3x + 9 is BELOW y =4 (remember to use sign! and flip if necessary!!)
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38
Q

box and whisper plot

A

gives data in 4 parts - each part = 25% of the data

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39
Q

scatter plot

A
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40
Q

break-even point

A

when income = expenses

  1. separate expenses from income
  2. write equations to model the situation for I and E
  3. find break-even point, when I = E
    1. use a graph
      1. graph the equations on same set of axes
      2. x-coord of itnersection = BEP
    2. use algebra
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41
Q

matrices

A

x + y = 10,000

7x + 15y = 86,000

[A] [xy] = [B]

A = coefficient matrix, B = constant matrix

[17115] [xy] = [10,00086,000]

use calc 2nd matrix → edit

[A]-1 [B]

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42
Q

linear equation with 3 variables

A

ax + by + cz = d

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43
Q

assigning

A
  1. connect
  2. look for something that connects with only one other thing
  3. narrow down
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44
Q

diagramming

A
  1. put the thing w/ the most things in the center
  2. when 2 vertices r connected by 2+ edges, draw @ least 1 edge as a curved line
  3. draw arrows to show “direction”
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45
Q

matrixing

A

each row = departure

each column = destination

1 = on, 0 = off

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46
Q

vertex of a network

A

a dot in the network

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47
Q

edge of a network

A

line connecting 2 dots in a network

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48
Q

maximizing and minimizing

A

x-value of vertex = -b/2a

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49
Q

finding the shortest route

A
  1. Draw a network diagram that models the map. Each vertex represents a city. Each edge represents an interstate highway. Distances do not need to be drawn to scale.
  2. Label the starting point with the ordered pair (-,0)
  3. For each edge that connects a labeled and an unlabeled vertex, find this sum:
    1. s = y-value of ordered pair for labeled vertex + length of edge
  4. Choose the edge from Step 3 that has the minimum sum s. Label the unlabeled vertex of that edge with this:
    1. (label of the other vertex of the edge, s)
  5. Repeat Steps 3 & 4 until the vertex for the destination is labeled. (Go all the way back to find shortest distance).
  6. When the vertex for Danville is labeled, use the ordered pairs to find the shortest route. (backtrack)
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50
Q

constraint

A

any condition that must be met by a variable or by a linear combination of variables

x > 0 → The # of AM ads can’t be negative

y > 0 → the # of PM ads can’t be negative

x + y < 20 → the total # of ads must be less than/equal to 20

200x + 50y < 2200 → the total cost of the ads must be less than or equal to $2200

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51
Q

feasible region

A

the graph of the solution of a system of inequalities that meets all given constraints

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52
Q

graphing feasible regions

A

use system of inequalities from “constraint” definition

  1. Since x > 0 and y > 0, the feasible region is in the first quadrant.
  2. Graph x + y < 20 in the first quadrant.
  3. Identify points inthe blue shaded region that also make 200x + 50y < 2200 true.
  4. The feasible region consists of all pts on or inside quadrilateral ABCO. u can find the coord’s of each vertex​ by solving a system of equations from the intersecting lines
    1. the origin (0,0) is the solution of the system: x = 0, y = 0
    2. solve this system: x + y = 20, x = 0 to get (0,20)
    3. Solve x + y = 20, 200x + 50y = 2200 to get (8,12)
    4. solve 200x + 50y = 2200, y = 0 to get (11,0)
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53
Q

linear programming

A

can be used

  • when u can represent the constraints on the variables with a system of linear inequalities
  • when the goal is to find the max/min value of a linear combo of the variables
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54
Q

corner-point principle

A

any max/min value of a linear combo of the variables will occur at one of the vertices of the feasible region

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55
Q

using the corner-point principle

A

AM ads heard by 90,000; PM ads heard by 30,000

@ most 20 ads, @ least as many AM ads as PM ads, at least 720,000 listeners

hows many of each ad should u run to minimize the total cost? how much will the ads cost?

  1. Represent the constraints with a system of linear inequalities
    • Let x = the number of AM ads, y = # of PM ads
    • x + y < 20
    • x > y
    • 90,000x + 30,000y > 720,000
    • x > 0
    • y > 0
  2. Graph and find vertices
  3. Write a linear combo that represents the total cost of the ads: 200x + 50y
  4. Use the corner-pt principle. Find the total cost for the combo of ads represented by each vertex.
    1. min turns out to be A(6,6)
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56
Q

even function

A

a function if its graph = symmetric w/ respect to y-axis

f(-x) = f(x)

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57
Q

odd function

A

a function if its graph is symmetric with respect to the origin

turn upside down to test - 180o

f(-x) = -f(x)

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58
Q

holes

A

when a value of x sets both the denom & the numer of a rational function equal to 0, there is a hole in the graph

a single pt in which the function has no value

to find:

f(x) = (x2[x-2])/(x-2)

look for repeating thingies like x -2

therefore, x = 2

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59
Q

finding asymptotes, holes, 0’s, x-int’s, & y’int’s

A

f(x) = x2+x-6 / x+3

  • vertical asymptote
    1. set denom to 0
    2. x + 3 = 0 → x = -3
  • holes
    • factor out the numerator to (x+3)(x-2)
    • x+3 is found on both the num & denom
    • therefore, x = -3
  • zeros (where y = 0)
    • set f(x) to 0 and solve for x
  • x-int’s → same as zeros
  • y-int (where x=0)
    • set all x’s to 0 and solve
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60
Q

control variable

A

a variable that determines, or controls, another variable

x

domain

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61
Q

dependent variable

A

a variable that is determined by, or depends on, another variable

y

range

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62
Q

function

A

a relationship for which each value of the control variable is paired with only one value of the dependent variable

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63
Q

domain

A

all possible values of the control variable

x-axis

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64
Q

range

A

all possible values of the dependent variable

y-axis

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65
Q

values of a function

A

the numbers in its range

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66
Q

vertical line test

A

if no vertical line crosses a graph in >2 points, the graph represents a function

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67
Q

one-to-one function

A

a function in which each member of the range = paired with exactly one member of the domain

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68
Q

horizontal line test

A

if no horizontal line crosses the graph, it’s

  • a one-to-one function
  • has an inverse
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69
Q

many-to-one function

A

a member of the range may be paired with more than 1 member of the domain

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70
Q

linear function

A

a function that has an equation of the form f(x) = mx + b, where m is the slope of its linear graph & b is the y-intercept

domain = all real #’s

range = all real #’s

f(x) = x + b → pos moves left, neg moves right

f(x) = mx → 0 < m < 1 less steep, m > 1 more steep

f(x) = -x → reflection

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71
Q

slope

A

change in f(x)

____________

change in x

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72
Q

long division

A

if no 3a, would still have to put 0a!!!

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73
Q

synthetic division

A

ignore green

inside numbers from coefficients

once have an x2 in answer, just factor normally to get x

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74
Q

degree

A

greatest power

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75
Q

horizontal asymptotes

A
  • if degree of num > degree of denom → NO hor. asympt.
  • n < d → hor. asympt. at y = 0 (x-axis)
    • if n = d, there is hor. asympt. @ y = an/bm where an = leading coeffcient of num & bm = leading coefficient of denom.
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76
Q

piecewise function

A

a function defined by 2+ equations

each equation applies to diff part of the function’s domain

Ex:

1/2lb or less → $13

more than 1/2 lb but less than 1 lb → $20

1lb + → $25

w = weight, c(w) = charge

c(w) = {13 if 0 < w < 1/2

{20 if 1/2 < w < 1

{25 if 1 < w

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77
Q

absolute value functions

A

|x| = x if x > 0; -x if x < 0

f(x) = |x| + b → pos moves up, neg moves down

f(x) = |x - 3| → neg moves right, pos moves left

f(x) = m|x| → 0 < m < 1 less steep, m > 1 more steep

f(x) = -|x| → reflection

graph: y = |x + 2|

y = x + 2 if x + 2 > 0 → x > -2

-(x + 2) if x + 2 < 0 → x < -2

Graph y = x + 2 and y = -(x + 2)

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78
Q

quadratic functions

A

f(x) = ax2 + bx + c

where a, b, and c are constants and a doesn’t equal 0

parabola

sign of a determines whether parabola opens upward or downward

f(x) = x2 + b → pos. up, neg. down

f(x) = (x - c)2 → c neg move right, pos. move left

f(x) = ax2 → same as others

f(x) = -x2 → same as others

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79
Q

height and distance formulas

A

t = time, v0 = initial speed, A = angle at which obj is thrown, h0 = initial height

distance fallen d(t) = 16t2

height after being thrown h(t) = -16t + (v0 sin A)t + h0

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80
Q

distance problems

A

a person drops a penny from a height of 50ft. express height of penny above ground as a function of time

let h(t) = height after t secs

(distance fallen) + (height above ground) = (initial height)

16t2 + h(t) = 50ft

h(t) = -16t2 + 50

domain = 0 < t <

range = 0 < h(t) < 50

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81
Q

height problems

A

lacrosse: ball leaves player’s stick from initial height of 7ft @ speed of 9ft/s & @ angle of 30o w/ respect to horizon
1. express height of ball as function of time

h(t) = -16t2 + (V0 sin A)t + h0

  • 16t2 + (90 sin 30)t + 7
  • 16t2 + 45t + 7 <- (quadratic)
    1. When is the ball 25ft above ground?

h(t) = 25 → 25 = -16t2 + 45t + 7

0 = -16t2 + 45t - 18

  • graph & find intersecting pts [h(t) = 25]
  • quadratic formula

​t = .5 sec and 2.3 sec

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82
Q

direct variation

A

has the form f(x) = kxn

where n is a pos integer & k doesn’t equal 0

the exponent n is also **degree **of the function

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83
Q

direct variation problems

A

solve for k to write a function

ball bearing varies directly w/ cube of radius, r = .4cm and w = 2.1g

write direct var function that describes weight in terms of radius

r = radius W(r) = weight

W(r) = kr3

2.1 = k(.4)3

k = 32.8

W(r) = 32.8r3

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84
Q

polynomial function

A

sum of 1+ direct variation functions

Ex: P(x) = x4 + 16x3 + 5x2 - 13x + 6

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85
Q

zeros of a function

A

values of x that make f(x) = 0

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86
Q

radical problems

A

sqrt(x + 7) - 1 = x

sqrt(x + 7) = x + 1

(sqrt[x+y])2 = (x+1)2

x + 7 = (x+1)(x+1)

x + 7 = x2 + 2x + 1

0 = x2+ x - 6

x = -3, x = 2

**check for extraneous **

x = 2

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87
Q

radical functions

A

functions that have a variable under a radical symbol

f(x) = √x + b → pos up, neg down

f(x) = √(x-c) → neg right, pos left

f(x) = a√x → 0 < a < 1 steeper, a > 1 less steep

f(x) = -√x → reflection

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88
Q

Properties of Radicals

A

n√b = x when xn = b

n√(bn) = {b when n is odd} { |b| when n is even}

n√(ab) = n√a • n√b {when n is odd, for all values of a & b; when n is even, for pos. values of a & b}

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89
Q

rational function

A

a function defined by a rational expression

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90
Q

rational expression

A

a quotient of 2 polynomials

graphing: find asymptotes

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91
Q

rational function problems

A

Eric took 2 exams & average was 63%. if gets 100 on rest of exams, how many does he need to take for aver 85%?

t = # of tests

(63 • 2 + 100t) / (t + 2) = 85

126 + 100t = 85(t + 2)

126 + 100t = 85t + 170

15t = 44

t = 2.93 → 3 exams

92
Q

composite of 2 functions

A

the composite of 2 functions f and g = the function f(g(x)), which is read “f of g of x”

u write (f○g)(x), which is read “the composite of f and g”

to find (f○g)(x), u can find g(x) first and then find f(x) for that value

93
Q

compositing problems

A

Given h(x) = x2 + x and k(x) = 6x, find

  1. (h○k)(x)

h(k(x) = h(6x) <- substitute in

h(x) = x2 + x → h(6x) = (6x)2 + (6x) = 36x2 + 6x

  1. (h○k)(1/3)

just use previous rule to plug in

4 + 2 = 6

  1. OPERATIONS ⇒ Given: f(x) = x2 + 1 and g(x) = x2 - 1
    1. f(x) + g(x) = (x2 + 1) + (x2 - 1) = 2x2
    2. f(x) - g(x) = (x2 + 1) - (x2 - 1) = (x2 + 1) + (-x2 + 1) ← flip the signs when subtracting = 2
    3. f(x) ÷ g(x) = (x2 + 1)(x2 - 1) = x4 - 1 ← (x² + 1) / (x² - 1)

only simplify if neg, √, or fraction in denom

  1. word problems - Chris pays 15% of quarterly earnings for tax and adds $50 to be on the safe side. earns $90 a day. days(d). x = payment on “x” dollars. find function for amount earned as function of days

e(d) = 90d ← income

E(x) = 0.15x + 50 ← tax

for d days, estimated tax payment = (E ○ e)(d) = E(90d)

E(0.15x + 50) → E(90d) = 0.15(90d) + 50 = 13.5d + 50

94
Q

synthetic proof

A

proof built using a system of postulates & theorems in which the prop’s of figures, but not their actual measurements r studied

95
Q

justifications of synthetic proof

A

given statements

definitions

postulates

previously proved theorems

96
Q

Bisecting Diagonals Th

A

if the 2 diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram

97
Q

implications

A

if-then statements that can be represented by symbols

Ex: p → q reads “If p, then q”

98
Q

logically equivalent

A

either both true or both false

original & contrapositive

99
Q

not logically equivalent

A

just cuz original = true, doesn’t mean converse & inverse r too

100
Q

converse

A

q → p

If q, then p

101
Q

inverse

A

~p → ~q

If not p, then not q

102
Q

contrapositive

A

~q → ~p

If not q, then not p

103
Q

median (of a triangle)

A

a segment joining a vertex to the midpt of the opp. side

104
Q

coordinate proof

A

a proof based on a coord. system in which all pts r represented by ordered pairs of #’s

105
Q

justifications for coordinate proof

A

distance & midpt formulas

parallel lines have the same slope

perp. lines have slopes that r neg. reciprocals of each other

a geometric figure may be placed anywhere in the coord. plane

106
Q

distance formula

A
107
Q

midpoint formula

A
108
Q

Isosceles Median Theorem

A

In an isosceles triangle, the medians drawn to the legs r equal in measure

109
Q

isosceles trapezoid

A

a trapezoid w/ a line of symmetry that passes through the midpts of the bases

110
Q

Isosceles Trapezoid Th

A

in an isosceles trapezoid

  1. the legs r equal in measure
  2. the diagonals r equal in measure
  3. the 2 angles@ each base r equal in measure
111
Q

inclusive definition

A

a definition that includes all possibilites

112
Q

exclusive definition

A

a definition that excludes some possibilities

113
Q

quadrilateral chart

A
114
Q

past postulates & theorems

A

Addition, Subtraction, Mult, Div Prop’s of Eq Post’s

Reflexive Prop of Eq Post

Subsitution

Distributive

If 2 angles r supp’s of same angle, then r equal in measure

if 2 angles r compl’s of same angle, then equal

Straight Angle Post - if the sides of an angle form a straight line, then the angle is a straight angle w/ measure 180º

Angle/Segment Addition Post - for any seg/angle, the measure of the whole = to the sum of the measures of its non-overlapping parts

vertical angles r =

the sum of the measures of the angles of a triangle = 180º

an exterior angle of a triangle = the sum of the measures of its 2 remote interior angles

if 2 sides of a triangle r =, then the angles opp those sides r =, and vice versa

if a tri is equilateral, then also equiangular, w/ three 60º angles, and vice versa

if 2 parallel lines r intersected by a trans, then corr angles r =, vv

if 2 parallel lines r intersected by a trans, then alt int angles r =, vv

if 2 par lines r inters’d by a trans, then co-int angles r supp, vv

if 2 lines r perp to the same trans, then they’re parallel

if a trans = perp to one of 2 par lines, then its perp to the other one as well

thru a pt not on a given line, there’s 1 and only 1 parallel line to the given line

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

a seg can be drawn perp to a given line from a pt not on the line

AA similiarity - if 2 angles of 1 tri r = to 2 angles of another tri, then the 2 tri’s r similar

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

in a tri, a seg that connects the midpts of 2 sides is parallel to the 3rd side & half as long

ASA, AAS th’s

SAS, SSS post’s

If the alt = drawn to the hyp of a right tri, then the 2 tri’s formed r similar to the orginal tri & to each other

Pythagorean Th

if the alt is drawn to the hyp of a right tri, then the measure of the alt is the geometric mean b/w the measures of the parts of the hyp

the sum of the lengths of any 2 sides of a tri > length of 3rd side

in an isosc tri, the medians drawn to the legs r equal in measure

in a parallelogram, the diagonals have the same midpt

in a rectangle, the diagonals r equal in measure

in a kite, the diagonals r perp to each other

in a parallelogram, opp sides r equal in measure

if a quadrilateral is a parallelogram, then consecutive angels r supp

if a quad is parallelogram, then opp angles r =

the sum of the measures of the angles of a quad = 360º

if both pairs of opp angles of a quad r equal in measure, then the quad = a parallelogram

115
Q

Interior Angle Measures in Polygons Th

A

the sum of the angle measures of an n-gon is given by the formula

S(n) = (n - 2)180º

116
Q

Exterior Angle Measures in Polygons Th

A

the sum of the exterior angle measures of an n-gon, 1 angle at each vertex, is 360º

117
Q

regular polygon

A

iff all its sides & angles r = in measure

118
Q

inscribed

A

drawn inside the figure

119
Q

circumscribed

A

drawn outside the figure

120
Q

chords

A

segments whose endpts r on the circle

121
Q

perp bisector of a chord th

A

the perp bisector of a chord of a circle passes thru the center of the circle

122
Q

central angle

A

an angle w/ its vertex @ the center of the circle

measure of an arc intercepted (cut off) by a central angle = measure of that central angle

123
Q

minor arc

A

< 180

can be named w/ 2/3 letter (just remember that a major arc is named w/ 3 letters to distinguish it from a minor arc w/ the same endpts)

124
Q

semicircle

A

= 180

named w/ 3 letters

outside letters = diameter

125
Q

major arc

A

180 < arc < 360

named w/ 3 letters

126
Q

inscribed angle

A

an angle formed by 2 chords that intersect at a point ON circle

127
Q

intercepted arc

A

the arc that lies w/in an inscribed angle

128
Q

inscribed angle measure th

A

the measure of an inscribed angle of a circle = 1/2 the measure of its intercepted arc

129
Q

inscribed right angle th

A

an inscribed angle whose intercepted arc is a semicircle is a right angle

130
Q

equal inscribed angles th

A

if 2 inscribed angles in the same circle intercept the same arc, then they r = in measure

131
Q

intersecting chords theorem

A

the measure of angle formed by 2 chords that intersect INSIDE a circle = 1/2 the sum of the measures of the intercepted arcs

.5(a + b)

132
Q

secants & tangents th

A

the measure of an angle formed by 2 secants, 2 tangents, or a secant & a tangent drawn from a pt outside the circle = 1/2 the difference of the measures of the intercepted arcs

½(big - small)

133
Q

tips for finding angles

A

continue radius to diameter

use systems of equations

remember perp rule

134
Q

tangent

A

a line in the plane of a cricle and intersecting the circle in exactly 1 pt

135
Q

secant

A

a line intersecting the circle in 2 pts

136
Q

Short Point Post

A

a seg can be drawn perp to a given line from a pt not on the line

the length of this seg is the shortest distance from the pt to the line

137
Q

Perp Tangent Th

A

if a line is tangent to a circle, then the line is perp to the radius drawn from the center to the pt of tangency

138
Q

Converse of Perp Tangent Th

A

if a line in the plane of a circle is perp to a radius at its center endpt, then then the line is tangent to the circle

139
Q

Equal Tangents Th

A

if 2 tangent segments are drawn from the same pt to the same circle, then they equal in measure

140
Q

semiperimeter

A

half the perimeter

141
Q

how to find area of circumscribed polygons using trig

A
  1. total degrees = 360º
  2. divide 360 by # of vertices to find all internal angles
  3. divide internal angles by 2 to find angle in new right triangle
  4. use trig to find sides (already have radius & angle & know its right tri)
  5. find area of original triangle
  6. multiply by # of vertices

Ex: pentagon w/ radius 30cm

  1. total degrees = 360
  2. 360÷5 = 72
  3. 72÷2 = 36
  4. tan36º = x/30; x = ~21.8
  5. 21.8 • 2 = 43.6; ½bh = ½(30)(43.6) = 654cm²
  6. 654 • 5 = 3,270cm3​​
142
Q

area of a circumscribed polygon

A

the area of any circumscribed polygon is the product of the radius(r) of the inscribed circle & the semiperimeter(s)

A = rs

143
Q

polyhedron

A

a space figure whose faces are all polygons

144
Q

semiregular polyhedron

A

a polyhedron w/ faces that r all regular polygons, & w/ the same # of faces of each type @ each vertex

145
Q

regular polyhedron

A

a polyhedron w/ faces that r all the same type of regular polygon, & w/ the same # of faces @ each vertex

5 regulars: tetrahedron, hexahedron, octahedron, dodecahedron, icosahedron

146
Q

tetrahedron

A

4 equilateral triangles

147
Q

hexahedron

A

6 squares

148
Q

octahedron

A

8 equilateral triangles

149
Q

dodecahedron

A

12 regular pentagons

150
Q

icosahedron

A

20 equilateral triangles

151
Q

Convex Polyhedron Postulate

A

in any convex polyhedron, the sum of the measures of the angles at each vertex is less than 360º

152
Q

net

A

a 2D drawing showing the connected faces of a space figure & how they r connected

can be cut out & “folded up” to form the space figure

153
Q

defect

A

the angle measure of the gap @ a vertex on a net for a polyhedron

can be found by subtracting the sum of the angles @ that vertex from 360

154
Q

Euler’s formula

A

F + V = E + 2

f = # of faces

v = # of vertices

e = # of edges

155
Q

formulas to know

A

explicit arithmetic → an = a1 + d(n - 1)

recursive arithmetic → a1 = first term; an = an-1 + d

explicit geometric → an = a1rn-1

recursive geometric → a1 = first term; an = (an-1)r

sum finite arithmetic series → S = n(a1+an) ÷ 2

sum finite geometric series → S = a1 - anr ÷ 1-r

sum geoemtric infinite series → S = a1 ÷ 1-r; |r| < 1

156
Q

sequence

A

an ordered list of numbers

157
Q

terms

A

the numbers in a sequence

158
Q

finite sequence

A

a sequence with a last term

159
Q

infinite sequence

A

a sequence with no last term

160
Q

graphing sequences

A

each term is paired w/ a number that gives its position in the sequence

by plotting pts with coord’s (position, term), u can graph a sequence on a coord plane

161
Q

limit of a sequence

A

when the terms of an infinite sequence get closer and closer to a single fixed number L, then L is called the limit

not all infinite sequences have one; repeating ones, for ex, like 1, 2, 3, 1, 2, 3, …

graphing it often helps

162
Q

explicit formula

A

gives the value of any term an in terms of n

163
Q

finding explicit formulas

A

Ex: Given 90, 83, 76, 69,…, find a formula for the sequence

  1. Find a pattern. || a repeated subtraction of 7
  2. write the 1st few terms. Show how u found each term. ||

a1 = 90

a2 = 83 = 90 - 1(7)

a3 = 76 = 90 - 2(7)

a4 = 69 = 90 - 3(7)

  1. Express the pattern in terms of n. || an = 90 - (n - 1)7
    2.
164
Q

subscript 0

A

when the 1st term of a sequence represents a starting value before any change occurs, subscript 0 is often used

Ex: monthly bank account balances, 1st term is v0 for initial deposit. next term = v1, for 1st month’s interest, etc. etc.

165
Q

percentage explicit formulas

A

Ex: bacteria count increases 10% each day; 10,000 now; Find formula for bacteria count after n days

  1. d + .1d {if annual interest, compounded monthly, then d + (.1/12)d}

d(1 + .1)

d(1.1)

  1. d0 = 10,000

d1 = 10,000(1.1)

d2 = d1(1.1)

= 10,000(1.1)(1.1)

= 10,000(1.1)²

d3 = d2(1.1)

= 10,000(1.1)2(1.1)

= 10,000(1.1)3

•
•

•

dn = 10,000(1.1)n

166
Q

self-similarity

A

the appearance of any part is similar to the whole shebang

167
Q

recursive formula

A

tells how to find the nth term from the term(s) before it

2 parts:

  1. a1 = 1 ↔ value(s) of 1st term(s) given
  2. an = 2an-1 ↔ recursion equation
168
Q

finding recursive formulas

A

Ex: 1, 2, 6, 24

  1. Write several terms of the sequence using subscripts. Then look for the relationship b/w each term & the term before it.

a1 = 1

a2 = 2 = 2 • 1

a3 = 6 = 3 • 2

a4 = 24 = 4 • 6

  1. Write in terms of a

a2 = 2 • a1

a3 = 3 • a2

a4= 4 • a3

  1. Write a recursion equation

an = nan-1

  1. Write recursive formula

a1 = 1

an = nan-1

169
Q

percentage recursive formulas

A

Ex: 650mg of aspirin every 6h; only 26% of aspirin remaining in body by the time of new dose; what happens to amount of aspirin in body if taken several days?

  1. write a recursion equation

amount aspiring after nth dose = 26% amount after prev. dose + new dose of 650mg

an = (0.26)(an-1) + 650

  1. Use a calc

Enter a1 → 650

Enter recursion equation using ANS for an-1 → .26ANS + 650

Keep pressing Enter

sequence appears to approach limit of about 878

170
Q

arithmetic sequence

A

a sequence in which the diff b/w any term & the term before it = a constant

Ex: 2,4, 6, 8

+2, +2, +2

171
Q

common difference

A

d

the constant value an- an-1

172
Q

geometric sequence

A

a sequence in which the ratio of any term to the term before it is a constant

Ex: 2, 4, 8, 16

x2, x2, x2`

173
Q

common ratio

A

r

the constant value an / an-1

174
Q

explicit arithmetic formula

A

an = a1 + (n - 1)d

175
Q

recursive arithmetic formula

A

a1 = first term

an = an-1 + d

176
Q

using explicit formulas to find out how many terms r in a finite sequence

A

Ex: 33, 29, 25, 21,…, 1

  1. arithmetic, geo, or neither?

d = -4 → arithmetic

  1. Use a formula

an = a1 + (n - 1)d

1 = 33 + (n - 1)(-4)

1 = 33 - 4n + 4

-36 = -4n

n = 9

there r 9 terms

177
Q

explicit geometric formula

A

an = a1rn-1

178
Q

recursive geometric formula

A

a1 = first term

an = (an-1)r

179
Q

geometric mean

A

√(ab)

Ex: find x in geo sequence 3, x, 18

  1. In geo sequence, an/aan-1 is a constant

a2/a1 = a3/a2

x/3 = 18/x

x2 = 54

x = + √54 = _+_3√6

x is 3√6 or -3√6

180
Q

series

A

the indicated sum of the terms when the terms of a sequence are added

sequence = 1, 2, 3, 4, 5, 6

series = 1 + 2 + 3 + 4 + 5 + 6

181
Q

finite series

A

has a last term

182
Q

infinite series

A

has no last term

183
Q

arithmetic series

A

its terms form an arithmetic sequence

184
Q

sum finite arithmetic series

A

S =

n(a1 + an)

________

2

185
Q

finding the sum of a finite arithmetic series

A

Ex: 7 + 12 + 17 + 22 +…+ 52

  1. arith, geo, or neither

d = 5, arithmetic

  1. find the # of terms

an = a1 + (n - 1)d

52 = 7 + (n - 1)5

52 = 7 + 5n - 5

n = 10 → 10 terms

  1. Use formula → S = 295
186
Q

sigma notation

A

uses summation symnbol Greek sigma Σ

Read: the sum of 2n for integer values of n from 1 to 6

if infinite series, number on top is ∞

187
Q

expanded form

A

when u substitute the values of n into the formula

188
Q

geometric series

A

its terms from a geometric sequence

189
Q

sum finite geometric series

A

S =

a1 - anr

________

1 - r

190
Q

evaluating sigma notation

A

if finite,

  1. write in expanded form
  2. decide if arithmetic, geo, or neither
  3. use formula for sum of whatever type it is

if infinite,

  1. see whether a sum even exists first by expanding first few terms
  2. if geometric, use ratio to see if sum exists. if arith, graph.
  3. find sum using formula or graphing
191
Q

partial sum

A

the sum of the first n terms of an infinite series

192
Q

sequence of partial sums

A

2, (2+4), (2+4+6), … forms a sequence of partial sums for the infinite series 2 + 4 + 6 +…

193
Q

sum of an infinite series

A

if the sequence of the partial sums of an infinite series has a limit, then that limit is the sum of the series

194
Q

finding sums of infinite series by graphing

A

Ex: 3 + (-1.5) + .75 + (-0.375) + …

  1. Find common ratio → r = 0.5
  2. Find 1st few partial sums

S1 = 3

S2 = 3 + (-1.5) = 1.5

S3 = 1.5 + 0.75 + (-0.375) = 1.875

S5 = 1.875 + .1875 = 2.0625

S6 = 2.0625 + (-0.09375) = 1.96875

  1. graph the partial sums (position, term)
  2. limit is about 2, so sum is about 2
195
Q

sum infinite geometric series

A

S = a1 / 1-r for |r| < 1

196
Q

discrete exponential growth

A

represented by points

Ex: most species only produce once a year

197
Q

continuous exponential growth

A

represented by a line, curve, etc.

Ex: over-lapping generations. some species bred thruout the year

198
Q

exponential function

A

y = abx

y - amount after x period

a - initial value

b - growth factor

x - time period

199
Q

how to write exponential functions and predict

A

2.9 mill ppl in 1980, increasing 1.7% each yr

  1. write an exponential function
  2. when will pop reach 4.5 mill?

a.
1. find growth factor b

pop @ end of yr = 100% of pop @ start + 1.7% of pop @ start

pop @ end of yr = 101.7% pop @ start of year

b = 1.017

  1. y = abx

y = (2.9)(1.017)x

b.

  1. Use a graph
    1. Graph the equation y = (2.9)(1.017)x
    2. x-value of 0 = 1980, x-value of 26 = 2006
  2. Use recursion
    1. Enter initial amount 2.9 & repeatedly multiply by growth factor (1.017)
    2. count # of times pressed Enter (26)
    3. 1980 + 26 = 2006
200
Q

exponential growthx

b > 1

A

y = ab

b > 1

201
Q

exponential decay

A

y = abx

0 < b < 1

202
Q

negative exponents

b

A

b-x = (1/b)x

Ex: y = 52(6/5)-x = 52(5/4)x

203
Q

half-life problems

A

1200 ppl; every ½ hour, ½ ppl leave

1st half-life → (1200)(½)

2nd half-life → (1200)(½)(½)

f(x) = 1200(½)x

**or **just use formula

how many ppl after 3 hours

about 18

204
Q

fractional exponent rules

A

a1/n = ⁿ√a

am/n =(ⁿ√a)m or ⁿ√(am)

IF n IS EVEN, MUST USE ABSOLUTE VALUE

205
Q

solving exponent algebraic equations

A

find the value of b when f(x) = 4bx and f(¾) = 32

4b¾ = 32

b¾ = 8

b¾(4/3)= 84/3

b = 84/3 = (3√8)4 = 24 = 16

206
Q

e

A

the number e is an irrational # that is approx. 2.718 ← this is also the limit of the sequence (1 + 1/1)1, (1+½)2, (1+1/3)3, …

207
Q

logistic growth function

A

a function in which the rate of growth of a quantity slows down after initial increasing/decreasing exponentially

208
Q

solving e problems

A

spread of flu in 1,000 ppl modeled by y = 1000/(1 + 990e-0.7x), where y = the # of ppl after x days

  1. how many ppl after 9 days?

Graph the equation

read y when x = 9 → abt 355 ppl

  1. horizontal asymptotes?

trace along same graph

min. y-value gets close to but never reaches 0
max. y-value gets close to but never reaches 1000

y = 0, y = 1000

  1. estimate max # of ppl → max. of y = 1000ppl
209
Q

investment formulas

A

A = P(1 + r/n)nt

A = Pert (for continuous)

P - initial value

r - rate (without 1)

n - # of time a year

t - years

210
Q

inverse functions

A

two functions f & g are inverse functions if g(b) = a whenever f(a) = b

graph = reflection of the graph of the original function over the line y = x

inverse of f(x) can be written f-1(x) or f-1

211
Q

graphing inverse functions

A

a. y = 2x
b. y = x2
1. plot the points to grpah each function. then interchange the coordinates of the original points and plot these points

212
Q

deciding existence of inverse functions

A

an inverse is a function. if a reflection of a function over the line y = x isn’t a function, then the inverse doesn’t exist

a. y = 2x b. y = x2
1. Use the vertical line test to decide whether a reflection is the graph of a function (inverse)
- or-
2. Just like vertical line tests, u can use the horizontal line test to see whether a function has an inverse (original)

213
Q

writing inverse equations

A

interchange x & y, and solve for y

214
Q

logarithmic function

A

the inverse of the exponential function f(x) = bx and written as f-1= logbx

x = ba ↔ a = logbx

x on outside, b in middle, a in center

base of a log can be any pos. # except 1

log4(-2) is undefined! can’t have neg log

215
Q

common logarithm

A

a log w/ base 10

common log of x written as log x

x = 10a ↔ a = log x

216
Q

natural logarithm

A

a log w/ base e

usually written as ln x

x = ea ↔ a = ln x

217
Q

evaluating logs

A

evaluate each. (Find a).

  1. log464

log464 = log443 → a = 3; has to be 43 cuz base = 4

  1. log(1/10,000)

log(1/10,000) = log10-4 → a = -4

  1. ln5.3 → use calc
  2. log145 → use calc
218
Q

solving logs

A

when solving these, remember that x isn’t necessarily the x in the formula. it can be a, b, or x

solve logx81 = 2

logb81 = 2

b2 = 81

b = 9

Solve. round decimal answers to nearest hundredth

a. 10x = 15

10a = 15

a = log 15 = abt 1.18

b. ea = 29

ea = 29

a = ln 29 = abt 3.37

c. log2x + log2(x-2) = 3

log2x + log2(x-2) = 3

log2x(x-2) = 3

x(x-2) = 23

x2 - 2x = 8

x2 - 2x - 8 = 0

(x+2)(x-4) = 0

x + 2 = 0 or x - 4 = 0

x = -2 x = 4

check for extraneous solutions! Sub possible solutions into the original equation to be sure they’re not extraneous

x = -2 is undefined, cuz no neg logs log2(-2)

219
Q

properties of logs

A

M, N, and P are pos. #’s w/ b not equaling 1, and k is any real #

220
Q

Product of Logarithms Property

A

logbMN = logbM + logbN

221
Q

Quotient of Logarithms Property

A

logbM/N = logbM - logbN

222
Q

Power of Logarithms Property

A

logbMk = k logb M

223
Q

writing in terms of logM and logN

A

a. logM2N3

logM2 + logN3 = 2logM + 3logN

b. log3√(M)/N4

log3√M - logN4 = logM1/3 - logN4 = 1/3logM - 4logN

224
Q

simplifying logs

A

ln18 - 2ln3 + ln4

ln18 - ln32 + ln4 → power prop.

ln18 - ln9 + ln4

ln18/9 + ln4 → quotient prop.

ln2 + ln4

ln(2•4) → product prop.

ln8

225
Q

log and exp word problems

A
  1. L = 10log(I/I0) What is the change in loudness(L) when I is doubled?

L1 = original loudness L2 = loudness after intensity is doubled

increase in loudness = L2 - L1

L2 - L1 = 10log(2I/I0) - 10log(I/I0)

10log(2•I/I0) - 10log(I/I0)

10(log2 + logI/I0) - 10log(I/I0)

10log2 + 10log(I/I0) - 10log(I/I0)

10log2 = abt 3

  1. A(t) = A0(0.883)t with A(t) - amount present t thousand yrs after death, A0 - amount @ time of death, and t - amount time since death (in 1000’s of yrs)
     found in 1968, died 8000 yrs ago, 37% present now; estimate age of the bone

A0(0.883)t = A(t)

A0(0.883)t = 0.37A0

0.883t = 0.37

log(0.883)t = log0.37

t log 0.883 = log0.37

t = log0.37/log0.883

t = 8

226
Q

Change of Base Property of Logs

A

for all pos. #’s with b and c not equaling 1,

logbM = (logcM) / (logcb)

227
Q

REMEMBER

A

when evaluating sigmas, or whatever, if it’s something like (3/4)k, then know that 3/4 is the common ratio! or if 7(-2)c, -2 is common ratio!

when calculating exponents with negative bases, be sure to use parenthesis! (-.75)3

Math II (17 decks)

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