Finals Flashcards

Question

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

Answer 1

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

Answer 2

"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

Answer 3

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

Answer 4

amplitude = none, go on forever in vertical directions period = pi one cycle occurs b/w -pi/2 and pi/2 (x-int = 0)

Answer 5

amplitude = none period = pi one cycle occurs b/w 0 and pi (x-int pi/2)

Answer 6

the x-int's of y = tan(x) are the asymptotes of y = cot(x) and vice versa

Answer 7

amplitude = none period = 2pi one cycle is between 0 & 2pi, with the center being @ pi/2

Answer 8

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)

Answer 9

amplitude = none period = 2pi one cycle occurs between -pi/2 and pi/2, with the center being at 0

Answer 10

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)

Answer 11

a set of step-by-step directions for a process simple sequence of steps that u follow in order

Answer 12

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

Answer 13

* 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!!)

Answer 14

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

Answer 15

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

Answer 16

x + y = 10,000 7x + 15y = 86,000 [A] [xy] = [B] A = coefficient matrix, B = constant matrix [¹₇¹₁₅] [^x_y] = [^10,000_86,000] use calc 2nd matrix → edit [A]-1 [B]

Answer 17

ax + by + cz = d

Answer 18

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

Answer 19

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"

Answer 20

each row = departure each column = destination 1 = on, 0 = off

Answer 21

a dot in the network

Answer 22

line connecting 2 dots in a network

Answer 23

x-value of vertex = -b/2a

Answer 24

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)

Answer 25

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

Answer 26

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

Answer 27

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)

Answer 28

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

Answer 29

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

Answer 30

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)

Answer 31

a function if its graph = symmetric w/ respect to y-axis f(-x) = f(x)

Answer 32

a function if its graph is symmetric with respect to the origin turn upside down to test - 180^o f(-x) = -f(x)

Answer 33

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

Answer 34

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

Answer 35

a variable that determines, or controls, another variable x domain

Answer 36

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

Answer 37

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

Answer 38

all possible values of the control variable x-axis

Answer 39

all possible values of the dependent variable y-axis

Answer 40

the numbers in its range

Answer 41

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

Answer 42

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

Answer 43

if no horizontal line crosses the graph, it's * a one-to-one function * has an inverse

Answer 44

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

Answer 45

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

Answer 46

change in f(x) \_\_\_\_\_\_\_\_\_\_\_\_ change in x

Answer 47

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

Answer 48

ignore green inside numbers from coefficients once have an x² in answer, just factor normally to get x

Answer 49

greatest power

Answer 50

* 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 = a_n/b_m where a_n = leading coeffcient of num & b_m = leading coefficient of denom.

Answer 51

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

Answer 52

|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)

Answer 53

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

Answer 54

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

Answer 55

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

Answer 56

lacrosse: ball leaves player's stick from initial height of 7ft @ speed of 9ft/s & @ angle of 30^o 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

Answer 57

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

Answer 58

**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

Answer 59

sum of 1+ direct variation functions Ex: P(x) = x⁴ + 16x³ + 5x² - 13x + 6

Answer 60

values of x that make f(x) = 0

Answer 61

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

Answer 62

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

Answer 63

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}

Answer 64

a function defined by a rational expression

Answer 65

a quotient of 2 polynomials graphing: find asymptotes

Answer 66

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

Answer 67

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

Answer 68

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

Answer 69

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

Answer 70

given statements definitions postulates previously proved theorems

Answer 71

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

Answer 72

if-then statements that can be represented by symbols Ex: p → q reads "If p, then q"

Answer 73

either both true or both false original & contrapositive

Answer 74

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

Answer 75

q → p If q, then p

Answer 76

~p → ~q If not p, then not q

Answer 77

~q → ~p If not q, then not p

Answer 78

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

Answer 79

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

Answer 80

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

Answer 81

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

Answer 82

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

Answer 83

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

Answer 84

a definition that includes all possibilites

Answer 85

a definition that excludes some possibilities

Answer 86

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

Answer 87

the sum of the angle measures of an n-gon is given by the formula S(n) = (n - 2)180º

Answer 88

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

Answer 89

iff all its sides & angles r = in measure

Answer 90

drawn inside the figure

Answer 91

drawn outside the figure

Answer 92

segments whose endpts r on the circle

Answer 93

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

Answer 94

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

Answer 95

\< 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)

Answer 96

= 180 named w/ 3 letters outside letters = diameter

Answer 97

180 \< arc \< 360 named w/ 3 letters

Answer 98

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

Answer 99

the arc that lies w/in an inscribed angle

Answer 100

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

Answer 101

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

Answer 102

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

Answer 103

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)

Answer 104

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)

Answer 105

continue radius to diameter use systems of equations remember perp rule

Answer 106

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

Answer 107

a line intersecting the circle in 2 pts

Answer 108

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

Answer 109

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

Answer 110

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

Answer 111

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

Answer 112

half the perimeter

Answer 113

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

Answer 114

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

Answer 115

a space figure whose faces are all polygons

Answer 116

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

Answer 117

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

Answer 118

4 equilateral triangles

Answer 119

8 equilateral triangles

Answer 120

12 regular pentagons

Answer 121

20 equilateral triangles

Answer 122

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

Answer 123

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

Answer 124

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

Answer 125

F + V = E + 2 f = # of faces v = # of vertices e = # of edges

Answer 126

explicit arithmetic → a_n = a₁ + d(n - 1) recursive arithmetic → a₁ = first term; a_n = a_n-1 + d explicit geometric → a_n = a₁r^n-1 recursive geometric → a₁ = first term; a_n = (a_n-1)r sum finite arithmetic series → S = n(a₁+a_n) ÷ 2 sum finite geometric series → S = a₁ - a_nr ÷ 1-r sum geoemtric infinite series → S = a₁ ÷ 1-r; |r| \< 1

Answer 127

an ordered list of numbers

Answer 128

the numbers in a sequence

Answer 129

a sequence with a last term

Answer 130

a sequence with no last term

Answer 131

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

Answer 132

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

Answer 133

gives the value of any term a_n in terms of n

Answer 134

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. || a_n = 90 - (n - 1)7 2.

Answer 135

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.

Answer 136

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 • • • d_n = 10,000(1.1)ⁿ

Answer 137

the appearance of any part is similar to the whole shebang

Answer 138

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. a_n = 2a_n-1 ↔ recursion equation

Answer 139

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 a_n = na_n-1 1. Write recursive formula a₁ = 1 a_n = na_n-1

Answer 140

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 a_n = (0.26)(a_n-1) + 650 1. Use a calc Enter a₁ → 650 Enter recursion equation using ANS for a_n-1 → .26ANS + 650 Keep pressing Enter sequence appears to approach limit of about 878

Answer 141

a sequence in which the diff b/w any term & the term before it = a constant Ex: 2,4, 6, 8 +2, +2, +2

Answer 142

d the constant value a_n- a_n-1

Answer 143

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`

Answer 144

r the constant value a_n / a_n-1

Answer 145

a_n = a₁ + (n - 1)d

Answer 146

a₁ = first term a_n = a_n-1 + d

Answer 147

Ex: 33, 29, 25, 21,..., 1 1. arithmetic, geo, or neither? d = -4 → arithmetic 1. Use a formula a_n = a₁ + (n - 1)d 1 = 33 + (n - 1)(-4) 1 = 33 - 4n + 4 -36 = -4n n = 9 there r 9 terms

Answer 148

a_n = a₁r^n-1

Answer 149

a₁ = first term a_n = (a_n-1)r

Answer 150

√(ab) Ex: find x in geo sequence 3, x, 18 1. In geo sequence, a_n/a_an-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

Answer 151

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

Answer 152

has a last term

Answer 153

has no last term

Answer 154

its terms form an arithmetic sequence

Answer 155

S = n(a₁ + a_n) \_\_\_\_\_\_\_\_ 2

Answer 156

Ex: 7 + 12 + 17 + 22 +...+ 52 1. arith, geo, or neither d = 5, arithmetic 1. find the # of terms a_n = a₁ + (n - 1)d 52 = 7 + (n - 1)5 52 = 7 + 5n - 5 n = 10 → 10 terms 1. Use formula → S = 295

Answer 157

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 ∞

Answer 158

when u substitute the values of n into the formula

Answer 159

its terms from a geometric sequence

Answer 160

S = a₁ - a_nr \_\_\_\_\_\_\_\_ 1 - r

Answer 161

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

Answer 162

the sum of the first n terms of an infinite series

Answer 163

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

Answer 164

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

Answer 165

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

Answer 166

S = a₁ / 1-r for |r| \< 1

Answer 167

represented by points Ex: most species only produce once a year

Answer 168

represented by a line, curve, etc. Ex: over-lapping generations. some species bred thruout the year

Answer 169

y = ab^x y - amount after x period a - initial value b - growth factor x - time period

Answer 170

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 = ab^x 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

Answer 171

y = ab b \> 1

Answer 172

y = ab^x 0 \< b \< 1

Answer 173

b^-x = (1/b)^x Ex: y = 52(6/5)^-x = 52(5/4)^x

Answer 174

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

Answer 175

a^1/n = ⁿ√a a^{m/n =}(ⁿ√a)^m or ⁿ√(a^m) **IF n IS EVEN, MUST USE ABSOLUTE VALUE**

Answer 176

find the value of b when f(x) = 4b^x and f(¾) = 32 4b^¾= 32 b^¾ = 8 b^¾(4/3)= 8^4/3 b = 8^4/3 = (³√8)⁴ = 2⁴ = 16

Answer 177

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, ...

Answer 178

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

Answer 179

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

Answer 180

A = P(1 + r/n)^nt A = Pe^rt (for continuous) P - initial value r - rate (without 1) n - # of time a year t - years

Answer 181

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

Answer 182

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

Answer 183

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 = 2^x 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)

Answer 184

interchange x & y, and solve for y

Answer 185

the inverse of the exponential function f(x) = b^x and written as f^-1= log_bx **x = b^a ↔ a = log_bx** 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

Answer 186

a log w/ base 10 common log of x written as log x x = 10^a ↔ a = log x

Answer 187

a log w/ base e usually written as ln x x = e^a ↔ a = ln x

Answer 188

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

Answer 189

**when solving these, remember that x isn't necessarily the x in the formula. it can be a, b, *or* x** solve log_x81 = 2 log_b81 = 2 b² = 81 b = 9 Solve. round decimal answers to nearest hundredth a. 10^x = 15 10^a = 15 a = log 15 = abt 1.18 b. e^a = 29 e^a = 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)

Answer 190

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

Answer 191

log_bMN = log_bM + log_bN

Answer 192

log_bM/N = log_bM - log_bN

Answer 193

log_bM^k = k log_b M

Answer 194

a. logM²N³ logM² + logN³ = 2logM + 3logN b. log³√(M)/N⁴ log³√M - logN⁴ = logM^1/3 - logN⁴ = 1/3logM - 4logN

Answer 195

ln18 - 2ln3 + ln4 ln18 - ln3² + ln4 → power prop. ln18 - ln9 + ln4 ln18/9 + ln4 → quotient prop. ln2 + ln4 ln(2•4) → product prop. ln8

Answer 196

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 2. 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 0.883^t = 0.37 log(0.883)^t = log0.37 t log 0.883 = log0.37 t = log0.37/log0.883 t = 8

Answer 197

for all pos. #'s with b and c not equaling 1, log_bM = (log_cM) / (log_cb)

Answer 198

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

Finals Flashcards

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