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MAT 1800 > final exam > Flashcards

final exam Flashcards

1
Q

triangle formulas

A 
A= 1/2bh 
P= a+b+c
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2
Q

rectangle formulas

A

A=lw

P=2(l+w)

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3
Q

trapezoid

A

A=1/2(a+b)h

P= a+b+c+d

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4
Q

ellipse

A

A=πab

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5
Q

square

A

A=a^2

P= 4a

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6
Q

parallelogram

A

A=bh

P= 2(a+b)

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7
Q

circle

A

A=πr^2

C=2πr

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8
Q

sector

A

A=1/2r^2θ

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9
Q

rectangular prism

A

SA= 2(wh+lw+lh) or x^2+4xh
V=lwh
*Rectangular Box

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10
Q

cylinder

A 
SA= 2πr^2+2πrh
V= πr^2h
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11
Q

guidelines for modeling w functions

A
  1. express model in words
  2. choose the variable
  3. set up the model
  4. use model
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12
Q

story problem steps

A
  1. get the question
  2. identify the variable(s)
  3. build equations
  4. solve
  5. ask “does my answer make sense?”
  6. state answer as a sentence
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13
Q

transformations steps

A
  1. sketch basic graph
  2. do horizontal shifts
  3. do reflections
  4. do vertical shifts
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14
Q

inverse functions

A
  1. f(x1) ≠ f(x2)
  2. f(x1) = f(x2)

f^-1(y)=x; f(x)=y

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15
Q

cancellation props of inverse functions

A 
if f and f^-1 are inverse functions,
f^-1(f(x))=x
f (f^-1(x))=x
if f and f^-1 have the property *, we say f and f^-1 are inverse functions 
*anything >1 is undefined
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16
Q

technique to find inverse functions

A
  1. interchange x+y roles
  2. solve for y in terms of x
  3. set f^-1(x)=y
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17
Q

average rate of change

A

f(b)-f(a)/b-a

*diff quotient (x=a; x=a+h)
f(a+h)-f(a)/h

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18
Q

domains for combining functions

A

(f+g), (f-g), and (fg) are D=A∩B

(f/g) is D{x∈A∩B I g(x) ≠ 0}

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19
Q

x= b _slope; y=a _slope

A

und; 0

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20
Q

inequalities steps

A
  1. move all nonzero terms to one side
  2. factor
  3. find interval
  4. sign chart w test values
  5. solve
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21
Q

complete the square steps

A
  1. make sure a=1
  2. subtract c (x^2+bx= -c)
  3. find (b/2)^2
  4. add that value to both sides
  5. factor
  6. take square rt of both sides
  7. solve
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22
Q

a^m/n =

A

(^n√a)^m

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23
Q

a^b+c =

A

a^b*a^c

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24
Q

(a^b)^c

A

a^b*c

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25
Q

e is about

A

2.71828

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26
Q

standard form

A

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

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27
Q

h=

A

-b/2a

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28
Q

k=

A

plug h in function

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29
Q

if a>0, there is a

A

minimum value of k at x=h

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30
Q

if a<0, there is a

A

maximum value of k at x=h

31
Q

graphing polynomial function steps

A

1) intercepts
2) sign check
3) end behavior
4) graph and label

32
Q

when there is a multiplicity of 1, it

A

crosses the x axis

33
Q

a^2+b^2=

A

(a+bi) (a-bi)

34
Q

i^2 is

A

-1

35
Q

use __ to determine real zeros of polynomials

A

long division

36
Q

if c is a zero of p(x),

A

then x-c is a factor of p(x)

37
Q

if a+bi is a zero of p(x),

A

a-bi is also a zero of p(x)

38
Q

coefficient

A

the number ex: 4

39
Q

term

A

the whole thing ex: 4x^2

40
Q

domain constraints

A
  1. 1/x ;x≠0
  2. √x ;x≥0 *n is even
  3. 1/√x ;x>0 *n is even *root is alone
  4. logaX; x>0
41
Q

graphing rational functions steps

A
  1. factor
  2. intercepts
    - x int on numerator
    - plug in 0 for y int
  3. VA (x=)
    - solve denominator
    * sign chart
  4. HA (y=)
  5. graph and label
42
Q

logarithmic properties

A 
loga1= 0
logaA= 1
logaA^x=x
a^logaX= x
logaA^c= ClogaA
loga(AB) = logA+logB
loga(A/B)= logA-logB
43
Q

basic unit circle values

A

x^2+y^2=1
π/6 (√3/2, 1/2)
π/4 (√2/2, √2/2)
π/3 (1/2, √3/2)

44
Q

exponential and log equations steps

A

1) put exponential on one side

2) solve

45
Q

trig functions

A

sin^2(x) + cos^2(x)= 1
tan^2(x)+1=sec^2(x)
1+cot^2(x)=csc^2(x)

46
Q

degrees to radians

A

multiply π/180

47
Q

radians to degrees

A

multiply 180/π

48
Q

coterminal angles

A

add or subtract 2π (360 degrees) to given angle

49
Q

trig functions of angles

A

SOH CAH TOA

a^2+b^2=c^2

50
Q

basic trig function domain and range

A

sin(θ)

domain: (-∞,∞)
range: [-1,1]

cos(θ)

domain: (-∞,∞)
range: [-1,1]

tan(θ)

domain: (-∞,∞) {π/2+nπ}
range: (-∞,∞)

51
Q

inverse trig functions domain and range

A

sin-1(θ)

domain: [-1,1]
range: [-π/2, π/2]

cos-1(θ)

domain: [-1,1]
range: [0, π]

tan-1(θ)

domain: (-∞,∞)
range: (-π/2, π/2)

52
Q

even-odd identities

A 
sin(-x)= -sinx 
cos(-x)= cosx
tan(-x)= -tanx
53
Q

guidelines for proving trig identities

A

1) start with one side
* indicate which side u choose to start with
2) use known identities
3) covert to sines and cosines

54
Q

formula for sine

A

sin(x+y)= sinXcosY + cosXsinY

sin (x-y)= sinXcosY - cosXsinY

55
Q

formula for cosine

A 
cos(x+y)= cosXcosY - sinXsinY
cos(x-y)= cosXcosY + sinXsinY
56
Q

double-angle formulas

A

sine: sin2x= 2sinxcosx
cosine: cos2x= cos^2x-sin^2x
= 1-2sin^2x
= 2cos^2x-1
tangent: tan2x=sin2x/cos2x

57
Q

sine graph

A 
sink(x-b)+c
period: 2π/k
amp: IaI 
phase shift: b
d=period/4
*always starts with "b"
*basic graph starts from 0
58
Q

cosine graph

A 
acosk(x-b)+c
period: 2π/k
amp: IaI 
phase shift: b
d=period/4
*always starts with "b"
*basic graph starts at highest point
59
Q

tangent graph

A 
atank(x-b)+c
period: π/k
d=period/4
*b in the middle
*2 asymptotes at the very ends
60
Q

basic trig equation steps

A

1) find primary solution in one complete period
sin [0,2π) cos [0,2π) tan (-π/2, π/2)
2) find general solution by adding the solution in step 1 by the multiple of the period
*sin and cos: add 2kπ
*tan: add kπ

61
Q

5-step strategy

A

1) write down in one function of one angle
2) find values of written function
3) solve for angle
4) solve for variable
5) check restrictions

62
Q

basic trig equations CHECK

A

1) factor
2) identities
3) formulas
- addition/subtraction
- double angle
* u substitution

63
Q

if inverse function is on the outside, look at the

A

domain

64
Q

if inverse function is on the inside, look at the

A

range

65
Q

solving exponential/log equations: exponential

A

1) isolate exp
2) take loga
3) solve for variable
* no check
* sometimes we can factor

66
Q

solving exponential/log equations: log

A 
way 1) 
1. isolate loga
2. write in exp form 
3. solve for variable 
4. check 
way 2)
logaX=logaY; X=Y
67
Q

whenever u see NONLINEAR inequalities you must solve by ___

A

sign chart

68
Q

half angle formulas

A

sin θ/2 = ± √(1-cosθ)÷2
cos θ/2= ± √(1+cosθ)÷2
tan θ/2= sin θ÷(1+cosθ)
= (1-cosθ)÷sinθ

69
Q

basic graph of e

A

above x-axis
increasing from left to right
crosses (0,1)

70
Q

basic graph of log

A 
increasing from - y values to + y values 
passes thru (1,0)
VA x=0
71
Q

a^2 x b^2 =

A

(ab)^2

72
Q

bad point

A

a point on the denominator of a non-linear inequality

73
Q

zero point

A

real zeros *set equal to zero

74
Q

how do u write factors that are 1 ± 2i

A

(x-(1-2i)) (x-(1+2i))

MAT 1800 (4 decks)

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