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B) Calculus > Functions > Flashcards

Functions Flashcards

0
Q

Domain

A

Set of x-values that yeild an output

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1
Q

Function

A

An continuous curve for which every input has an output

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2
Q

Range

A

Set of possible outputs of a function

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3
Q

Independent variable

A

X-value, associated with the domain

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4
Q

Dependent variable

A

Y-value, associated with range

‘Depends’ on x-value

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5
Q

Graph

A

Set of all points (x,y) represented on a plane

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6
Q

Argument

A

The function expression

Represented by ‘f(x)’

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7
Q

Vertical line test

A

When examining a graph, if there is more than one output for any one input, the curve cannot represent a function

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8
Q

Interval notation

A

Exclusive ()

Inclusive [ ]

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9
Q

Composite function

A 
Function whose input depends on the output of a second function g(x)
Writen as (f o g)(x)
Or f(g(x))
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10
Q

Symmetric to x-axis

A

The graph is the same when flipped upside down, folded on x

Cannot be a function, fails the vertical line test

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11
Q

Symmetric to the y-axis

A

The graph looks the same if viewed backwards, folded on y

Occupies adjacent quadrants

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12
Q

Symmetric to the origin

A

Looks the same when rotated 180 degrees on the paper

Occupies diagonal quadrants

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13
Q

Even functions

A

f(x)=f(-x)

Looks the same which viewed backwards

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14
Q

Odd functions

A

f(-x)=-f(x)

Looks the same which viewed from the origin

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15
Q

Polynomials

A

Algebraic functions represented by terms with descending powers

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16
Q

Rational functions

A

Algebraic function in which one polynomial divided by another

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17
Q

Algebraic Functions

A

Use only +,-, x, /, ^, or √

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18
Q

Exponential functions

A

Transcendental functions in which the variable is an exponent to a given base.
Infinite domain
Range>0
As x→0, f(x)=1

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19
Q

Logarithmic functions

A

Transcendental functions in the form Log-base exponent

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20
Q

Trigonometric function

A

Transcendental functions Involving trigonometric expressions

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21
Q

Transcendental Functions

A

Non-algebraic functions

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22
Q

Linear function

A

Algebraic function that take the form ‘y=mx+b’

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23
Q

Peicewise functions

A

A function in which the argument is different on a variety of intervals
Writen as f(x)={argument

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24
Q

Power function

A

Algebraic function in which the variable is raised to a given power

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25
Q

Root functions

A

Algebraic function in which the variable is down to a √ or ^(1/n)

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26
Q

Function transformation

A

y=cf(a(x-b))+d

a- horizontal stretch
b- horizontal shift
c- vertical stretch
d- vertical shift

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27
Q

Vertical stretch

A

Factor multiplied by the function output, (could be a fraction)
c(f(x))

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28
Q

Vertical shift

A

Factor added or subtracted from function output

f(x)±d

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29
Q

Natural exponential function

A

f(x)=e^x

e is the base in the exponential function

30
Q

Inverse function

A

The argument for f^(-1)(x)

Calculated by isolating the x-variable on the =

31
Q

One-to-one function

A

Each output has only one x-value

Use a ‘horizontal-line test’

32
Q

Horizontal-line test

A

Test to determine whether function is one-to-one

33
Q

Change of base formula

A

Log-f(x) = [log-i(x)]/[log-i(f)]

34
Q

Radians

A

Number of ‘radius lengths’ an arc completes
π for one full circle
Number of circles is described in trig-functions

35
Q

Angle measure, from radians

A

θ=s/r

s- radians
r- radius

36
Q

Hypotenuse

A

Longest side of the triangle
Radius when represented by a circle
H=√(x^2+y^2)

37
Q

Cosine θ

A

Cosθ= adj/hyp= x/r

38
Q

Sine θ

A

Sinθ= opp/hyp= y/r

39
Q

Tangent θ

A

Tanθ= opp/adj= y/x

40
Q

Cotangent θ

A

Cotθ= adj/opp= x/y

41
Q

Secant θ

A

Secθ= hyp/adj= r/x

42
Q

Cosecant θ

A

Cscθ= hyp/opp= r/y

43
Q

Reciprocal identities (tangent)

A

Tanθ= sinθ/cosθ

44
Q

Reciprocal identities (Cotangent)

A

Cotθ= cosθ/sinθ

45
Q

Reciprocal identities (Cosecant)

A

Cscθ= 1/sinθ

46
Q

Reciprocal identities (Secant)

A

Secθ= 1/cosθ

47
Q

Reciprocal identities (sine)

A

Sinθ= 1/cscθ

48
Q

Reciprocal identities (cosine)

A

Cosθ= 1/secθ

49
Q

Pythagorean Identities [Sin^2(θ)]

A

Sin^2(θ)=1-cos^2(θ)

50
Q

Pythagorean Identities [cos^2(θ)]

A

cos^2(θ)=1-Sin^2(θ)

51
Q

Pythagorean Identities [tan^2(θ)]

A

tan^2(θ)=Sec^2(θ)-1

52
Q

Pythagorean Identities [cot^2(θ)]

A

cot^2(θ)=csc^2(θ)-1

53
Q

Pythagorean Identities [csc^2(θ)]

A

csc^2(θ)=cot^2(θ)+1

54
Q

Pythagorean Identities [sec^2(θ)]

A

Sec^2(θ)=1+tan^2(θ)

55
Q

Double-half Angle formulas [sin^2(θ)]

A

sin^2(θ)=(1-cos(2*θ))/2

56
Q

Double-half Angle formulas [cos(2*θ)]

A

cos(2*θ)=cos^2(θ)-sin^2(θ)

57
Q

Horizontal Stretch

A

Factor multiplied by the x-variable, (could be a fraction)

f(ax)

58
Q

Arc length (radians)

A

S=θ*radius

59
Q

Radius, from radians

A

Radius=Radians/θ

60
Q

Period (sec/cyc)

A

Length of a single trigonometric cycle
Period=2π/B
Where B is y=sin(B*x)
Also Period=1/frequency

61
Q

Frequency (cyc/sec)

A

Number of cycles that occurs per x-unit
Frequency=B/2π
Where B is y=sin(B*x)
Also Frequency=1/period

62
Q

Double-half Angle formulas [cos^2(θ)]

A

cos^2(θ)=(1+cos(2*θ))/2

63
Q

Inverse Trig functions

A

y=trig^-1(x)
x=trig(y)
Reflexive over the y=x line, to their original function

64
Q

Double-half Angle formulas [sin(2*θ)]

A

sin(2θ)=2sinθ*cosθ

65
Q

Horizontal shift

A

Factor added or subtracted from variable

f(x±b)

66
Q

Modeling Growth

A

Always as:
A(t)=Pe^(rt)
A- actual amount as a function of ‘t’
P- principal, the value with which you started
r- rate, new output units per unit of time
t- time

67
Q

Graphing complex trig functions

A

1) Create graph with respect to time
2) Start at x=hShift. If sine, y=vShift. If cosine, y=amplitude+vShift
3) Calculate period from the frequency. Mark the above y-value at every frequency multiple on x
4) If sine, mark that y-value between the frequency multiples too. If cosine, mark those frequency midpoints with yValue-(2*amplitude)
5) If sine, mark the first frequency quarter point with (amplitude+vShift), alternating between positive and negative for each half-frequency measure thereafter. If cosine, mark the frequency quarter points with the y-value in the middle of the y-values on either side.
6) Connect all of the points with a smooth curve

68
Q

Sinθ times the cosθ

A

Sinθ*cosθ=(1/2)sin(2θ)

69
Q

Reduction of cos^2(θ)-1

A

cos^2(θ)-1=1/2*cos(2a)

70
Q

Reduction of 1-sin^2(θ)

A

1-sin^2(θ)=1/2*cos(2a)

71
Q

Reduction of cos^2(θ)-sin^2(θ)

A

cos^2(θ)-sin^2(θ)=cos(2a)

72
Q

Reduction of 3sin(θ)-4sin^3(θ)

A

3sin(θ)-4sin^3(θ)=sin(3*θ)

73
Q

Reduction of 4cos^3(θ)-3cos(θ)

A

4cos^3(θ)-3cos(θ)=cos(3*θ)

B) Calculus (17 decks)

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