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Cal 1 > To Remember > Flashcards

To Remember Flashcards

1
Q

Sec(x)

A

1/cos(x)

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2
Q

Tan(x)

A

Sin(x)/Cos(x)

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3
Q

Sin^2 (x) + Cos^2 (x)

A

1

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4
Q

Cot (°)

A

a/o

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5
Q

Sec (°)

A

H/a

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6
Q

Csc (°)

A

H/o

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7
Q

Csc(x)

A

1/Sin(x)

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8
Q

Trig Circle

A 
0° = (1 , 0)
30° = (sqrt3/2 , 1/2)
45° = (sqrt2/2 , sqrt2/2)
60° = (1/2 , sqrt3/2)
90° = (0 , 1)
120° = (-1/2 , sqrt3/2)
135° = (-sqrt2/2 , sqrt2/2)
150° = (-sqrt3/2 , 1/2)
180° = (-1 , 0)
210° = (-sqrt3/2 , -1/2)
225° = (-sqrt2/2 , -sqrt2/2)
240° = (-1/2 , -sqrt3/2)
270° = (0, -1)
300° = (1/2 , -sqrt3/2)
315° = (sqrt2/2 , -sqrt2/2)
330° = (sqrt3/2 , -1/2)
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9
Q

Sin (x+2pi)

A

Sin (x)

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10
Q

Cos (x + 2pi)

A

Cos (x)

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11
Q

Log function =

A

Log b (X) = Y

Y^b = X

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12
Q

b^(x+y)

A

b^x b^y

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13
Q

b^-n

A

1/b^n

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14
Q

b^x/y

A

ysqrt of b^x or (ysqrt(b))^x

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15
Q

b^x-y

A

b^x / b^y

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16
Q

(b^x)^y

A

b^xy

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17
Q

(ab)^x

A

a^x b^x

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18
Q

log b (b^x)

A

X

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19
Q

b ^log x

A

X

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20
Q

log b (xy)

A

log b (x) + log b (y)

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21
Q

log b (x/y)

A

log b (x) - log b (y)

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22
Q

log b (X^r)

A

r log b (X)

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23
Q

log e (X)

A

ln (x)

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24
Q

ln (e^x)

A

X

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25
Q

log b (x) when b >=0

A

ln(x) / ln(b)

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26
Q

Average velocity

A

f(x2)-f(x1) / x2-x1

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27
Q

Lim x—> a f(x) = + or - inf

A

Vertical asymptote

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28
Q

Lim x—>a ( fx + gx )

A

Lim x—>a f(x) + Lim x—>a g(x)

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29
Q

Lim x—>a (C fx)

A

C Lim x—>a f(x)

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30
Q

Lim x—>a (fx*gx)

A

Lim x—>a f(x) * Lim x—>a g(x)

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31
Q

Lim x—>a (fx/gx)

A

Lim x—>a f(x) / Lim x—>a g(x)

Where g(x) is not 0

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32
Q

Lim x—>a (fx)^n

A

(Lim x—>a fx)^n

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33
Q

Lim x—>a (sqrt fx)

A

Sqrt (Lim x—>a fx)

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34
Q

Direct substitution property

A

Plug in X valued into s polynomial

Lim x—>a f(x) = f(a)

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35
Q

Rationalizing

A

Sqrt(A)-Sqrt(B) * (sqrtA+sqrtB)/(sqrtA+sqrtB)

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36
Q

Squeeze theorem

A

f(x) =< g(x) =< h(x)

Lim x—>a f(x) = Lim x—>a h(x) = L

& Lim x—>a g(x) = L

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37
Q

Lim x—>+ or - inf

A

Horizontal asymptote

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38
Q

Lim x—>+ or - inf (1/x^r)

***positive r

A

0

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39
Q

Example:

Lim x—>2+ arctan(1 / x+2)

A

Arctan (1/2-2) =

Arctsn (inf)

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40
Q

Continuous if:

A
  1. Limit is defined (a = domain f)
  2. Lim x—>a f(x) exists
  3. Lim x—>a f(x) = f(a)
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41
Q

F id discontinuous when there is:

A
  • hole
  • kink
  • corner
  • break
  • jump
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42
Q

Lim x—>a+ f(x) = f(a)

A

Continuous from the right

43
Q

Lim x—>a- f(x) = f(a)

A

Continuous from the left

44
Q

Polynomials are continuous on

A

(-inf , inf)

45
Q

Tan (x) is continuous id continuous

A

Whenever cos(x) does not equal to 0

Pi/2 , etc

46
Q

Lim x—>a f(gx)

A

F( lim x—>a gx )

47
Q

Slope of the tangent line (limit law)

A

Lim x—>a [f(x)-f(a)] / (x-a)

= Lim h—>0 [f(a+h) - f(a)] / h

48
Q

Instantaneous rate of change

A

Lim delta x—>0 (delta y) / (delta x)

= Lim x2–>x1 (fx2-fx1) / x2-x1

49
Q

Velocity f’(a)

A

Lim h—>0

f(a+h)-f(a)
—————–
h

50
Q

Differentiable

A

= continuous

51
Q

Discontinuous when there is (therefore not differentiable)

A

A corner or vertical tangent

52
Q

d/dx (C)

A

0

53
Q

d/dx (x)

A

1

54
Q

d/dx (X^n)

A

n * x^n-1

55
Q

d/dx [c*f(x)]

A

C d/dx f(x)

56
Q

d/dx [f(x)+g(x)]

A

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

57
Q

d/dx e^x

A

e^x

58
Q

d/dx b^x

A

(b^x) ln (b)

59
Q

d/dx ln(x)

A

1/x

60
Q

d/dx log b (x)

A

1 / (X*lnb)

61
Q

Product rule derivative

A

f’g+fg’

62
Q

Quotient rule derivative

A

f’g-fg’
———
g^2

63
Q

d/dx sin(x)

A

Cos(x)

64
Q

d/dx cos(x)

A

-sin(x)

65
Q

d/dx tan(x)

A

Sec^2 (x)

66
Q

Cot (x)

A

Cos(x) / sin(x)

67
Q

d/dx csc(x)

A

-csc(x)cot(c)

68
Q

d/dx sec(x)

A

Sec(x)tan(x)

69
Q

d/dx cot(x)

A

-csc^2(x)

70
Q

Lim x—>0 (sin X / X)

A

1

71
Q

Lim x—>0 (cos X / X)

A

0

72
Q

Chain rule

A

f’(g) * g’

73
Q

Power rule and chain rule

A

d/dx (g)^n =

n * (g)^n-1 * g’

74
Q

d/dx arcsin (x)

A

1
——————
Sqrt(1+x^2)

75
Q

d/dx arccos (x)

A

1
- ——————
Sqrt(1+x^2)

76
Q

d/dx arctan (x)

A

1
—————
1+x^2

77
Q

d/dx arccot (x)

A

1
- ————
1+x^2

78
Q

d/dx arcsec (x)

A

1
————————
X*sqrt(x^2 - 1)

79
Q

d/dx arccsc (x)

A

1
- ————————
X*sqrt(x^2 - 1)

80
Q

d/dx ln (gx)

A

g’ / g

81
Q

Log differentiation (4 steps)

A

1- ln both sides
2- log laws to simplify
3- derive both sides
4- solve for y

82
Q

Lim x—>0 (e^x - 1) / x

A

1

83
Q

e =

A

Lim x—>0 (1+x)^(1/x)

Lim n—>inf (1+1/n)^n

84
Q

Rates of change model

A

P(t) = Po * e^kt

Where t is time

85
Q

Area of triangle

A

h•b
——
2

86
Q

Area of circle

A

Pi•r^2

87
Q

Critical numbers =

A

When derivative is =0 or DNE

88
Q

Mean Value Theorem

  1. Is continuous on [a,b]
  2. Is differentiable on (a,b)
A

f(b)-f(a)
f’(c) = —————
b-a

Or f(b)-f(a)=f’(c)*(b-a)

89
Q

f’(x)>0

A

Increasing on interval

90
Q

f’(x)<0

A

Decreasing on interval

91
Q

F’ = Increasing to decreasing

A

Local max

92
Q

F’ = Decreasing to increasing

A

Local min

93
Q

f’’(x)>0

A

Concave up

94
Q

f’’(x)<0

A

Concave down

95
Q

f’(c)=0 and f’’(c)>0

A

Local min

96
Q

f’(c)=0 and f’’(x)<0

A

Local max

97
Q

Inflection points

A

Change in concavity

98
Q

1+tan^2 (x)

A

Sec^2 (x)

99
Q

1+cot^2 (x)

A

Csc^2 (x)

100
Q

L’Hôpital’s Rule

A

If Lim x—>a f(x)/g(x) is undetermined ( = inf ):

Lim x—>a f’(x) / g’(x)

101
Q

Transforming indeterminate products

A

fg = f / (1/g)

Or

fg = g / (1/f)

102
Q

Transforming indeterminate powers

2 steps

A

1- ln both sides and simplify

2- put as a fraction

103
Q

Linear approximation:

A

f(x) ≈ f(a) + f’(a)•(x-a)

L(x) = f(a) + f’(a)•(x-a)

Cal 1 (1 decks)

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