Calculus and Hyperbolic Functions Flashcards

1
Q

State the definition of cosh x

A

1/2(e^x + e^-x)

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2
Q

State the definition of sinh x

A

1/2(e^x - e^-x)

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3
Q

State the definition of tanh x

A

sinh x / cosh x

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4
Q

State the definition of sech x

A

1 / cosh x

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5
Q

State the domain and range of cosh x

A

Domain: x ∈ R
Range: y > 0

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6
Q

State the definition of cosech x

A

1 / sinh x

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7
Q

State the definition of coth x

A

cosh x / sinh x

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8
Q

State the domain and range of sinh x

A

Domain: x ∈ R
Range: y ∈ R

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9
Q

State the domain and range of tanh x

A

Domain: x ∈ R
Range: y > 1, y < -1

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10
Q

State the domain and range of sech x

A

Domain: x ∈ R
Range: 0 < y <= 1

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11
Q

State the domain and range of cosech x

A

Domain: x ∈ R, x ≠ 0
Range: y ∈ R, y ≠ 0

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12
Q

State the domain and range of coth x

A

Domain: x ∈ R, x ≠ 0
Range: y > 1, y < -1

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13
Q

State the identity which is equivalent to 1

A

cosh²x - sinh²x ≡ 1

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14
Q

Finish the identity sech²x ≡

A

sech²x ≡ 1 - tanh²x

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15
Q

Finish the identity cosech²x ≡

A

cosech²x ≡ coth²x - 1

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16
Q

Finish the identity cosh 2x ≡

A

cosh 2x ≡ cosh²x + sinh²x

17
Q

Finish the identity sinh 2x ≡

A

sinh 2x ≡ 2sinhxcoshx

18
Q

d (cosh x)/ dx =

19
Q

d (sinh x)/ dx =

20
Q

∫ cosh x dx =

A

sinh x + c

21
Q

∫ sinh x dx =

A

cosh x + c

22
Q

State the logarithmic form of arcosh x

A

ln(x + √(x² - 1))

23
Q

State the logarithmic form of arsinh x

A

ln(x + √(x² + 1))

24
Q

State the logarithmic form of arsech x

A

ln((1/x) + √((1/x²) - 1))

25
State the logarithmic form of arcosech x
ln((1/x) + √((1/x²) + 1))
26
State the logarithmic form of artanh x
1/2 ln((1 + x)/ (1 - x))
27
State the logarithmic form of arcoth x
1/2 ln((x + 1)/ (x - 1))
28
∫ 1/√(x² - a² ) dx =
arcosh (x/a) + c OR ln ( x + √(x² - a² )) + c
29
∫ 1/√(x² + a² ) dx =
arsinh (x/a) + c OR ln ( x + √(x² + a² )) + c
30
State the domain and range of arcosh x
Domain: x >= 1 Range: y >= 0
31
State the domain and range of arsinh x
Domain: x ∈ R Range: y ∈ R
32
State the domain and range of arcosech x
Domain: x ∈ R Range: y ∈ R
33
State the domain and range of artanh x
Domain: -1 < x < 1 Range: y ∈ R
34
State the domain and range of arcoth x
Domain: x < -1, x > 1 Range: y ∈ R
35
State the domain and range of arsech x
Domain: 0 < x <= 1 Range: y >= 0