Calculus and Hyperbolic Functions Flashcards
State the definition of cosh x
1/2(e^x + e^-x)
State the definition of sinh x
1/2(e^x - e^-x)
State the definition of tanh x
sinh x / cosh x
State the definition of sech x
1 / cosh x
State the domain and range of cosh x
Domain: x ∈ R
Range: y > 0
State the definition of cosech x
1 / sinh x
State the definition of coth x
cosh x / sinh x
State the domain and range of sinh x
Domain: x ∈ R
Range: y ∈ R
State the domain and range of tanh x
Domain: x ∈ R
Range: y > 1, y < -1
State the domain and range of sech x
Domain: x ∈ R
Range: 0 < y <= 1
State the domain and range of cosech x
Domain: x ∈ R, x ≠ 0
Range: y ∈ R, y ≠ 0
State the domain and range of coth x
Domain: x ∈ R, x ≠ 0
Range: y > 1, y < -1
State the identity which is equivalent to 1
cosh²x - sinh²x ≡ 1
Finish the identity sech²x ≡
sech²x ≡ 1 - tanh²x
Finish the identity cosech²x ≡
cosech²x ≡ coth²x - 1
Finish the identity cosh 2x ≡
cosh 2x ≡ cosh²x + sinh²x
Finish the identity sinh 2x ≡
sinh 2x ≡ 2sinhxcoshx
d (cosh x)/ dx =
sinh x
d (sinh x)/ dx =
cosh x
∫ cosh x dx =
sinh x + c
∫ sinh x dx =
cosh x + c
State the logarithmic form of arcosh x
ln(x + √(x² - 1))
State the logarithmic form of arsinh x
ln(x + √(x² + 1))
State the logarithmic form of arsech x
ln((1/x) + √((1/x²) - 1))
State the logarithmic form of arcosech x
ln((1/x) + √((1/x²) + 1))
State the logarithmic form of artanh x
1/2 ln((1 + x)/ (1 - x))
State the logarithmic form of arcoth x
1/2 ln((x + 1)/ (x - 1))
∫ 1/√(x² - a² ) dx =
arcosh (x/a) + c
OR
ln ( x + √(x² - a² )) + c
∫ 1/√(x² + a² ) dx =
arsinh (x/a) + c
OR
ln ( x + √(x² + a² )) + c
State the domain and range of arcosh x
Domain: x >= 1
Range: y >= 0
State the domain and range of arsinh x
Domain: x ∈ R
Range: y ∈ R
State the domain and range of arcosech x
Domain: x ∈ R
Range: y ∈ R
State the domain and range of artanh x
Domain: -1 < x < 1
Range: y ∈ R
State the domain and range of arcoth x
Domain: x < -1, x > 1
Range: y ∈ R
State the domain and range of arsech x
Domain: 0 < x <= 1
Range: y >= 0
