Calculus and Hyperbolic Functions

Question 1

State the definition of cosh x

Answer 1

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

Question 2

State the definition of sinh x

Answer 2

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

Question 3

State the definition of tanh x

Answer 3

sinh x / cosh x

Question 4

State the definition of sech x

Answer 4

1 / cosh x

Question 5

State the domain and range of cosh x

Answer 5

Domain: x ∈ R
Range: y > 0

Question 6

State the definition of cosech x

Answer 6

1 / sinh x

Question 7

State the definition of coth x

Answer 7

cosh x / sinh x

Question 8

State the domain and range of sinh x

Answer 8

Domain: x ∈ R
Range: y ∈ R

Question 9

State the domain and range of tanh x

Answer 9

Domain: x ∈ R
Range: y > 1, y < -1

Question 10

State the domain and range of sech x

Answer 10

Domain: x ∈ R
Range: 0 < y <= 1

Question 11

State the domain and range of cosech x

Answer 11

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

Question 12

State the domain and range of coth x

Answer 12

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

Question 13

State the identity which is equivalent to 1

Answer 13

cosh²x - sinh²x ≡ 1

Question 14

Finish the identity sech²x ≡

Answer 14

sech²x ≡ 1 - tanh²x

Question 15

Finish the identity cosech²x ≡

Answer 15

cosech²x ≡ coth²x - 1

Question 16

Finish the identity cosh 2x ≡

Answer 16

cosh 2x ≡ cosh²x + sinh²x

Question 17

Finish the identity sinh 2x ≡

Answer 17

sinh 2x ≡ 2sinhxcoshx

Question 18

d (cosh x)/ dx =

Answer 18

sinh x

Question 19

d (sinh x)/ dx =

Answer 19

cosh x

Question 20

∫ cosh x dx =

Answer 20

sinh x + c

Question 21

∫ sinh x dx =

Answer 21

cosh x + c

Question 22

State the logarithmic form of arcosh x

Answer 22

ln(x + √(x² - 1))

Question 23

State the logarithmic form of arsinh x

Answer 23

ln(x + √(x² + 1))

Question 24

State the logarithmic form of arsech x

Answer 24

ln((1/x) + √((1/x²) - 1))

Question 25

State the logarithmic form of arcosech x

Answer 25

ln((1/x) + √((1/x²) + 1))

Question 26

State the logarithmic form of artanh x

Answer 26

1/2 ln((1 + x)/ (1 - x))

Question 27

State the logarithmic form of arcoth x

Answer 27

1/2 ln((x + 1)/ (x - 1))

Question 28

∫ 1/√(x² - a² ) dx =

Answer 28

arcosh (x/a) + c

OR

ln ( x + √(x² - a² )) + c

Question 29

∫ 1/√(x² + a² ) dx =

Answer 29

arsinh (x/a) + c

OR

ln ( x + √(x² + a² )) + c

Question 30

State the domain and range of arcosh x

Answer 30

Domain: x >= 1
Range: y >= 0

Question 31

State the domain and range of arsinh x

Answer 31

Domain: x ∈ R
Range: y ∈ R

Question 32

State the domain and range of arcosech x

Answer 32

Domain: x ∈ R
Range: y ∈ R

Question 33

State the domain and range of artanh x

Answer 33

Domain: -1 < x < 1
Range: y ∈ R

Question 34

State the domain and range of arcoth x

Answer 34

Domain: x < -1, x > 1
Range: y ∈ R

Question 35

State the domain and range of arsech x

Answer 35

Domain: 0 < x <= 1
Range: y >= 0