Hyperbolic Functions

Question 1

What is the exponential form of cosh x?

Answer 1

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

Question 2

What is the exponential form of sinh x?

Answer 2

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

Question 3

What is the exponential form of tanh x?

Answer 3

Tanh x = e^x - e^-x / e^x + e^-x

Question 4

What are the properties of the cosh x graph?

Answer 4

Minimum at (0,1)
Symmetrical about y axis
Domain: Real
Range: y > 1
Points are the mean of y = e^x and y = e^-x

Question 5

What are the properties of the sinh x graph?

Answer 5

Points are mean of y = e^x and y = -e^-x
Passes through origin with non-stationary points of inflection
Odd function
Domain and Range are Real

Question 6

What are the properties of the tanh x graph?

Answer 6

Odd function
Domain: Real
Range: -1 < y < 1

Question 7

What is the rule for converting trig identities to hyperbolic identities?

Answer 7

Add the ‘h’ and flip the sign of any term involving a product of 2 sinh functions

Question 8

What is the derivative and integral of sinh and cosh?

Answer 8

Each other

Question 9

What is the derivative of tanh x?

Answer 9

dy/dx = sech²x
I = ln(cosh x)

Question 10

What is the difference between Rcos(x + a) questions and Rcosh(x + a)?

Answer 10

The identities are different and you calculate R by finding the square root of the difference between each coefficient squared

Question 11

What is the domain and range of the artanh x?

Answer 11

Domain: -1 < x < 1
Range: y is Real

Question 12

What is the domain and range of the arcosh x?

Answer 12

Domain: x > 1
Range: y > 0

Question 13

What is the domain and range of the arcsinh x?

Answer 13

Domain: x is Real
Range: y is Real

Question 14

What is the formula for artanh x?

Answer 14

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

Question 15

What is the formula for arcosh x?

Answer 15

ln(x + sqrt( x² - 1 ))

Question 16

What is the formula for arcsinh x?

Answer 16

ln(x + sqrt( x² + 1 ))

Question 17

When is the ln formula of an inverse hyperbolic function +-?

Answer 17

When the cosh function is used and you are asked to solve for cosh x = a

Question 18

When is the sqrt in the ln formula of an inverse hyperbolic function +-?

Answer 18

When the value outside the sqrt is greater than the value of the sqrt

Question 19

What is the derivative of arsinh(x)?

Answer 19

1 / sqrt( 1 + x² )

Question 20

What is the derivative of arcosh(x)?

Answer 20

1 / sqrt( x² -1 )

Question 21

What is the derivative of artanh(x)?

Answer 21

1 / 1 - x²

Question 22

What is the integral of 1 / sqrt( x² - a² )?

Answer 22

Arcosh( x / a ) + C

Ln ( x + sqrt( x² - a² ) ) + K

Question 23

What is the integral of 1 / sqrt( x² + a² )?

Answer 23

Arsinh ( x / a ) + C

Ln ( x + sqrt( x² + a² ) ) + K

Question 24

What is the integral of 1 / a² - x² ?

Answer 24

1 / a * artanh( x / a ) + C

1 / 2a * ln ( a + x / a - x ) + K

Question 25

What is the formula for a revolution integral about the 'x' axis?

Answer 25

I^b_a (Pi)(y²)

Question 26

What is the formula for a revolution integral about the 'y' axis?

Answer 26

I^b_a (Pi)(x²)

Question 27

What is the formula for the mean value of a function?

Answer 27

(1 / b - a ) I^b_a f(x)