Index FAQ Hyperbolic functions

Index FAQ Hyperbolic functions Hungarian and English notation

Index FAQ Groupwork in 4 groups For each function : - find domain - discuss parity - find limits at the endpoints of the domain -find zeros if any -find intervals such that the function is cont. -find local and global extremas if any -find range -find asymptotes

Index FAQ Summary: cosh What are the asymptotes of cosh(x) -in the infinity (2) -negative infiniy (2) PROVE YOUR STATEMENT!

Index FAQ Application of the use of hyperbolic cosine to describe the shape of a hanging wire/chain. Summary: cosh

Index FAQ Background So, cables like power line cables, which hang freely, hang in curves called hyperbolic cosine curves.

Index FAQ Chaincurve-catentity

Index FAQ Background Suspension cables like those of the Golden Gate Bridge, which support a constant load per horizontal foot, hang in parabolas.

Index FAQ Which shape do you suppose in this case?

Index FAQ Application: we will solve it SOON! Electric wires suspended between two towers form a catenary with the equation If the towers are 120 ft apart, what is the length of the suspended wire? Use the arc length formula 120'

Index FAQ What are the asymptotes of cosh(x) -in the infinity (2) -negative infiniy (2) PROVE YOUR STATEMENT! Summary: sinh

Index FAQ Analogy between trigonometric and hyperbolic functions If t is any real number, then the point P(cos t, sin t) lies on the unit circle x 2 + y 2 = 1 because cos 2 t + sin 2 t = 1. T is the OPQ angle measured in radian Trigonometric functions are also called CIRCULAR functions If t is any real number, then the point P(cosh t, sinh t) lies on the right branch of the hyperbola x 2 - y 2 = 1 because cosh 2 t - sin 2 t = 1 and cosh t ≥ 1. t does not represent the measure of an angle. HYPERBOLIC functions

Index FAQ It turns out that t represents twice the area of the shaded hyperbolic sector HYPERBOLIC FUNCTIONS In the trigonometric case t represents twice the area of the shaded circular sector

Index FAQ Identities Except for the one above. if we have “trig-like” functions, it follows that we will have “trig-like” identities. For example:

Index FAQ Proof of

Index FAQ Other identities HW: Prove all remainder ones in your cheatsheet!

Index FAQ Surprise, this is positive! Derivatives

Index FAQ Summary: Tanh(x) What are the asymptotes of tanh(x) -in the infinity (2) -In the negative infiniy (2) PROVE YOUR STATEMENT! Find the derivative!

Index FAQ The velocity of a water wave with length L moving across a body of water with depth d is modeled by the function where g is the acceleration due to gravity. Application of tanh: description of ocean waves

Index FAQ Hyperbolic cotangent What are the asymptotes of cotanh(x) -in the infinity (2) -In the negative infiniy (2) -At 0? PROVE YOUR STATEMENT! Find the derivative!

Index FAQ Summary: Hyperbolic Functions

Index FAQ The sinh is one-to-one function. So, it has inverse function denoted by sinh -1 INVERSE HYPERBOLIC FUNCTIONS

Index FAQ INVERSE HYPERBOLIC FUNCTIONS The tanh is one-to-one function. So, it has inverse function denoted by tanh -1

Index FAQ INVERSE FUNCTIONS This figure shows that cosh is not one-to- one.However, when restricted to the domain [0, ∞ ], it becomes one-to-one. The inverse hyperbolic cosine function is defined as the inverse of this restricted function

Index FAQ Inverse hyperbolic functions HW.: Define the inverse of the coth(x) function

Index FAQ INVERSE FUNCTIONS

Index FAQ ey >0ey >0 INVERSE FUNCTIONS e y – 2x – e -y = 0 multiplying by e z. e 2y – 2xe y – 1 = 0 (e y ) 2 – 2x(e y ) – 1 = 0

Index FAQ DERIVATIVES The formulas for the derivatives of tanh -1 x and coth -1 x appear to be identical. However, the domains of these functions have no numbers in common: tanh -1 x is defined for | x | < 1. coth -1 x is defined for | x | >1.

Index FAQ Sources: workbooks/mathcentre/hyperbolicfunctio ns.pdf workbooks/mathcentre/hyperbolicfunctio ns.pdf