Hyperbolic & Inverse Hyperbolic FunctionsCatenary CurveHyperbolic FunctionsDifferentiationIntegrationExampleApplicationIntegrals Involving Inverse Hyperbolic FunctionsTry It!Slide 10AssignmentHyperbolic & Inverse Hyperbolic FunctionsLesson 7.8Catenary Curve•The curve formed by a hangingcable is called a catenary•They behave similar to trig functions•They are related to the hyperbola in similar manner as trig functions to the circle•Thus are called hyperbolic functions( )/ /( )2x a x aaf x e e-= +Hyperbolic Functions•Definitions•Note: domainis all real numbers•Note properties, Theorem 7.2, pg 482sinh2cosh2sinhtanhcoshx xx xx xx xe exe exx e exx e e-----=+=-= =+Differentiation•Rules for differentiating hyperbolic functions•Note others on pg 483sinh ' coshcosh ' sinhy x y xy x y x= == =Integration•Formulas for integration22sinh cosh csch cothcosh sinh sech tanh sech sech tanh csc h coth csch u du u C u du u Cu du u C u u du u Cu du u C u u du u C= + =- += + =- += + =- +� �� �� �Example•Try •What should be the u, the du ?•Substitute, integratesinh1 coshxdxx+�1 cosh , sinhu x du x dx= + =ln ln 1 coshduu C x Cu= + = + +�Application•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 60cosh60xy =120'[ ]21 '( )biaL f x dx= +�Integrals Involving Inverse Hyperbolic Functions12 212 2-12 212 21sinh1cosh1 1tanh1 1sechudu Cau audu Cau audu Ca u a audu Ca au a u---= ++= +-= +-= +-����Try It!•Note the definite integral•What is the a, the u, the du?a = 3, u = 2x, du = 2 dx42119 4dxx-+�4211 229 4dxx-+�4111 2sinh2 3x--=Application•Find the area enclosed by x = -¼, x = ¼, y = 0, and•Which pattern does this match?•What is the a, the u, the du?211 4yx=-Assignment•Lesson 7.8•Page 486•Exercises 1 – 45
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