5.10 Hyperbolic Functions 5.8 Even Answers: - 68) False: The range of y arcsin x is 2 , 2 2 70) 2 2 False: arcsin 0 arccos 0 0 1 2 5.10 Hyperbolic Functions Definition of the Hyperbolic Functions e e sinh x 2 x x 1 csch x ,x 0 sinh x x 1 sech x cosh x e e cosh x 2 x sinh x tanh x cosh x 1 coth x ,x 0 tanh x Problem Problem Problem 1 Problem 2 5.10 Hyperbolic Functions Many of the trigonometric identities have corresponding hyperbolic identities. sin 2x 2sin x cos x 2sinh x cosh x Substitute and see if we get sinh2x e x e x e x e x 2 2 2 e e 2 2x 2 x sinh 2x Definitions 5.10 Hyperbolic Functions Many of the trigonometric identities have corresponding hyperbolic identities. cos x sin x 1 2 2 e e e e 2 2 cosh x sinh x 2 2 x e 2e 4 2x 4 4 1 x 2 x 2 x e 2e 4 2x x 2 x 2 5.10 Hyperbolic Functions 5.10 Hyperbolic Functions Problem 1 Problem 2 5.10 Hyperbolic Functions Differentiation of Hyperbolic Functions d sinh x 2 3 dx 2 x cosh x 3 2 d u ' sinh x tanh x ln cosh x dx u cosh x Derivative of sinhx and coshx 5.10 Hyperbolic Functions Differentiation of Hyperbolic Functions d x sinh x cosh x dx sinh x x cosh x sinh x x cosh x Derivative of sinhx and coshx 5.10 Hyperbolic Functions Find the relative extrema of f ( x) ( x 1) cosh x sinh x. f '( x) cosh x ( x 1)sinh x cosh x ( x 1) sinh x 0 x 1, 0 2nd Derivative Test: f ''( x) sinh x ( x 1) cosh x f ''(1) 0 1, sinh1 is a relative min. f ''(0) 0 (0, cosh 0 sinh 0) is a relative max. 0, 1 Graph of coshx Graph of sinhx 5.10 Hyperbolic Functions Power cables are suspended between two towers, forming the catenary shown in the picture. The equation for this catenary is x y a cosh . a The distance between the two towers is 2b. Find the slope of the catenary at the point where the cable meets the righthand tower. 5.10 Hyperbolic Functions x y a cosh . a 5.10 Hyperbolic Functions x y a cosh . a x 1 x y ' a sinh sinh a a a b @ (b, a cosh ), the slope (from the left) is a b m sinh . a Graph 5.10 Hyperbolic Functions Solve cosh 2 x sinh 2 x dx 2 u ? u sinh 2x du 2cosh 2xdx 1 2 1 u u du 2 2 3 3 3 C 3 sinh 2 x u C C 6 6 5.10 Hyperbolic Functions Read pages 396-397 Problem 5.10 Hyperbolic Functions A person is holding a rope that is tied to a boat. As the person walks along the dock, the boat travels along a tractrix, given by the equation x 2 2 y a sec h a x a 1 where a is the length of the rope. If a 20 feet, find the distance the person must walk to bring the boat 5 feet from the dock. Problem 1 Problem 2 x 2 2 y a sec h a x a 1 5.10 Hyperbolic Functions Picture The distance the person has walked is y1 y 20 x 2 2 1 x 2 2 2 2 20sec h 20 x 20 x 20 x 20sec h 20 1 When x 5, this distance is 2 5 1 1 1/ 4 20ln 4 4 1 1/16 y1 20sec h 1 20 ln 20 1/ 4 Sech^-1=? 41.27 feet 5.10 Hyperbolic Functions Problem 5.10 Hyperbolic Functions Problem 5.10 Hyperbolic Functions Show that the boat is always pointing towards the person. For a point x, y on a tractrix, the slope of the graph gives the direction of the boat. d 1 x 2 2 y' 20sec h 20 x dx 20 1 1 1 2 x 20 2 2 2 20 x / 20 1 x / 20 2 20 x Picture d/dx[sech^-1]=? 1 1 1 2 x 20 2 2 2 20 x / 20 1 x / 20 2 20 x 20 x 1 x / 20 2 20 2 400 x x 400 400 x 20 x 2 2 x 20 x 2 2 20 x 202 x 2 20 x 20 x 2 2 20 x 202 x 2 20 20 x 20 x 20 202 x 2 2 2 2 x 2 x 2 20 x x 2 2 x 20 x 2 2 x 20 x 2 x 2 2 m 202 x 2 / x the boat is always pointing towards the person. 5.10 Hyperbolic Functions Evaluate x dx 4 9x Integrals 2 1 a a u ln C u a2 u 2 a u du a2 u 3x 2 2 du 3dx u 2 u 2 du 1 2 4 9x ln C 2 3x 2 2 5.10 Hyperbolic Functions HW 5.10/1-77 EOO Quiz : Evaluate cosh 2 x 1 sinh x 1 dx