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Unformatted text preview: 7.8 THE HYPERBOLIC SINE AND COSINE Certain combinations of the exponential functions e x and e − x occur so frequently in mathematical applications that they are given special names. The hyperbolic sine (sinh) and hyperbolic cosine (cosh) are the functions defined by (7.8.1) sinh x = 1 2 ( e x − e − x ), cosh x = 1 2 ( e x + e − x ). The reasons for these names will become apparent as we go on. Since d dx ( sinh x ) = d dx 1 2 ( e x − e − x ) = 1 2 ( e x + e − x ) and d dx ( cosh x ) = d dx [ 1 2 ( e x + e − x )] = 1 2 ( e x − e − x ), we have (7.8.2) d dx ( sinh x ) = cosh x , d dx ( cosh x ) = sinh x . In short, each of these functions is the derivative of the other. The Graphs We begin with the hyperbolic sine. Since sinh ( − x ) = 1 2 ( e − x − e x ) = − 1 2 ( e x − e − x ) = − sinh x , the hyperbolic sine is an odd function. The graph is therefore symmetric about the origin. Since d dx ( sinh x ) = cosh x = 1 2 ( e x + e − x ) > for all real x , the hyperbolic sine increases everywhere. Since d 2 dx 2 ( sinh x ) = d dx ( cosh x ) = sinh x = 1 2 ( e x − e − x ), you can see that 436 CHAPTER 7 THE TRANSCENDENTAL FUNCTIONS d 2 dx 2 ( sinh x ) is negative, for x < 0, at x = positive, for x > 0. The graph is therefore concave down on ( −∞ , 0) and concave up on (0, ∞ ). The point (0, sinh 0) = (0, 0) is the only point of inflection. The slope at the origin is cosh 0 = 1. A sketch of the graph appears in Figure 7.8.1. We turn now to the hyperbolic cosine. Since cosh ( − x ) = 1 2 ( e − x + e x ) = 1 2 ( e x + e − x ) = cosh x , y O x y = sinh x Figure 7.8.1 x y y = cosh x (0,1) Figure 7.8.2 y = sinh x (0, 1) y = cosh x 2 1 y = e x x y Figure 7.8.3 x y Figure 7.8.4 the hyperbolic cosine is an even function. The graph is therefore symmetric about the y-axis. Since d dx ( cosh x ) = sinh x , you can see that d dx ( cosh x ) is negative, for x < 0, at x = positive, for x > 0. The function therefore decreases on ( −∞ , 0] and increases on [0, ∞ ). The number cosh 0 = 1 2 ( e + e − ) = 1 2 (1 + 1) = 1 is a local and absolute minimum. There are no other extreme values. Since d 2 dx 2 ( cosh x ) = d dx ( sinh x ) = cosh x > for all real x , the graph is everywhere concave up. (See Figure 7.8.2.)the graph is everywhere concave up....
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This note was uploaded on 10/12/2010 for the course MATH 12345 taught by Professor Smith during the Spring '10 term at University of Houston - Downtown.
- Spring '10