SalasSV_07_08 - 7.8 THE HYPERBOLIC SINE AND COSINE Certain...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

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.

Page1 / 5

SalasSV_07_08 - 7.8 THE HYPERBOLIC SINE AND COSINE Certain...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online