SalasSV_07_08 - 7.8 THE HYPERBOLIC SINE AND COSINE 435...

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

View Full Document Right Arrow Icon
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 de fi ned 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 ) > 0 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
Image of page 1

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

View Full Document Right Arrow Icon
436 CHAPTER 7 THE TRANSCENDENTAL FUNCTIONS d 2 dx 2 ( sinh x ) is negative, for x < 0 0, at x = 0 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 in fl ection. 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 0, at x = 0 positive, for x > 0. The function therefore decreases on ( − ∞ , 0] and increases on [0, ). The number cosh 0 = 1 2 ( e 0 + e 0 ) = 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 > 0 for all real x , the graph is everywhere concave up. (See Figure 7.8.2.) Figure 7.8.3 shows the graphs of three functions y = sinh x = 1 2 ( e x e x ), y = 1 2 e x , y = cosh x = 1 2 ( e x + e x ).
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern