This file is part of a distribution of the Calculus Applets website (http://www.calculusapplets.com) (v1.1) which has been reformatted for the needs of this OpenCourseware course.

We define three hyperbolic functions as follows: , , and .The applet shows the graphs of these functions and their derivatives.

Try the following:

- The applet initially shows the graph of cosh(
*x*) on the left and its derivative on the right. The hyperbolic cosine looks sort of like a parabola, but looking at the derivative (which for a parabola is a straight line) you can see that the curvature isn't quite the same as a parabola.

- Now select the second example, showing sinh(
*x*) and its derivative. Do these look familiar? Switch back and forth between the first and second examples. What do you notice? You should see that and that . Notice that, unlike the case with the regular trigonometric sine and cosine, there is no additional minus sign introduced when taking the derivative of cosh.

- Select the third example, showing tanh(
*x*) and its derivative.You should be able to calculate the derivative from the definition of tanh(*x*) and the quotient rule. It is .

For more information on rights and downloading, refer to http://www.calculusapplets.com/download.html.

© Copyright 2001 David J. Eck

© Copyright 2007 Thomas S. Downey

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

You must cause any files that you modify to carry prominent notices stating that you changed the files and the date of any change, and modified files must be put into a Java package different from edu.hws.

THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

Citation: tdowney. (2009, August 06). Hyperbolic Functions. Retrieved September 30, 2014, from Notre Dame OpenCourseWare Web site: http://ocw.nd.edu/mathematics/elements-of-calculus-i/calculus-applets-website/Hyperbolic%20Functions.html.

Copyright 2012,
Thomas S. Downey.
This work is licensed under a
Creative Commons Attribution 3.0 License.