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5.8 Hyperbolic Functions 383 5.8 Hyperbolic Functions 383 5.8 Hyperbolic Functions Develop properties of hyperbolic functions. Differentiate and integrate hyperbolic functions. Develop properties of inverse hyperbolic functions. Differentiate and integrate functions involving inverse hyperbolic functions. Hyperbolic Functions In this section, you will look briefly at a special class of exponential functions called hyperbolic functions. The name hyperbolic function arose from comparison of the area of a semicircular region, as shown in Figure 5.29, with the area of a region under a hyperbola, as shown in Figure 5.30. y y y = 1 + x2 2 2 y = 1 − x2 x x JOHANN HEINRICH LAMBERT −11−11 (1728–1777) 2 ϩ 2 ϭ Ϫ 2 ϩ 2 ϭ The first person to publish a Circle:x y 1 Hyperbola: x y 1 comprehensive study on hyperbolic Figure 5.29 Figure 5.30 functions was Johann Heinrich Lambert, a Swiss-German mathe- The integral for the semicircular region involves an inverse trigonometric (circular) matician and colleague of Euler. See LarsonCalculus.com to read function: more of this biography. 1 1 1 ␲ ͵ Ί1 Ϫ x2 dx ϭ ΄xΊ1 Ϫ x2 ϩ arcsin x ΅ ϭ Ϸ 1.571. Ϫ1 2 Ϫ1 2 The integral for the hyperbolic region involves an inverse hyperbolic function: 1 1 1 ͵ Ί ϩ 2 ϭ Ί ϩ 2 ϩ Ϫ1 Ϸ 1 x dx ΄x 1 x sinh x ΅ 2.296. Ϫ1 2 Ϫ1 This is only one of many ways in which the hyperbolic functions are similar to the trigonometric functions. Definitions of the Hyperbolic Functions REMARK The notation e x Ϫ eϪx 1 sinh x ϭ csch x ϭ , x 0 sinh x is read as “the hyperbolic 2 sinh x sine of x,” cosh x as “the e x ϩ eϪx 1 cosh x ϭ sech x ϭ hyperbolic cosine of x,” and 2 cosh x so on. sinh x 1 tanh x ϭ coth x ϭ , x 0 cosh x tanh x FOR FURTHER INFORMATION For more information on the development of hyperbolic functions, see the article “An Introduction to Hyperbolic Functions in Elementary Calculus” by Jerome Rosenthal in Mathematics Teacher. To view this article, go to MathArticles.com. American Institute of Physics (AIP) (use Emilio Serge Visual Archive) Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 384 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions The graphs of the six hyperbolic functions and their domains and ranges are shown in Figure 5.31. Note that the graph of sinh x can be obtained by adding the corresponding ͑ ͒ ϭ 1 x ͑ ͒ ϭϪ1 Ϫx y-coordinates of the exponential functions f x 2e and g x 2e . Likewise, the graph of cosh x can be obtained by adding the corresponding y-coordinates of the ͑ ͒ ϭ 1 x ͑ ͒ ϭ 1 Ϫx exponential functions f x 2e and h x 2e . y y y y = cosh x 2 2 2 x y = tanh x f(x) = e 2 1 y = sinh x −x ex 1 h(x) = e f(x) = 2 2 x x x −2 −1 1 2 −2 −112 −2 −1 1 2 −1 −x −1 −1 g(x) = −e 2 −2 −2 −2 Domain: ͑Ϫϱ, ϱ͒ Domain: ͑Ϫϱ, ϱ͒ Domain: ͑Ϫϱ, ϱ͒ Range: ͑Ϫϱ, ϱ͒ Range: ͓1, ϱ͒ Range: ͑Ϫ1, 1͒ y y y y = csch x = 1 2 sinh x 2 1 y = sech x = 1 y = coth x = cosh x tanh x 1 1 x x x −112 −21−1 2 −2 −1 1 2 −1 −1 −1 −2 Domain: ͑Ϫϱ, 0͒ ʜ ͑0, ϱ͒ Domain: ͑Ϫϱ, ϱ͒ Domain: ͑Ϫϱ, 0͒ ʜ ͑0, ϱ͒ Range: ͑Ϫϱ, 0͒ ʜ ͑0, ϱ͒ Range: ͑0, 1͔ Range: ͑Ϫϱ, Ϫ1͒ ʜ ͑1, ϱ͒ Figure 5.31 Many of the trigonometric identities have corresponding hyperbolic identities. For instance, ex ϩ eϪx 2 ex Ϫ eϪx 2 cosh2 x Ϫ sinh2 x ϭ ΂ ΃ Ϫ ΂ ΃ 2 2 e2x ϩ 2 ϩ eϪ2x e2x Ϫ 2 ϩ eϪ2x ϭ Ϫ 4 4 4 ϭ 4 ϭ 1. HYPERBOLIC IDENTITIES 2 Ϫ 2 ϭ ͑ ϩ ͒ ϭ ϩ FOR FURTHER INFORMATION cosh x sinh x 1 sinh x y sinh x cosh y cosh x sinh y To understand geometrically the tanh2 x ϩ sech2 x ϭ 1 sinh͑x Ϫ y͒ ϭ sinh x cosh y Ϫ cosh x sinh y relationship between the hyperbolic coth2 x Ϫ csch2 x ϭ 1 cosh͑x ϩ y͒ ϭ cosh x cosh y ϩ sinh x sinh y and exponential functions, see the cosh͑x Ϫ y͒ ϭ cosh x cosh y Ϫ sinh x sinh y article “A Short Proof Linking Ϫ1 ϩ cosh 2x 1 ϩ cosh 2x the Hyperbolic and Exponential sinh2 x ϭ cosh2 x ϭ Functions” by Michael J. Seery 2 2 in The AMATYC Review. sinh 2x ϭ 2 sinh x cosh x cosh 2x ϭ cosh2 x ϩ sinh2 x Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 5.8 Hyperbolic Functions 385 Differentiation and Integration of Hyperbolic Functions Because the hyperbolic functions are written in terms of e x and e Ϫx, you can easily derive rules for their derivatives. The next theorem lists these derivatives with the corresponding integration rules. THEOREM 5.18 Derivatives and Integrals of Hyperbolic Functions Let u be a differentiable function of x. d ͓sinh u͔ ϭ ͑cosh u͒uЈ ͵cosh u du ϭ sinh u ϩ C dx d ͓cosh u͔ ϭ ͑sinh u͒uЈ ͵sinh u du ϭ cosh u ϩ C dx d ͓tanh u͔ ϭ ͑sech2 u͒uЈ ͵sech2 u du ϭ tanh u ϩ C dx d ͓coth u͔ ϭϪ͑csch2 u͒uЈ ͵csch2 u du ϭϪcoth u ϩ C dx d ͓sech u͔ ϭϪ͑sech u tanh u͒uЈ ͵sech u tanh u du ϭϪsech u ϩ C dx d ͓csch u͔ ϭϪ͑csch u coth u͒uЈ ͵csch u coth u du ϭϪcsch u ϩ C dx Proof Here is a proof of two of the differentiation rules. (You are asked to prove some of the other differentiation rules in Exercises 103–105.) d d ex Ϫ eϪx ͓sinh x͔ ϭ ΄ ΅ dx dx 2 ex ϩ eϪx ϭ 2 ϭ cosh x d d sinh x ͓tanh x͔ ϭ ΄ ΅ dx dx cosh x cosh x͑cosh x͒ Ϫ sinh x͑sinh x͒ ϭ cosh2 x 1 ϭ cosh2 x ϭ sech2 x See LarsonCalculus.com for Bruce Edwards’s video of this proof. Differentiation of Hyperbolic Functions d a. ͓sinh͑x 2 Ϫ 3͔͒ ϭ 2x cosh͑x 2 Ϫ 3͒ dx d sinh x b. ͓ln͑cosh x͔͒ ϭ ϭ tanh x dx cosh x d c. ͓x sinh x Ϫ cosh x͔ ϭ x cosh x ϩ sinh x Ϫ sinh x ϭ x cosh x dx d d. ͓͑x Ϫ 1͒ cosh x Ϫ sin x͔ ϭ ͑x Ϫ 1͒ sinh x ϩ cosh x Ϫ cosh x ϭ ͑x Ϫ 1͒ sinh x dx Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 386 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions Finding Relative Extrema f(x) = (x − 1) cosh x − sinh x Find the relative extrema of y f͑x͒ ϭ ͑x Ϫ 1͒ cosh x Ϫ sinh x. 1 Solution Using the result of Example 1(d), set the first derivative of f equal to 0. x ͑x Ϫ 1͒ sinh x ϭ 0 −2 −1 1 3 (0, −1) So, the critical numbers are x ϭ 1 and x ϭ 0. Using the Second Derivative Test, you (1, −sinh 1) can verify that the point ͑0, Ϫ1͒ yields a relative maximum and the point ͑1, Ϫsinh 1͒ −2 yields a relative minimum, as shown in Figure 5.32. Try using a graphing utility to −3 confirm this result. If your graphing utility does not have hyperbolic functions, you can use exponential functions, as shown. fЉ ͑0͒ 0, so ͑0, Ϫ1͒ is a relative < 1 1 maximum. fЉ ͑1͒ > 0, so ͑1, Ϫsinh 1͒ f͑x͒ ϭ ͑x Ϫ 1͒΂ ΃͑e x ϩ eϪx͒ Ϫ ͑ex Ϫ eϪx͒ is a relative minimum. 2 2 Figure 5.32 1 ϭ ͑xe x ϩ xeϪx Ϫ e x Ϫ eϪx Ϫ e x ϩ eϪx͒ 2 1 ϭ ͑xe x ϩ xeϪx Ϫ 2e x͒ 2 When a uniform flexible cable, such as a telephone wire, is suspended from two points, it takes the shape of a catenary, as discussed in Example 3. Hanging Power Cables See LarsonCalculus.com for an interactive version of this type of example. Power cables are suspended between two towers, forming the catenary shown in y Figure 5.33.
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