{"id":167,"date":"2021-02-03T22:20:44","date_gmt":"2021-02-03T22:20:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=167"},"modified":"2022-03-11T21:46:52","modified_gmt":"2022-03-11T21:46:52","slug":"hyperbolic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/hyperbolic-functions\/","title":{"raw":"Hyperbolic Functions","rendered":"Hyperbolic Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify the hyperbolic functions, their graphs, and basic identities<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe <strong>hyperbolic functions<\/strong> are defined in terms of certain combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex]. These functions arise naturally in various engineering and physics applications, including the study of water waves and vibrations of elastic membranes. Another common use for a hyperbolic function is the representation of a hanging chain or cable, also known as a catenary. If we introduce a coordinate system so that the low point of the chain lies along the [latex]y[\/latex]-axis, we can describe the height of the chain in terms of a hyperbolic function. First, we define the hyperbolic functions.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202700\/CNX_Calc_Figure_01_05_009.jpg\" alt=\"A photograph of a spider web collecting dew drops.\" width=\"488\" height=\"403\" \/> Figure 6. The shape of a strand of silk in a spider\u2019s web can be described in terms of a hyperbolic function. The same shape applies to a chain or cable hanging from two supports with only its own weight. (credit: \u201cMtpaley\u201d, Wikimedia Commons)[\/caption]\r\n\r\n<div id=\"fs-id1170572467985\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1170572467989\"><strong>Hyperbolic cosine<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170572467996\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\cosh x=\\large \\frac{e^x+e^{\u2212x}}{2}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572468030\"><strong>Hyperbolic sine<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170572468036\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sinh x=\\large \\frac{e^x-e^{\u2212x}}{2}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572468070\"><strong>Hyperbolic tangent<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170572468077\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\tanh x=\\large \\frac{\\sinh x}{\\cosh x} \\normalsize = \\large \\frac{e^x-e^{\u2212x}}{e^x+e^{\u2212x}}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572431436\"><strong>Hyperbolic cosecant<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170572431443\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\text{csch} \\, x=\\large \\frac{1}{\\sinh x} \\normalsize = \\large \\frac{2}{e^x-e^{\u2212x}}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572431488\"><strong>Hyperbolic secant<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170572431494\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\text{sech} \\, x=\\large \\frac{1}{\\cosh x} \\normalsize = \\large \\frac{2}{e^x+e^{\u2212x}}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572431539\"><strong>Hyperbolic cotangent<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170572431545\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\coth x=\\large \\frac{\\cosh x}{\\sinh x} \\normalsize = \\large \\frac{e^x+e^{\u2212x}}{e^x-e^{\u2212x}}[\/latex]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572431609\">The name <em>cosh<\/em> rhymes with \u201cgosh,\u201d whereas the name <em>sinh<\/em> is pronounced \u201ccinch.\u201d <em>Tanh<\/em>, <em>sech<\/em>, <em>csch<\/em>, and <em>coth<\/em> are pronounced \u201ctanch,\u201d \u201cseech,\u201d \u201ccoseech,\u201d and \u201ccotanch,\u201d respectively.<\/p>\r\n<p id=\"fs-id1170572234293\">Using the definition of [latex]\\cosh(x)[\/latex] and principles of physics, it can be shown that the height of a hanging chain, such as the one in Figure 6, can be described by the function [latex]h(x)=a \\cosh(x\/a)+c[\/latex] for certain constants [latex]a[\/latex] and [latex]c[\/latex].<\/p>\r\n<p id=\"fs-id1170572234364\">But why are these functions called <em>hyperbolic functions<\/em>? To answer this question, consider the quantity [latex]\\cosh^2 t-\\sinh^2 t[\/latex]. Using the definition of [latex]\\cosh[\/latex] and [latex]\\sinh[\/latex], we see that<\/p>\r\n\r\n<div id=\"fs-id1170572234411\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\cosh^2 t-\\sinh^2 t=\\large \\frac{e^{2t}+2+e^{-2t}}{4}-\\frac{e^{2t}-2+e^{-2t}}{4} \\normalsize =1[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572171610\">This identity is the analog of the trigonometric identity [latex]\\cos^2 t+\\sin^2 t=1[\/latex]. Here, given a value [latex]t[\/latex], the point [latex](x,y)=(\\cosh t,\\sinh t)[\/latex] lies on the unit hyperbola [latex]x^2-y^2=1[\/latex] (Figure 7).<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202702\/CNX_Calc_Figure_01_05_007.jpg\" alt=\"An image of a graph. The x axis runs from -1 to 3 and the y axis runs from -3 to 3. The graph is of the relation \u201c(x squared) - (y squared) -1\u201d. The left most point of the relation is at the x intercept, which is at the point (1, 0). From this point the relation both increases and decreases in curves as x increases. This relation is known as a hyperbola and it resembles a sideways \u201cU\u201d shape. There is a point plotted on the graph of the relation labeled \u201c(cosh(1), sinh(1))\u201d, which is at the approximate point (1.5, 1.2).\" width=\"325\" height=\"275\" \/> Figure 7. The unit hyperbola [latex]\\cosh^2 t-\\sinh^2 t=1[\/latex].[\/caption]\r\n<div id=\"fs-id1170572171758\" class=\"bc-section section\">\r\n<h3>Graphs of Hyperbolic Functions<\/h3>\r\n<p id=\"fs-id1170572171763\">To graph [latex]\\cosh x[\/latex] and [latex]\\sinh x[\/latex], we make use of the fact that both functions approach [latex]\\left(\\frac{1}{2}\\right)e^x[\/latex] as [latex]x \\to \\infty[\/latex], since [latex]e^{\u2212x} \\to 0[\/latex] as [latex]x \\to \\infty[\/latex]. As [latex]x \\to \u2212\\infty, \\, \\cosh x[\/latex] approaches [latex]\\frac{1}{2}e^{\u2212x}[\/latex], whereas [latex]\\sinh x[\/latex] approaches [latex]-\\frac{1}{2}e^{\u2212x}[\/latex]. Therefore, using the graphs of [latex]\\frac{1}{2}e^x, \\, \\frac{1}{2}e^{\u2212x}[\/latex], and [latex]\u2212\\frac{1}{2}e^{\u2212x}[\/latex] as guides, we graph [latex]\\cosh x[\/latex] and [latex]\\sinh x[\/latex]. To graph [latex]\\tanh x[\/latex], we use the fact that [latex]\\tanh(0)=0, \\, -1&lt;\\tanh(x)&lt;1[\/latex] for all [latex]x, \\, \\tanh x \\to 1[\/latex] as [latex]x \\to \\infty[\/latex], and [latex]\\tanh x \\to \u22121[\/latex] as [latex]x \\to \u2212\\infty[\/latex]. The graphs of the other three hyperbolic functions can be sketched using the graphs of [latex]\\cosh x, \\, \\sinh x[\/latex], and [latex]\\tanh x[\/latex] (Figure 8).<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"573\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202706\/CNX_Calc_Figure_01_05_011.jpg\" alt=\"An image of six graphs. Each graph has an x axis that runs from -3 to 3 and a y axis that runs from -4 to 4. The first graph is of the function \u201cy = cosh(x)\u201d, which is a hyperbola. The function decreases until it hits the point (0, 1), where it begins to increase. There are also two functions that serve as a boundary for this function. The first of these functions is \u201cy = (1\/2)(e to power of -x)\u201d, a decreasing curved function and the second of these functions is \u201cy = (1\/2)(e to power of x)\u201d, an increasing curved function. The function \u201cy = cosh(x)\u201d is always above these two functions without ever touching them. The second graph is of the function \u201cy = sinh(x)\u201d, which is an increasing curved function. There are also two functions that serve as a boundary for this function. The first of these functions is \u201cy = (1\/2)(e to power of x)\u201d, an increasing curved function and the second of these functions is \u201cy = -(1\/2)(e to power of -x)\u201d, an increasing curved function that approaches the x axis without touching it. The function \u201cy = sinh(x)\u201d is always between these two functions without ever touching them. The third graph is of the function \u201cy = sech(x)\u201d, which increases until the point (0, 1), where it begins to decrease. The graph of the function has a hump. The fourth graph is of the function \u201cy = csch(x)\u201d. On the left side of the y axis, the function starts slightly below the x axis and decreases until it approaches the y axis, which it never touches. On the right side of the y axis, the function starts slightly to the right of the y axis and decreases until it approaches the x axis, which it never touches. The fifth graph is of the function \u201cy = tanh(x)\u201d, an increasing curved function. There are also two functions that serve as a boundary for this function. The first of these functions is \u201cy = 1\u201d, a horizontal line function and the second of these functions is \u201cy = -1\u201d, another horizontal line function. The function \u201cy = tanh(x)\u201d is always between these two functions without ever touching them. The sixth graph is of the function \u201cy = coth(x)\u201d. On the left side of the y axis, the function starts slightly below the boundary line \u201cy = 1\u201d and decreases until it approaches the y axis, which it never touches. On the right side of the y axis, the function starts slightly to the right of the y axis and decreases until it approaches the boundary line \u201cy = -1\u201d, which it never touches.\" width=\"573\" height=\"929\" \/> Figure 8. The hyperbolic functions involve combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex].[\/caption]<\/div>\r\n<div id=\"fs-id1170572433248\" class=\"bc-section section\">\r\n<h3>Identities Involving Hyperbolic Functions<\/h3>\r\n<p id=\"fs-id1170572433254\">The identity [latex]\\cosh^2 t-\\sinh^2 t[\/latex], shown in Figure 7, is one of several identities involving the hyperbolic functions, some of which are listed next. The first four properties follow easily from the definitions of hyperbolic sine and hyperbolic cosine. Except for some differences in signs, most of these properties are analogous to identities for trigonometric functions.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Identities Involving Hyperbolic Functions<\/h3>\r\n\r\n<hr \/>\r\n\r\n<ol id=\"fs-id1170572433294\">\r\n \t<li>[latex]\\cosh(\u2212x)=\\cosh x[\/latex]<\/li>\r\n \t<li>[latex]\\sinh(\u2212x)=\u2212\\sinh x[\/latex]<\/li>\r\n \t<li>[latex]\\cosh x+\\sinh x=e^x[\/latex]<\/li>\r\n \t<li>[latex]\\cosh x-\\sinh x=e^{\u2212x}[\/latex]<\/li>\r\n \t<li>[latex]\\cosh^2 x-\\sinh^2 x=1[\/latex]<\/li>\r\n \t<li>[latex]1-\\tanh^2 x=\\text{sech}^2 x[\/latex]<\/li>\r\n \t<li>[latex]\\coth^2 x-1=\\text{csch}^2 x[\/latex]<\/li>\r\n \t<li>[latex]\\sinh(x \\pm y)=\\sinh x \\cosh y \\pm \\cosh x \\sinh y[\/latex]<\/li>\r\n \t<li>[latex]\\cosh (x \\pm y)=\\cosh x \\cosh y \\pm \\sinh x \\sinh y[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1170572433248\" class=\"bc-section section\">\r\n<div id=\"fs-id1170572443393\" class=\"textbook exercises\">\r\n<h3>Example: Evaluating Hyperbolic Functions<\/h3>\r\n<ol id=\"fs-id1170572443403\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>Simplify [latex]\\sinh(5 \\ln x)[\/latex].<\/li>\r\n \t<li>If [latex]\\sinh x=\\frac{3}{4}[\/latex], find the values of the remaining five hyperbolic functions.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1170572443462\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572443462\"]\r\n<ol id=\"fs-id1170572443462\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>Using the definition of the [latex]\\sinh[\/latex] function, we write\r\n<div id=\"fs-id1170570995857\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sinh(5 \\ln x)=\\large \\frac{e^{5 \\ln x}-e^{-5 \\ln x}}{2} \\normalsize = \\large \\frac{e^{\\ln(x^5)}-e^{\\ln(x^{-5})}}{2} \\normalsize =\\large \\frac{x^5-x^{-5}}{2}[\/latex].<\/div><\/li>\r\n \t<li>Using the identity [latex]\\cosh^2 x-\\sinh^2 x=1[\/latex], we see that\r\n<div id=\"fs-id1170573388429\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\cosh^2 x=1+\\big(\\frac{3}{4}\\big)^2=\\frac{25}{16}[\/latex].<\/div>\r\nSince [latex]\\cosh x \\ge 1[\/latex] for all [latex]x[\/latex], we must have [latex]\\cosh x=5\/4[\/latex]. Then, using the definitions for the other hyperbolic functions, we conclude that [latex]\\tanh x=3\/5, \\, \\text{csch} \\, x=4\/3, \\, \\text{sech} \\, x=4\/5[\/latex], and [latex]\\coth x=5\/3[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to Example: Evaluating Hyperbolic Functions[\/caption]\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tOkk_pSFpzk?controls=0&amp;start=1498&amp;end=1738&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266834\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.5ExponentialAndLogarithmicFunctions1498to1738_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"1.5 Exponential and Logarithmic Functions\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1170572548314\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170572549916\">Simplify [latex]\\cosh(2 \\ln x)[\/latex].<\/p>\r\n[reveal-answer q=\"473309\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"473309\"]\r\n<p id=\"fs-id1165039563336\">Use the definition of the cosh function and the power property of logarithm functions.<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572549946\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572549946\"]\r\n<p id=\"fs-id1170572549946\">[latex]\\frac{(x^2+x^{-2})}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Inverse Hyperbolic Functions<\/h3>\r\n<p id=\"fs-id1170572549987\">From the graphs of the hyperbolic functions, we see that all of them are one-to-one except [latex]\\cosh x[\/latex] and [latex]\\text{sech} \\, x[\/latex]. If we restrict the domains of these two functions to the interval [latex][0,\\infty)[\/latex], then all the hyperbolic functions are one-to-one, and we can define the <strong>inverse hyperbolic functions<\/strong>. Since the hyperbolic functions themselves involve exponential functions, the inverse hyperbolic functions involve logarithmic functions.<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1170572550036\">Inverse Hyperbolic Functions:<\/p>\r\n\r\n<div id=\"fs-id1170572550043\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{cccc}\\sinh^{-1} x=\\text{arcsinh } x=\\ln(x+\\sqrt{x^2+1})\\hfill &amp; &amp; &amp; \\cosh^{-1} x=\\text{arccosh } x=\\ln(x+\\sqrt{x^2-1})\\hfill \\\\ \\tanh^{-1} x=\\text{arctanh } x=\\frac{1}{2}\\ln\\big(\\frac{1+x}{1-x}\\big)\\hfill &amp; &amp; &amp; \\coth^{-1} x=\\text{arccot } x=\\frac{1}{2}\\ln\\big(\\frac{x+1}{x-1}\\big)\\hfill \\\\ \\text{sech}^{-1} x=\\text{arcsech } x=\\ln\\big(\\frac{1+\\sqrt{1-x^2}}{x}\\big)\\hfill &amp; &amp; &amp; \\text{csch}^{-1} x=\\text{arccsch } x=\\ln\\big(\\frac{1}{x}+\\frac{\\sqrt{1+x^2}}{|x|}\\big)\\hfill \\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170572235136\">Let\u2019s look at how to derive the first equation. The others follow similarly. Suppose [latex]y=\\sinh^{-1} x[\/latex]. Then, [latex]x=\\sinh y[\/latex] and, by the definition of the hyperbolic sine function, [latex]x=\\frac{e^y-e^{\u2212y}}{2}[\/latex]. Therefore,<\/p>\r\n\r\n<div id=\"fs-id1170572235202\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]e^y-2x-e^{\u2212y}=0[\/latex]<\/div>\r\nMultiplying this equation by [latex]e^y[\/latex], we obtain\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]e^{2y}-2xe^y-1=0[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572235286\">This can be solved like a quadratic equation, with the solution<\/p>\r\n\r\n<div id=\"fs-id1170572235290\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]e^y=\\large \\frac{2x \\pm \\sqrt{4x^2+4}}{2} \\normalsize =x \\pm \\sqrt{x^2+1}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572482176\">Since [latex]e^y&gt;0[\/latex], the only solution is the one with the positive sign. Applying the natural logarithm to both sides of the equation, we conclude that<\/p>\r\n\r\n<div id=\"fs-id1170572482196\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y=\\ln(x+\\sqrt{x^2+1})[\/latex]<\/div>\r\n&nbsp;\r\n<div id=\"fs-id1170572482234\" class=\"textbook exercises\">\r\n<h3>Example: Evaluating Inverse Hyperbolic Functions<\/h3>\r\nEvaluate each of the following expressions\r\n\r\n[latex]\\sinh^{-1}(2)[\/latex]\r\n[latex]\\tanh^{-1}\\left(\\frac{1}{4}\\right)[\/latex]\r\n\r\n[reveal-answer q=\"277655\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"277655\"]\r\n<p id=\"fs-id1170572482306\">[latex]\\sinh^{-1}(2)=\\ln(2+\\sqrt{2^2+1})=\\ln(2+\\sqrt{5}) \\approx 1.4436[\/latex]<\/p>\r\n<p id=\"fs-id1170572176029\">[latex]\\tanh^{-1}(\\frac{1}{4})=\\frac{1}{2}\\ln(\\frac{1+1\/4}{1-1\/4})=\\frac{1}{2}\\ln(\\frac{5\/4}{3\/4})=\\frac{1}{2}\\ln(\\frac{5}{3}) \\approx 0.2554[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1170572176157\" class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170572176165\">Evaluate [latex]\\tanh^{-1}\\left(\\frac{1}{2}\\right)[\/latex].<\/p>\r\n[reveal-answer q=\"3088722\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"3088722\"]\r\n\r\nUse the definition of [latex]\\tanh^{-1} x[\/latex] and simplify.\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1170572176215\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572176215\"]\r\n\r\n[latex]\\frac{1}{2}\\ln(3) \\approx 0.5493[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify the hyperbolic functions, their graphs, and basic identities<\/li>\n<\/ul>\n<\/div>\n<p>The <strong>hyperbolic functions<\/strong> are defined in terms of certain combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex]. These functions arise naturally in various engineering and physics applications, including the study of water waves and vibrations of elastic membranes. Another common use for a hyperbolic function is the representation of a hanging chain or cable, also known as a catenary. If we introduce a coordinate system so that the low point of the chain lies along the [latex]y[\/latex]-axis, we can describe the height of the chain in terms of a hyperbolic function. First, we define the hyperbolic functions.<\/p>\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202700\/CNX_Calc_Figure_01_05_009.jpg\" alt=\"A photograph of a spider web collecting dew drops.\" width=\"488\" height=\"403\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 6. The shape of a strand of silk in a spider\u2019s web can be described in terms of a hyperbolic function. The same shape applies to a chain or cable hanging from two supports with only its own weight. (credit: \u201cMtpaley\u201d, Wikimedia Commons)<\/p>\n<\/div>\n<div id=\"fs-id1170572467985\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\n<p id=\"fs-id1170572467989\"><strong>Hyperbolic cosine<\/strong><\/p>\n<div id=\"fs-id1170572467996\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\cosh x=\\large \\frac{e^x+e^{\u2212x}}{2}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572468030\"><strong>Hyperbolic sine<\/strong><\/p>\n<div id=\"fs-id1170572468036\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sinh x=\\large \\frac{e^x-e^{\u2212x}}{2}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572468070\"><strong>Hyperbolic tangent<\/strong><\/p>\n<div id=\"fs-id1170572468077\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\tanh x=\\large \\frac{\\sinh x}{\\cosh x} \\normalsize = \\large \\frac{e^x-e^{\u2212x}}{e^x+e^{\u2212x}}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572431436\"><strong>Hyperbolic cosecant<\/strong><\/p>\n<div id=\"fs-id1170572431443\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\text{csch} \\, x=\\large \\frac{1}{\\sinh x} \\normalsize = \\large \\frac{2}{e^x-e^{\u2212x}}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572431488\"><strong>Hyperbolic secant<\/strong><\/p>\n<div id=\"fs-id1170572431494\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\text{sech} \\, x=\\large \\frac{1}{\\cosh x} \\normalsize = \\large \\frac{2}{e^x+e^{\u2212x}}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572431539\"><strong>Hyperbolic cotangent<\/strong><\/p>\n<div id=\"fs-id1170572431545\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\coth x=\\large \\frac{\\cosh x}{\\sinh x} \\normalsize = \\large \\frac{e^x+e^{\u2212x}}{e^x-e^{\u2212x}}[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1170572431609\">The name <em>cosh<\/em> rhymes with \u201cgosh,\u201d whereas the name <em>sinh<\/em> is pronounced \u201ccinch.\u201d <em>Tanh<\/em>, <em>sech<\/em>, <em>csch<\/em>, and <em>coth<\/em> are pronounced \u201ctanch,\u201d \u201cseech,\u201d \u201ccoseech,\u201d and \u201ccotanch,\u201d respectively.<\/p>\n<p id=\"fs-id1170572234293\">Using the definition of [latex]\\cosh(x)[\/latex] and principles of physics, it can be shown that the height of a hanging chain, such as the one in Figure 6, can be described by the function [latex]h(x)=a \\cosh(x\/a)+c[\/latex] for certain constants [latex]a[\/latex] and [latex]c[\/latex].<\/p>\n<p id=\"fs-id1170572234364\">But why are these functions called <em>hyperbolic functions<\/em>? To answer this question, consider the quantity [latex]\\cosh^2 t-\\sinh^2 t[\/latex]. Using the definition of [latex]\\cosh[\/latex] and [latex]\\sinh[\/latex], we see that<\/p>\n<div id=\"fs-id1170572234411\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\cosh^2 t-\\sinh^2 t=\\large \\frac{e^{2t}+2+e^{-2t}}{4}-\\frac{e^{2t}-2+e^{-2t}}{4} \\normalsize =1[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572171610\">This identity is the analog of the trigonometric identity [latex]\\cos^2 t+\\sin^2 t=1[\/latex]. Here, given a value [latex]t[\/latex], the point [latex](x,y)=(\\cosh t,\\sinh t)[\/latex] lies on the unit hyperbola [latex]x^2-y^2=1[\/latex] (Figure 7).<\/p>\n<div style=\"width: 335px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202702\/CNX_Calc_Figure_01_05_007.jpg\" alt=\"An image of a graph. The x axis runs from -1 to 3 and the y axis runs from -3 to 3. The graph is of the relation \u201c(x squared) - (y squared) -1\u201d. The left most point of the relation is at the x intercept, which is at the point (1, 0). From this point the relation both increases and decreases in curves as x increases. This relation is known as a hyperbola and it resembles a sideways \u201cU\u201d shape. There is a point plotted on the graph of the relation labeled \u201c(cosh(1), sinh(1))\u201d, which is at the approximate point (1.5, 1.2).\" width=\"325\" height=\"275\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 7. The unit hyperbola [latex]\\cosh^2 t-\\sinh^2 t=1[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1170572171758\" class=\"bc-section section\">\n<h3>Graphs of Hyperbolic Functions<\/h3>\n<p id=\"fs-id1170572171763\">To graph [latex]\\cosh x[\/latex] and [latex]\\sinh x[\/latex], we make use of the fact that both functions approach [latex]\\left(\\frac{1}{2}\\right)e^x[\/latex] as [latex]x \\to \\infty[\/latex], since [latex]e^{\u2212x} \\to 0[\/latex] as [latex]x \\to \\infty[\/latex]. As [latex]x \\to \u2212\\infty, \\, \\cosh x[\/latex] approaches [latex]\\frac{1}{2}e^{\u2212x}[\/latex], whereas [latex]\\sinh x[\/latex] approaches [latex]-\\frac{1}{2}e^{\u2212x}[\/latex]. Therefore, using the graphs of [latex]\\frac{1}{2}e^x, \\, \\frac{1}{2}e^{\u2212x}[\/latex], and [latex]\u2212\\frac{1}{2}e^{\u2212x}[\/latex] as guides, we graph [latex]\\cosh x[\/latex] and [latex]\\sinh x[\/latex]. To graph [latex]\\tanh x[\/latex], we use the fact that [latex]\\tanh(0)=0, \\, -1<\\tanh(x)<1[\/latex] for all [latex]x, \\, \\tanh x \\to 1[\/latex] as [latex]x \\to \\infty[\/latex], and [latex]\\tanh x \\to \u22121[\/latex] as [latex]x \\to \u2212\\infty[\/latex]. The graphs of the other three hyperbolic functions can be sketched using the graphs of [latex]\\cosh x, \\, \\sinh x[\/latex], and [latex]\\tanh x[\/latex] (Figure 8).<\/p>\n<div style=\"width: 583px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202706\/CNX_Calc_Figure_01_05_011.jpg\" alt=\"An image of six graphs. Each graph has an x axis that runs from -3 to 3 and a y axis that runs from -4 to 4. The first graph is of the function \u201cy = cosh(x)\u201d, which is a hyperbola. The function decreases until it hits the point (0, 1), where it begins to increase. There are also two functions that serve as a boundary for this function. The first of these functions is \u201cy = (1\/2)(e to power of -x)\u201d, a decreasing curved function and the second of these functions is \u201cy = (1\/2)(e to power of x)\u201d, an increasing curved function. The function \u201cy = cosh(x)\u201d is always above these two functions without ever touching them. The second graph is of the function \u201cy = sinh(x)\u201d, which is an increasing curved function. There are also two functions that serve as a boundary for this function. The first of these functions is \u201cy = (1\/2)(e to power of x)\u201d, an increasing curved function and the second of these functions is \u201cy = -(1\/2)(e to power of -x)\u201d, an increasing curved function that approaches the x axis without touching it. The function \u201cy = sinh(x)\u201d is always between these two functions without ever touching them. The third graph is of the function \u201cy = sech(x)\u201d, which increases until the point (0, 1), where it begins to decrease. The graph of the function has a hump. The fourth graph is of the function \u201cy = csch(x)\u201d. On the left side of the y axis, the function starts slightly below the x axis and decreases until it approaches the y axis, which it never touches. On the right side of the y axis, the function starts slightly to the right of the y axis and decreases until it approaches the x axis, which it never touches. The fifth graph is of the function \u201cy = tanh(x)\u201d, an increasing curved function. There are also two functions that serve as a boundary for this function. The first of these functions is \u201cy = 1\u201d, a horizontal line function and the second of these functions is \u201cy = -1\u201d, another horizontal line function. The function \u201cy = tanh(x)\u201d is always between these two functions without ever touching them. The sixth graph is of the function \u201cy = coth(x)\u201d. On the left side of the y axis, the function starts slightly below the boundary line \u201cy = 1\u201d and decreases until it approaches the y axis, which it never touches. On the right side of the y axis, the function starts slightly to the right of the y axis and decreases until it approaches the boundary line \u201cy = -1\u201d, which it never touches.\" width=\"573\" height=\"929\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 8. The hyperbolic functions involve combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex].<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572433248\" class=\"bc-section section\">\n<h3>Identities Involving Hyperbolic Functions<\/h3>\n<p id=\"fs-id1170572433254\">The identity [latex]\\cosh^2 t-\\sinh^2 t[\/latex], shown in Figure 7, is one of several identities involving the hyperbolic functions, some of which are listed next. The first four properties follow easily from the definitions of hyperbolic sine and hyperbolic cosine. Except for some differences in signs, most of these properties are analogous to identities for trigonometric functions.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Identities Involving Hyperbolic Functions<\/h3>\n<hr \/>\n<ol id=\"fs-id1170572433294\">\n<li>[latex]\\cosh(\u2212x)=\\cosh x[\/latex]<\/li>\n<li>[latex]\\sinh(\u2212x)=\u2212\\sinh x[\/latex]<\/li>\n<li>[latex]\\cosh x+\\sinh x=e^x[\/latex]<\/li>\n<li>[latex]\\cosh x-\\sinh x=e^{\u2212x}[\/latex]<\/li>\n<li>[latex]\\cosh^2 x-\\sinh^2 x=1[\/latex]<\/li>\n<li>[latex]1-\\tanh^2 x=\\text{sech}^2 x[\/latex]<\/li>\n<li>[latex]\\coth^2 x-1=\\text{csch}^2 x[\/latex]<\/li>\n<li>[latex]\\sinh(x \\pm y)=\\sinh x \\cosh y \\pm \\cosh x \\sinh y[\/latex]<\/li>\n<li>[latex]\\cosh (x \\pm y)=\\cosh x \\cosh y \\pm \\sinh x \\sinh y[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1170572433248\" class=\"bc-section section\">\n<div id=\"fs-id1170572443393\" class=\"textbook exercises\">\n<h3>Example: Evaluating Hyperbolic Functions<\/h3>\n<ol id=\"fs-id1170572443403\" style=\"list-style-type: lower-alpha;\">\n<li>Simplify [latex]\\sinh(5 \\ln x)[\/latex].<\/li>\n<li>If [latex]\\sinh x=\\frac{3}{4}[\/latex], find the values of the remaining five hyperbolic functions.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572443462\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572443462\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572443462\" style=\"list-style-type: lower-alpha;\">\n<li>Using the definition of the [latex]\\sinh[\/latex] function, we write\n<div id=\"fs-id1170570995857\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\sinh(5 \\ln x)=\\large \\frac{e^{5 \\ln x}-e^{-5 \\ln x}}{2} \\normalsize = \\large \\frac{e^{\\ln(x^5)}-e^{\\ln(x^{-5})}}{2} \\normalsize =\\large \\frac{x^5-x^{-5}}{2}[\/latex].<\/div>\n<\/li>\n<li>Using the identity [latex]\\cosh^2 x-\\sinh^2 x=1[\/latex], we see that\n<div id=\"fs-id1170573388429\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\cosh^2 x=1+\\big(\\frac{3}{4}\\big)^2=\\frac{25}{16}[\/latex].<\/div>\n<p>Since [latex]\\cosh x \\ge 1[\/latex] for all [latex]x[\/latex], we must have [latex]\\cosh x=5\/4[\/latex]. Then, using the definitions for the other hyperbolic functions, we conclude that [latex]\\tanh x=3\/5, \\, \\text{csch} \\, x=4\/3, \\, \\text{sech} \\, x=4\/5[\/latex], and [latex]\\coth x=5\/3[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Evaluating Hyperbolic Functions<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tOkk_pSFpzk?controls=0&amp;start=1498&amp;end=1738&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266834\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266834\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.5ExponentialAndLogarithmicFunctions1498to1738_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;1.5 Exponential and Logarithmic Functions&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1170572548314\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170572549916\">Simplify [latex]\\cosh(2 \\ln x)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q473309\">Hint<\/span><\/p>\n<div id=\"q473309\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165039563336\">Use the definition of the cosh function and the power property of logarithm functions.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572549946\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572549946\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572549946\">[latex]\\frac{(x^2+x^{-2})}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Inverse Hyperbolic Functions<\/h3>\n<p id=\"fs-id1170572549987\">From the graphs of the hyperbolic functions, we see that all of them are one-to-one except [latex]\\cosh x[\/latex] and [latex]\\text{sech} \\, x[\/latex]. If we restrict the domains of these two functions to the interval [latex][0,\\infty)[\/latex], then all the hyperbolic functions are one-to-one, and we can define the <strong>inverse hyperbolic functions<\/strong>. Since the hyperbolic functions themselves involve exponential functions, the inverse hyperbolic functions involve logarithmic functions.<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\n<p id=\"fs-id1170572550036\">Inverse Hyperbolic Functions:<\/p>\n<div id=\"fs-id1170572550043\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{cccc}\\sinh^{-1} x=\\text{arcsinh } x=\\ln(x+\\sqrt{x^2+1})\\hfill & & & \\cosh^{-1} x=\\text{arccosh } x=\\ln(x+\\sqrt{x^2-1})\\hfill \\\\ \\tanh^{-1} x=\\text{arctanh } x=\\frac{1}{2}\\ln\\big(\\frac{1+x}{1-x}\\big)\\hfill & & & \\coth^{-1} x=\\text{arccot } x=\\frac{1}{2}\\ln\\big(\\frac{x+1}{x-1}\\big)\\hfill \\\\ \\text{sech}^{-1} x=\\text{arcsech } x=\\ln\\big(\\frac{1+\\sqrt{1-x^2}}{x}\\big)\\hfill & & & \\text{csch}^{-1} x=\\text{arccsch } x=\\ln\\big(\\frac{1}{x}+\\frac{\\sqrt{1+x^2}}{|x|}\\big)\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<p id=\"fs-id1170572235136\">Let\u2019s look at how to derive the first equation. The others follow similarly. Suppose [latex]y=\\sinh^{-1} x[\/latex]. Then, [latex]x=\\sinh y[\/latex] and, by the definition of the hyperbolic sine function, [latex]x=\\frac{e^y-e^{\u2212y}}{2}[\/latex]. Therefore,<\/p>\n<div id=\"fs-id1170572235202\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]e^y-2x-e^{\u2212y}=0[\/latex]<\/div>\n<p>Multiplying this equation by [latex]e^y[\/latex], we obtain<\/p>\n<div class=\"equation unnumbered\" style=\"text-align: center;\">[latex]e^{2y}-2xe^y-1=0[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572235286\">This can be solved like a quadratic equation, with the solution<\/p>\n<div id=\"fs-id1170572235290\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]e^y=\\large \\frac{2x \\pm \\sqrt{4x^2+4}}{2} \\normalsize =x \\pm \\sqrt{x^2+1}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572482176\">Since [latex]e^y>0[\/latex], the only solution is the one with the positive sign. Applying the natural logarithm to both sides of the equation, we conclude that<\/p>\n<div id=\"fs-id1170572482196\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]y=\\ln(x+\\sqrt{x^2+1})[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div id=\"fs-id1170572482234\" class=\"textbook exercises\">\n<h3>Example: Evaluating Inverse Hyperbolic Functions<\/h3>\n<p>Evaluate each of the following expressions<\/p>\n<p>[latex]\\sinh^{-1}(2)[\/latex]<br \/>\n[latex]\\tanh^{-1}\\left(\\frac{1}{4}\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q277655\">Show Solution<\/span><\/p>\n<div id=\"q277655\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572482306\">[latex]\\sinh^{-1}(2)=\\ln(2+\\sqrt{2^2+1})=\\ln(2+\\sqrt{5}) \\approx 1.4436[\/latex]<\/p>\n<p id=\"fs-id1170572176029\">[latex]\\tanh^{-1}(\\frac{1}{4})=\\frac{1}{2}\\ln(\\frac{1+1\/4}{1-1\/4})=\\frac{1}{2}\\ln(\\frac{5\/4}{3\/4})=\\frac{1}{2}\\ln(\\frac{5}{3}) \\approx 0.2554[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572176157\" class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170572176165\">Evaluate [latex]\\tanh^{-1}\\left(\\frac{1}{2}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q3088722\">Hint<\/span><\/p>\n<div id=\"q3088722\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the definition of [latex]\\tanh^{-1} x[\/latex] and simplify.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572176215\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572176215\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{1}{2}\\ln(3) \\approx 0.5493[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-167\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>1.5 Exponential and Logarithmic Functions. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":24,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"1.5 Exponential and Logarithmic 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