{"id":2205,"date":"2018-01-11T21:34:01","date_gmt":"2018-01-11T21:34:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/calculus-of-the-hyperbolic-functions\/"},"modified":"2018-02-14T18:57:20","modified_gmt":"2018-02-14T18:57:20","slug":"calculus-of-the-hyperbolic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/chapter\/calculus-of-the-hyperbolic-functions\/","title":{"raw":"6.9 Calculus of the Hyperbolic Functions","rendered":"6.9 Calculus of the Hyperbolic Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Apply the formulas for derivatives and integrals of the hyperbolic functions.<\/li>\r\n \t<li>Apply the formulas for the derivatives of the inverse hyperbolic functions and their associated integrals.<\/li>\r\n \t<li>Describe the common applied conditions of a catenary curve.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1167793385448\">We were introduced to hyperbolic functions in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/introduction\/\">Introduction to Functions and Graphs<\/a>, along with some of their basic properties. In this section, we look at differentiation and integration formulas for the hyperbolic functions and their inverses.<\/p>\r\n\r\n<div id=\"fs-id1167793586339\" class=\"bc-section section\">\r\n<h1>Derivatives and Integrals of the Hyperbolic Functions<\/h1>\r\n<p id=\"fs-id1167793637684\">Recall that the hyperbolic sine and hyperbolic cosine are defined as<\/p>\r\n\r\n<div id=\"fs-id1167793288534\" class=\"equation unnumbered\">[latex]\\text{sinh}x=\\frac{{e}^{x}-{e}^{\\text{\u2212}x}}{2}\\text{ and }\\text{cosh}x=\\frac{{e}^{x}+{e}^{\\text{\u2212}x}}{2}.[\/latex]<\/div>\r\n<p id=\"fs-id1167793233791\">The other hyperbolic functions are then defined in terms of [latex]\\text{sinh}x[\/latex] and [latex]\\text{cosh}x.[\/latex] The graphs of the hyperbolic functions are shown in the following figure.<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_06_09_001\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"958\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213342\/CNX_Calc_Figure_06_09_001.jpg\" alt=\"This figure has six graphs. The first graph labeled \u201ca\u201d is of the function y=sinh(x). It is an increasing function from the 3rd quadrant, through the origin to the first quadrant. The second graph is labeled \u201cb\u201d and is of the function y=cosh(x). It decreases in the second quadrant to the intercept y=1, then becomes an increasing function. The third graph labeled \u201cc\u201d is of the function y=tanh(x). It is an increasing function from the third quadrant, through the origin, to the first quadrant. The fourth graph is labeled \u201cd\u201d and is of the function y=coth(x). It has two pieces, one in the third quadrant and one in the first quadrant with a vertical asymptote at the y-axis. The fifth graph is labeled \u201ce\u201d and is of the function y=sech(x). It is a curve above the x-axis, increasing in the second quadrant, to the y-axis at y=1 and then decreases. The sixth graph is labeled \u201cf\u201d and is of the function y=csch(x). It has two pieces, one in the third quadrant and one in the first quadrant with a vertical asymptote at the y-axis.\" width=\"958\" height=\"749\" \/> Figure 1. Graphs of the hyperbolic functions.[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1167793925306\">It is easy to develop differentiation formulas for the hyperbolic functions. For example, looking at [latex]\\text{sinh}x[\/latex] we have<\/p>\r\n\r\n<div id=\"fs-id1167794049237\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill \\frac{d}{dx}(\\text{sinh}x)&amp; =\\frac{d}{dx}(\\frac{{e}^{x}-{e}^{\\text{\u2212}x}}{2})\\hfill \\\\ &amp; =\\frac{1}{2}\\left[\\frac{d}{dx}({e}^{x})-\\frac{d}{dx}({e}^{\\text{\u2212}x})\\right]\\hfill \\\\ &amp; =\\frac{1}{2}\\left[{e}^{x}+{e}^{\\text{\u2212}x}\\right]=\\text{cosh}x.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1167793278441\">Similarly, [latex](d\\text{\/}dx)\\text{cosh}x=\\text{sinh}x.[\/latex] We summarize the differentiation formulas for the hyperbolic functions in the following table.<\/p>\r\n\r\n<table id=\"fs-id1167793363804\" summary=\"This is a table of two columns. The first column is labeled f(x). Its entries are the hyperbolic trigonometric functions. The second column is labeled d\/dx f(x) and is the corresponding derivatives of the hyperbolic trigonometric functions.\"><caption>Derivatives of the Hyperbolic Functions<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]f(x)[\/latex]<\/th>\r\n<th>[latex]\\frac{d}{dx}f(x)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]\\text{sinh}x[\/latex]<\/td>\r\n<td>[latex]\\text{cosh}x[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\text{cosh}x[\/latex]<\/td>\r\n<td>[latex]\\text{sinh}x[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\text{tanh}x[\/latex]<\/td>\r\n<td>[latex]{\\text{sech}}^{2}x[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\text{coth}x[\/latex]<\/td>\r\n<td>[latex]\\text{\u2212}{\\text{csch}}^{2}x[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\text{sech}x[\/latex]<\/td>\r\n<td>[latex]\\text{\u2212}\\text{sech}x\\text{tanh}x[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\text{csch}x[\/latex]<\/td>\r\n<td>[latex]\\text{\u2212}\\text{csch}x\\text{coth}x[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167794139383\">Let\u2019s take a moment to compare the derivatives of the hyperbolic functions with the derivatives of the standard trigonometric functions. There are a lot of similarities, but differences as well. For example, the derivatives of the sine functions match: [latex](d\\text{\/}dx) \\sin x= \\cos x[\/latex] and [latex](d\\text{\/}dx)\\text{sinh}x=\\text{cosh}x.[\/latex] The derivatives of the cosine functions, however, differ in sign: [latex](d\\text{\/}dx) \\cos x=\\text{\u2212} \\sin x,[\/latex] but [latex](d\\text{\/}dx)\\text{cosh}x=\\text{sinh}x.[\/latex] As we continue our examination of the hyperbolic functions, we must be mindful of their similarities and differences to the standard trigonometric functions.<\/p>\r\n<p id=\"fs-id1167794334421\">These differentiation formulas for the hyperbolic functions lead directly to the following integral formulas.<\/p>\r\n\r\n<div id=\"fs-id1167793944458\" class=\"equation unnumbered\">[latex]\\begin{array}{cccccccc}\\hfill \\int \\text{sinh}udu&amp; =\\hfill &amp; \\text{cosh}u+C\\hfill &amp; &amp; &amp; \\hfill \\int {\\text{csch}}^{2}udu&amp; =\\hfill &amp; \\text{\u2212}\\text{coth}u+C\\hfill \\\\ \\hfill \\int \\text{cosh}udu&amp; =\\hfill &amp; \\text{sinh}u+C\\hfill &amp; &amp; &amp; \\hfill \\int \\text{sech}u\\text{tanh}udu&amp; =\\hfill &amp; \\text{\u2212}\\text{sech}u+C\\hfill \\\\ \\hfill \\int {\\text{sech}}^{2}udu&amp; =\\hfill &amp; \\text{tanh}u+C\\hfill &amp; &amp; &amp; \\hfill \\int \\text{csch}u\\text{coth}udu&amp; =\\hfill &amp; \\text{\u2212}\\text{csch}u+C\\hfill \\end{array}[\/latex]<\/div>\r\n<div id=\"fs-id1167793852238\" class=\"textbox examples\">\r\n<h3>Differentiating Hyperbolic Functions<\/h3>\r\n<div id=\"fs-id1167793936071\" class=\"exercise\">\r\n<div id=\"fs-id1167793876199\" class=\"textbox\">\r\n<p id=\"fs-id1167793278457\">Evaluate the following derivatives:<\/p>\r\n\r\n<ol id=\"fs-id1167794060382\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dx}(\\text{sinh}({x}^{2}))[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dx}{(\\text{cosh}x)}^{2}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167794098810\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794098810\"]\r\n<p id=\"fs-id1167794098810\">Using the formulas in <a class=\"autogenerated-content\" href=\"#fs-id1167793363804\">(Figure)<\/a> and the chain rule, we get<\/p>\r\n\r\n<ol id=\"fs-id1167793500488\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dx}(\\text{sinh}({x}^{2}))=\\text{cosh}({x}^{2})\u00b72x[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dx}{(\\text{cosh}x)}^{2}=2\\text{cosh}x\\text{sinh}x[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793970793\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1167794145876\" class=\"exercise\">\r\n<div id=\"fs-id1167794126285\" class=\"textbox\">\r\n<p id=\"fs-id1167794028505\">Evaluate the following derivatives:<\/p>\r\n\r\n<ol id=\"fs-id1167794043812\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dx}(\\text{tanh}({x}^{2}+3x))[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dx}(\\frac{1}{{(\\text{sinh}x)}^{2}})[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1167794187159\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1167794187159\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794187159\"]\r\n<ol style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dx}(\\text{tanh}({x}^{2}+3x))=({\\text{sech}}^{2}({x}^{2}+3x))(2x+3)[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dx}(\\frac{1}{{(\\text{sinh}x)}^{2}})=\\frac{d}{dx}{(\\text{sinh}x)}^{-2}=-2{(\\text{sinh}x)}^{-3}\\text{cosh}x[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1167793372558\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1167794072646\">Use the formulas in <a class=\"autogenerated-content\" href=\"#fs-id1167793363804\">(Figure)<\/a> and apply the chain rule as necessary.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793279424\" class=\"textbox examples\">\r\n<h3>Integrals Involving Hyperbolic Functions<\/h3>\r\n<div id=\"fs-id1167794051555\" class=\"exercise\">\r\n<div id=\"fs-id1167794127457\" class=\"textbox\">\r\n<p id=\"fs-id1167793965336\">Evaluate the following integrals:<\/p>\r\n\r\n<ol id=\"fs-id1167794050806\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\int x\\text{cosh}({x}^{2})dx[\/latex]<\/li>\r\n \t<li>[latex]\\int \\text{tanh}xdx[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793254882\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793254882\"]\r\n<p id=\"fs-id1167793254882\">We can use [latex]u[\/latex]-substitution in both cases.<\/p>\r\n\r\n<ol id=\"fs-id1167794042760\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Let [latex]u={x}^{2}.[\/latex] Then, [latex]du=2xdx[\/latex] and\r\n<div id=\"fs-id1167794337482\" class=\"equation unnumbered\">[latex]\\int x\\text{cosh}({x}^{2})dx=\\int \\frac{1}{2}\\text{cosh}udu=\\frac{1}{2}\\text{sinh}u+C=\\frac{1}{2}\\text{sinh}({x}^{2})+C.[\/latex]<\/div><\/li>\r\n \t<li>Let [latex]u=\\text{cosh}x.[\/latex] Then, [latex]du=\\text{sinh}xdx[\/latex] and\r\n<div id=\"fs-id1167794043265\" class=\"equation unnumbered\">[latex]\\int \\text{tanh}xdx=\\int \\frac{\\text{sinh}x}{\\text{cosh}x}dx=\\int \\frac{1}{u}du=\\text{ln}|u|+C=\\text{ln}|\\text{cosh}x|+C.[\/latex]<\/div>\r\nNote that [latex]\\text{cosh}x&gt;0[\/latex] for all [latex]x,[\/latex] so we can eliminate the absolute value signs and obtain\r\n<div id=\"fs-id1167793358384\" class=\"equation unnumbered\">[latex]\\int \\text{tanh}xdx=\\text{ln}(\\text{cosh}x)+C.[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793504384\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1167794146347\" class=\"exercise\">\r\n<div id=\"fs-id1167794330193\" class=\"textbox\">\r\n<p id=\"fs-id1167793819956\">Evaluate the following integrals:<\/p>\r\n\r\n<ol id=\"fs-id1167793985861\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\int {\\text{sinh}}^{3}x\\text{cosh}xdx[\/latex]<\/li>\r\n \t<li>[latex]\\int {\\text{sech}}^{2}(3x)dx[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793373829\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793373829\"]\r\n<ol id=\"fs-id1167793373829\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\int {\\text{sinh}}^{3}x\\text{cosh}xdx=\\frac{{\\text{sinh}}^{4}x}{4}+C[\/latex]<\/li>\r\n \t<li>[latex]\\int {\\text{sech}}^{2}(3x)dx=\\frac{\\text{tanh}(3x)}{3}+C[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1167793245855\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1167794199986\">Use the formulas above and apply [latex]u[\/latex]-substitution as necessary.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794077052\" class=\"bc-section section\">\r\n<h1>Calculus of Inverse Hyperbolic Functions<\/h1>\r\nLooking at the graphs of the hyperbolic functions, we see that with appropriate range restrictions, they all have inverses. Most of the necessary range restrictions can be discerned by close examination of the graphs. The domains and ranges of the inverse hyperbolic functions are summarized in the following table.\r\n<table id=\"fs-id1167793362111\" summary=\"This table has three columns. The first column is labeled function and has the inverse hyperbolic functions listed in the column. The second column is labeled domain and has the domains of the inverse hyperbolic functions. The third column is labeled range and has the ranges of the inverse hyperbolic functions.\"><caption>Domains and Ranges of the Inverse Hyperbolic Functions<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Function<\/th>\r\n<th>Domain<\/th>\r\n<th>Range<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]{\\text{sinh}}^{-1}x[\/latex]<\/td>\r\n<td>[latex](\\text{\u2212}\\infty ,\\infty )[\/latex]<\/td>\r\n<td>[latex](\\text{\u2212}\\infty ,\\infty )[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]{\\text{cosh}}^{-1}x[\/latex]<\/td>\r\n<td>[latex](1,\\infty )[\/latex]<\/td>\r\n<td>[latex]\\left[0,\\infty )[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]{\\text{tanh}}^{-1}x[\/latex]<\/td>\r\n<td>[latex](-1,1)[\/latex]<\/td>\r\n<td>[latex](\\text{\u2212}\\infty ,\\infty )[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]{\\text{coth}}^{-1}x[\/latex]<\/td>\r\n<td>[latex](\\text{\u2212}\\infty ,-1)\\cup (1,\\infty )[\/latex]<\/td>\r\n<td>[latex](\\text{\u2212}\\infty ,0)\\cup (0,\\infty )[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]{\\text{sech}}^{-1}x[\/latex]<\/td>\r\n<td>[latex](0\\text{, 1})[\/latex]<\/td>\r\n<td>[latex]\\left[0,\\infty )[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]{\\text{csch}}^{-1}x[\/latex]<\/td>\r\n<td>[latex](\\text{\u2212}\\infty ,0)\\cup (0,\\infty )[\/latex]<\/td>\r\n<td>[latex](\\text{\u2212}\\infty ,0)\\cup (0,\\infty )[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167793998256\">The graphs of the inverse hyperbolic functions are shown in the following figure.<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_06_09_002\" class=\"wp-caption aligncenter\"><span id=\"fs-id1167793245836\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213346\/CNX_Calc_Figure_06_09_002.jpg\" alt=\"This figure has six graphs. The first graph labeled \u201ca\u201d is of the function y=sinh^-1(x). It is an increasing function from the 3rd quadrant, through the origin to the first quadrant. The second graph is labeled \u201cb\u201d and is of the function y=cosh^-1(x). It is in the first quadrant, beginning on the x-axis at 2 and increasing. The third graph labeled \u201cc\u201d is of the function y=tanh^-1(x). It is an increasing function from the third quadrant, through the origin, to the first quadrant. The fourth graph is labeled \u201cd\u201d and is of the function y=coth^-1(x). It has two pieces, one in the third quadrant and one in the first quadrant with a vertical asymptote at the y-axis. The fifth graph is labeled \u201ce\u201d and is of the function y=sech^-1(x). It is a curve decreasing in the first quadrant and stopping on the x-axis at x=1. The sixth graph is labeled \u201cf\u201d and is of the function y=csch^-1(x). It has two pieces, one in the third quadrant and one in the first quadrant with a vertical asymptote at the y-axis.\" \/><\/span><\/div>\r\n<p id=\"fs-id1167793259261\">To find the derivatives of the inverse functions, we use implicit differentiation. We have<\/p>\r\n\r\n<div id=\"fs-id1167793959214\" class=\"equation unnumbered\">[latex]\\begin{array}{ccc}\\hfill y&amp; =\\hfill &amp; {\\text{sinh}}^{-1}x\\hfill \\\\ \\hfill \\text{sinh}y&amp; =\\hfill &amp; x\\hfill \\\\ \\hfill \\frac{d}{dx}\\text{sinh}y&amp; =\\hfill &amp; \\frac{d}{dx}x\\hfill \\\\ \\hfill \\text{cosh}y\\frac{dy}{dx}&amp; =\\hfill &amp; 1.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1167793589391\">Recall that [latex]{\\text{cosh}}^{2}y-{\\text{sinh}}^{2}y=1,[\/latex] so [latex]\\text{cosh}y=\\sqrt{1+{\\text{sinh}}^{2}y}.[\/latex] Then,<\/p>\r\n\r\n<div id=\"fs-id1167794042534\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}=\\frac{1}{\\text{cosh}y}=\\frac{1}{\\sqrt{1+{\\text{sinh}}^{2}y}}=\\frac{1}{\\sqrt{1+{x}^{2}}}.[\/latex]<\/div>\r\n<p id=\"fs-id1167793638871\">We can derive differentiation formulas for the other inverse hyperbolic functions in a similar fashion. These differentiation formulas are summarized in the following table.<\/p>\r\n\r\n<table id=\"fs-id1167794020892\" summary=\"This table has two columns. The first column is labeled f(x) and has the inverse hyperbolic functions as entries. The second column is labeled d\/dx f(x) and is the derivatives of the inverse hyperbolic functions.\"><caption>Derivatives of the Inverse Hyperbolic Functions<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]f(x)[\/latex]<\/th>\r\n<th>[latex]\\frac{d}{dx}f(x)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]{\\text{sinh}}^{-1}x[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{\\sqrt{1+{x}^{2}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]{\\text{cosh}}^{-1}x[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{\\sqrt{{x}^{2}-1}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]{\\text{tanh}}^{-1}x[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{1-{x}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]{\\text{coth}}^{-1}x[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{1-{x}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]{\\text{sech}}^{-1}x[\/latex]<\/td>\r\n<td>[latex]\\frac{-1}{x\\sqrt{1-{x}^{2}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]{\\text{csch}}^{-1}x[\/latex]<\/td>\r\n<td>[latex]\\frac{-1}{|x|\\sqrt{1+{x}^{2}}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167794119173\">Note that the derivatives of [latex]{\\text{tanh}}^{-1}x[\/latex] and [latex]{\\text{coth}}^{-1}x[\/latex] are the same. Thus, when we integrate [latex]1\\text{\/}(1-{x}^{2}),[\/latex] we need to select the proper antiderivative based on the domain of the functions and the values of [latex]x.[\/latex] Integration formulas involving the inverse hyperbolic functions are summarized as follows.<\/p>\r\n\r\n<div id=\"fs-id1167793951927\" class=\"equation unnumbered\">[latex]\\begin{array}{cccccccc}\\hfill \\int \\frac{1}{\\sqrt{1+{u}^{2}}}du&amp; =\\hfill &amp; {\\text{sinh}}^{-1}u+C\\hfill &amp; &amp; &amp; \\hfill \\int \\frac{1}{u\\sqrt{1-{u}^{2}}}du&amp; =\\hfill &amp; \\text{\u2212}{\\text{sech}}^{-1}|u|+C\\hfill \\\\ \\hfill \\int \\frac{1}{\\sqrt{{u}^{2}-1}}du&amp; =\\hfill &amp; {\\text{cosh}}^{-1}u+C\\hfill &amp; &amp; &amp; \\hfill \\int \\frac{1}{u\\sqrt{1+{u}^{2}}}du&amp; =\\hfill &amp; \\text{\u2212}{\\text{csch}}^{-1}|u|+C\\hfill \\\\ \\hfill \\int \\frac{1}{1-{u}^{2}}du&amp; =\\hfill &amp; \\bigg\\{\\begin{array}{c}{\\text{tanh}}^{-1}u+C\\text{ if }|u|&lt;1\\hfill \\\\ {\\text{coth}}^{-1}u+C\\text{ if }|u|&gt;1\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\end{array}[\/latex]<\/div>\r\n<div id=\"fs-id1167793984979\" class=\"textbox examples\">\r\n<h3>Differentiating Inverse Hyperbolic Functions<\/h3>\r\n<div id=\"fs-id1167793984981\" class=\"exercise\">\r\n<div id=\"fs-id1167793829064\" class=\"textbox\">\r\n<p id=\"fs-id1167793372563\">Evaluate the following derivatives:<\/p>\r\n\r\n<ol id=\"fs-id1167794155431\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dx}({\\text{sinh}}^{-1}(\\frac{x}{3}))[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dx}{({\\text{tanh}}^{-1}x)}^{2}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793949787\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793949787\"]\r\n<p id=\"fs-id1167793949787\">Using the formulas in <a class=\"autogenerated-content\" href=\"#fs-id1167794020892\">(Figure)<\/a> and the chain rule, we obtain the following results:<\/p>\r\n\r\n<ol id=\"fs-id1167793298499\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dx}({\\text{sinh}}^{-1}(\\frac{x}{3}))=\\frac{1}{3\\sqrt{1+\\frac{{x}^{2}}{9}}}=\\frac{1}{\\sqrt{9+{x}^{2}}}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dx}{({\\text{tanh}}^{-1}x)}^{2}=\\frac{2({\\text{tanh}}^{-1}x)}{1-{x}^{2}}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793626577\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1167793626580\" class=\"exercise\">\r\n<div id=\"fs-id1167794332393\" class=\"textbox\">\r\n<p id=\"fs-id1167794332395\">Evaluate the following derivatives:<\/p>\r\n\r\n<ol id=\"fs-id1167793590727\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dx}({\\text{cosh}}^{-1}(3x))[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dx}{({\\text{coth}}^{-1}x)}^{3}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793498614\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793498614\"]\r\n<ol id=\"fs-id1167793498614\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\frac{d}{dx}({\\text{cosh}}^{-1}(3x))=\\frac{3}{\\sqrt{9{x}^{2}-1}}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{d}{dx}{({\\text{coth}}^{-1}x)}^{3}=\\frac{3{({\\text{coth}}^{-1}x)}^{2}}{1-{x}^{2}}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1167794037907\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1167794137287\">Use the formulas in <a class=\"autogenerated-content\" href=\"#fs-id1167794020892\">(Figure)<\/a> and apply the chain rule as necessary.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793285224\" class=\"textbox examples\">\r\n<h3>Integrals Involving Inverse Hyperbolic Functions<\/h3>\r\n<div id=\"fs-id1167793285226\" class=\"exercise\">\r\n<div id=\"fs-id1167793285228\" class=\"textbox\">\r\n<p id=\"fs-id1167793871641\">Evaluate the following integrals:<\/p>\r\n\r\n<ol id=\"fs-id1167793937415\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\int \\frac{1}{\\sqrt{4{x}^{2}-1}}dx[\/latex]<\/li>\r\n \t<li>[latex]\\int \\frac{1}{2x\\sqrt{1-9{x}^{2}}}dx[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793361818\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793361818\"]\r\n<p id=\"fs-id1167793361818\">We can use [latex]u\\text{-substitution}[\/latex] in both cases.<\/p>\r\n\r\n<ol id=\"fs-id1167794296597\" style=\"list-style-type: lower-alpha\">\r\n \t<li>Let [latex]u=2x.[\/latex] Then, [latex]du=2dx[\/latex] and we have\r\n<div id=\"fs-id1167793883719\" class=\"equation unnumbered\">[latex]\\int \\frac{1}{\\sqrt{4{x}^{2}-1}}dx=\\int \\frac{1}{2\\sqrt{{u}^{2}-1}}du=\\frac{1}{2}{\\text{cosh}}^{-1}u+C=\\frac{1}{2}{\\text{cosh}}^{-1}(2x)+C.[\/latex]<\/div><\/li>\r\n \t<li>Let [latex]u=3x.[\/latex] Then, [latex]du=3dx[\/latex] and we obtain\r\n<div id=\"fs-id1167793444519\" class=\"equation unnumbered\">[latex]\\int \\frac{1}{2x\\sqrt{1-9{x}^{2}}}dx=\\frac{1}{2}\\int \\frac{1}{u\\sqrt{1-{u}^{2}}}du=-\\frac{1}{2}{\\text{sech}}^{-1}|u|+C=-\\frac{1}{2}{\\text{sech}}^{-1}|3x|+C.[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793925131\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1167793969727\" class=\"exercise\">\r\n<div id=\"fs-id1167793969729\" class=\"textbox\">\r\n<p id=\"fs-id1167793265985\">Evaluate the following integrals:<\/p>\r\n\r\n<ol id=\"fs-id1167793265989\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\int \\frac{1}{\\sqrt{{x}^{2}-4}}dx,\\text{}x&gt;2[\/latex]<\/li>\r\n \t<li>[latex]\\int \\frac{1}{\\sqrt{1-{e}^{2x}}}dx[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793371557\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793371557\"]\r\n<ol id=\"fs-id1167793371557\" style=\"list-style-type: lower-alpha\">\r\n \t<li>[latex]\\int \\frac{1}{\\sqrt{{x}^{2}-4}}dx={\\text{cosh}}^{-1}(\\frac{x}{2})+C[\/latex]<\/li>\r\n \t<li>[latex]\\int \\frac{1}{\\sqrt{1-{e}^{2x}}}dx=\\text{\u2212}{\\text{sech}}^{-1}({e}^{x})+C[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1167794333942\">Use the formulas above and apply [latex]u\\text{-substitution}[\/latex] as necessary.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794046158\" class=\"bc-section section\">\r\n<h1>Applications<\/h1>\r\n<p id=\"fs-id1167793964561\">One physical application of hyperbolic functions involves <span class=\"no-emphasis\">hanging cables<\/span>. If a cable of uniform density is suspended between two supports without any load other than its own weight, the cable forms a curve called a <strong>catenary<\/strong>. High-voltage power lines, chains hanging between two posts, and strands of a spider\u2019s web all form catenaries. The following figure shows chains hanging from a row of posts.<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_06_09_003\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213354\/CNX_Calc_Figure_06_09_003.jpg\" alt=\"An image of chains hanging between posts that all take the shape of a catenary.\" width=\"975\" height=\"731\" \/> Figure 3. Chains between these posts take the shape of a catenary. (credit: modification of work by OKFoundryCompany, Flickr)[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1167793591346\">Hyperbolic functions can be used to model catenaries. Specifically, functions of the form [latex]y=a\\text{cosh}(x\\text{\/}a)[\/latex] are catenaries. <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_06_09_004\">(Figure)<\/a> shows the graph of [latex]y=2\\text{cosh}(x\\text{\/}2).[\/latex]<\/p>\r\n\r\n<div id=\"CNX_Calc_Figure_06_09_004\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"342\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213358\/CNX_Calc_Figure_06_09_004.jpg\" alt=\"This figure is a graph. It is of the function f(x)=2cosh(x\/2). The curve decreases in the second quadrant to the y-axis. It intersects the y-axis at y=2. Then the curve becomes increasing.\" width=\"342\" height=\"347\" \/> Figure 4. A hyperbolic cosine function forms the shape of a catenary.[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1167793370895\" class=\"textbox examples\">\r\n<h3>Using a Catenary to Find the Length of a Cable<\/h3>\r\n<div id=\"fs-id1167793884153\" class=\"exercise\">\r\n<div id=\"fs-id1167793884155\" class=\"textbox\">\r\n<p id=\"fs-id1167793926505\">Assume a hanging cable has the shape [latex]10\\text{cosh}(x\\text{\/}10)[\/latex] for [latex]-15\\le x\\le 15,[\/latex] where [latex]x[\/latex] is measured in feet. Determine the length of the cable (in feet).<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793307481\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793307481\"]\r\n<p id=\"fs-id1167793307481\">Recall from Section 6.4 that the formula for arc length is<\/p>\r\n\r\n<div id=\"fs-id1167793943925\" class=\"equation unnumbered\">[latex]\\text{Arc Length}={\\int }_{a}^{b}\\sqrt{1+{\\left[{f}^{\\prime }(x)\\right]}^{2}}dx.[\/latex]<\/div>\r\n<p id=\"fs-id1167794160035\">We have [latex]f(x)=10\\text{cosh}(x\\text{\/}10),[\/latex] so [latex]{f}^{\\prime }(x)=\\text{sinh}(x\\text{\/}10).[\/latex] Then<\/p>\r\n\r\n<div id=\"fs-id1167794337061\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill \\text{Arc Length}&amp; ={\\int }_{a}^{b}\\sqrt{1+{\\left[{f}^{\\prime }(x)\\right]}^{2}}dx\\hfill \\\\ &amp; ={\\int }_{-15}^{15}\\sqrt{1+{\\text{sinh}}^{2}(\\frac{x}{10})}dx.\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1167793964837\">Now recall that [latex]1+{\\text{sinh}}^{2}x={\\text{cosh}}^{2}x,[\/latex] so we have<\/p>\r\n\r\n<div id=\"fs-id1167794333245\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill \\text{Arc Length}&amp; ={\\int }_{-15}^{15}\\sqrt{1+{\\text{sinh}}^{2}(\\frac{x}{10})}dx\\hfill \\\\ &amp; ={\\int }_{-15}^{15}\\text{cosh}(\\frac{x}{10})dx\\hfill \\\\ &amp; =10\\text{sinh}{(\\frac{x}{10})|}_{-15}^{15}=10\\left[\\text{sinh}(\\frac{3}{2})-\\text{sinh}(-\\frac{3}{2})\\right]=20\\text{sinh}(\\frac{3}{2})\\hfill \\\\ &amp; \\approx 42.586\\text{ft}\\text{.}\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794326134\" class=\"textbox exercises checkpoint\">\r\n<div id=\"fs-id1167793414872\" class=\"exercise\">\r\n<div id=\"fs-id1167793414874\" class=\"textbox\">\r\n<p id=\"fs-id1167794043358\">Assume a hanging cable has the shape [latex]15\\text{cosh}(x\\text{\/}15)[\/latex] for [latex]-20\\le x\\le 20.[\/latex] Determine the length of the cable (in feet).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1167793292342\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793292342\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793292342\"][latex]52.95\\text{ft}[\/latex]<\/div>\r\n<div id=\"fs-id1167793237822\" class=\"commentary\">\r\n<h4>Hint<\/h4>\r\n<p id=\"fs-id1167794022005\">Use the procedure from the previous example.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793269537\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1167793618899\">\r\n \t<li>Hyperbolic functions are defined in terms of exponential functions.<\/li>\r\n \t<li>Term-by-term differentiation yields differentiation formulas for the hyperbolic functions. These differentiation formulas give rise, in turn, to integration formulas.<\/li>\r\n \t<li>With appropriate range restrictions, the hyperbolic functions all have inverses.<\/li>\r\n \t<li>Implicit differentiation yields differentiation formulas for the inverse hyperbolic functions, which in turn give rise to integration formulas.<\/li>\r\n \t<li>The most common physical applications of hyperbolic functions are calculations involving catenaries.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1167794126430\" class=\"textbox exercises\">\r\n<div id=\"fs-id1167794126434\" class=\"exercise\">\r\n<div id=\"fs-id1167793940609\" class=\"textbox\">\r\n<p id=\"fs-id1167793940611\"><strong>[T]<\/strong> Find expressions for [latex]\\text{cosh}x+\\text{sinh}x[\/latex] and [latex]\\text{cosh}x-\\text{sinh}x.[\/latex] Use a calculator to graph these functions and ensure your expression is correct.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793552000\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793552000\"]\r\n<p id=\"fs-id1167793552000\">[latex]{e}^{x}\\text{ and }{e}^{\\text{\u2212}x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794051638\" class=\"exercise\">\r\n<div id=\"fs-id1167794051640\" class=\"textbox\">\r\n<p id=\"fs-id1167793930753\">From the definitions of [latex]\\text{cosh}(x)[\/latex] and [latex]\\text{sinh}(x),[\/latex] find their antiderivatives.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793956239\" class=\"exercise\">\r\n<div id=\"fs-id1167793956241\" class=\"textbox\">\r\n<p id=\"fs-id1167793956243\">Show that [latex]\\text{cosh}(x)[\/latex] and [latex]\\text{sinh}(x)[\/latex] satisfy [latex]y\\text{\u2033}=y.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793416604\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793416604\"]\r\n<p id=\"fs-id1167793416604\">Answers may vary<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793269055\" class=\"exercise\">\r\n<div id=\"fs-id1167793269058\" class=\"textbox\">\r\n\r\nUse the quotient rule to verify that [latex]\\text{tanh}(x)\\prime ={\\text{sech}}^{2}(x).[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793514512\" class=\"exercise\">\r\n<div id=\"fs-id1167793514514\" class=\"textbox\">\r\n<p id=\"fs-id1167793944608\">Derive [latex]{\\text{cosh}}^{2}(x)+{\\text{sinh}}^{2}(x)=\\text{cosh}(2x)[\/latex] from the definition.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793870399\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793870399\"]\r\n<p id=\"fs-id1167793870399\">Answers may vary<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794037737\" class=\"exercise\">\r\n<div id=\"fs-id1167794037739\" class=\"textbox\">\r\n<p id=\"fs-id1167793219307\">Take the derivative of the previous expression to find an expression for [latex]\\text{sinh}(2x).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794141049\" class=\"exercise\">\r\n<div id=\"fs-id1167794141051\" class=\"textbox\">\r\n<p id=\"fs-id1167794141053\">Prove [latex]\\text{sinh}(x+y)=\\text{sinh}(x)\\text{cosh}(y)+\\text{cosh}(x)\\text{sinh}(y)[\/latex] by changing the expression to exponentials.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167794137118\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794137118\"]\r\n<p id=\"fs-id1167794137118\">Answers may vary<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794336126\" class=\"exercise\">\r\n<div id=\"fs-id1167794336128\" class=\"textbox\">\r\n<p id=\"fs-id1167794336130\">Take the derivative of the previous expression to find an expression for [latex]\\text{cosh}(x+y).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793929899\">For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct.<\/p>\r\n\r\n<div id=\"fs-id1167793630189\" class=\"exercise\">\r\n<div id=\"fs-id1167793630191\" class=\"textbox\">\r\n<p id=\"fs-id1167793630193\"><strong>[T]<\/strong>[latex]\\text{cosh}(3x+1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793318463\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793318463\"]\r\n<p id=\"fs-id1167793318463\">[latex]3\\text{sinh}(3x+1)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793382588\" class=\"exercise\">\r\n<div id=\"fs-id1167793382590\" class=\"textbox\">\r\n<p id=\"fs-id1167793382592\"><strong>[T]<\/strong>[latex]\\text{sinh}({x}^{2})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793416764\" class=\"exercise\">\r\n<div id=\"fs-id1167794329964\" class=\"textbox\">\r\n<p id=\"fs-id1167794329966\"><strong>[T]<\/strong>[latex]\\frac{1}{\\text{cosh}(x)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793268310\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793268310\"]\r\n<p id=\"fs-id1167793268310\">[latex]\\text{\u2212}\\text{tanh}(x)\\text{sech}(x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793985952\" class=\"exercise\">\r\n<div id=\"fs-id1167793985955\" class=\"textbox\">\r\n<p id=\"fs-id1167793985957\"><strong>[T]<\/strong>[latex]\\text{sinh}(\\text{ln}(x))[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793384557\" class=\"exercise\">\r\n<div id=\"fs-id1167793384559\" class=\"textbox\">\r\n<p id=\"fs-id1167793246794\"><strong>[T]<\/strong>[latex]{\\text{cosh}}^{2}(x)+{\\text{sinh}}^{2}(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793947964\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793947964\"]\r\n<p id=\"fs-id1167793947964\">[latex]4\\text{cosh}(x)\\text{sinh}(x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793301148\" class=\"exercise\">\r\n<div id=\"fs-id1167793301150\" class=\"textbox\">\r\n<p id=\"fs-id1167793301152\"><strong>[T]<\/strong>[latex]{\\text{cosh}}^{2}(x)-{\\text{sinh}}^{2}(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794284874\" class=\"exercise\">\r\n<div id=\"fs-id1167793455832\" class=\"textbox\">\r\n<p id=\"fs-id1167793455835\"><strong>[T]<\/strong>[latex]\\text{tanh}(\\sqrt{{x}^{2}+1})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167794293324\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794293324\"]\r\n<p id=\"fs-id1167794293324\">[latex]\\frac{x{\\text{sech}}^{2}(\\sqrt{{x}^{2}+1})}{\\sqrt{{x}^{2}+1}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794199184\" class=\"exercise\">\r\n<div id=\"fs-id1167794199186\" class=\"textbox\">\r\n<p id=\"fs-id1167794199188\"><strong>[T]<\/strong>[latex]\\frac{1+\\text{tanh}(x)}{1-\\text{tanh}(x)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794043454\" class=\"exercise\">\r\n<div id=\"fs-id1167793883113\" class=\"textbox\">\r\n<p id=\"fs-id1167793883115\"><strong>[T]<\/strong>[latex]{\\text{sinh}}^{6}(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167794039862\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794039862\"]\r\n<p id=\"fs-id1167794039862\">[latex]6{\\text{sinh}}^{5}(x)\\text{cosh}(x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793637691\" class=\"exercise\">\r\n<div id=\"fs-id1167793385422\" class=\"textbox\">\r\n<p id=\"fs-id1167793385424\"><strong>[T]<\/strong>[latex]\\text{ln}(\\text{sech}(x)+\\text{tanh}(x))[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793281078\">For the following exercises, find the antiderivatives for the given functions.<\/p>\r\n\r\n<div id=\"fs-id1167793281082\" class=\"exercise\">\r\n<div id=\"fs-id1167793281084\" class=\"textbox\">\r\n<p id=\"fs-id1167793638839\">[latex]\\text{cosh}(2x+1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793372486\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793372486\"]\r\n<p id=\"fs-id1167793372486\">[latex]\\frac{1}{2}\\text{sinh}(2x+1)+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794178053\" class=\"exercise\">\r\n<div id=\"fs-id1167794178055\" class=\"textbox\">\r\n<p id=\"fs-id1167794178057\">[latex]\\text{tanh}(3x+2)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794336738\" class=\"exercise\">\r\n<div id=\"fs-id1167794336741\" class=\"textbox\">\r\n<p id=\"fs-id1167794336743\">[latex]x\\text{cosh}({x}^{2})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793950952\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793950952\"]\r\n<p id=\"fs-id1167793950952\">[latex]\\frac{1}{2}{\\text{sinh}}^{2}({x}^{2})+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793956996\" class=\"exercise\">\r\n<div id=\"fs-id1167793958193\" class=\"textbox\">\r\n<p id=\"fs-id1167793958195\">[latex]3{x}^{3}\\text{tanh}({x}^{4})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793372582\" class=\"exercise\">\r\n<div id=\"fs-id1167793316107\" class=\"textbox\">\r\n<p id=\"fs-id1167793316109\">[latex]{\\text{cosh}}^{2}(x)\\text{sinh}(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793393695\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793393695\"]\r\n<p id=\"fs-id1167793393695\">[latex]\\frac{1}{3}{\\text{cosh}}^{3}(x)+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793265440\" class=\"exercise\">\r\n<div id=\"fs-id1167793265442\" class=\"textbox\">\r\n\r\n[latex]{\\text{tanh}}^{2}(x){\\text{sech}}^{2}(x)[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793450621\" class=\"exercise\">\r\n<div id=\"fs-id1167793450623\" class=\"textbox\">\r\n<p id=\"fs-id1167793450625\">[latex]\\frac{\\text{sinh}(x)}{1+\\text{cosh}(x)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n<p id=\"fs-id1167794118195\">[latex]\\text{ln}(1+\\text{cosh}(x))+C[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793275006\" class=\"exercise\">\r\n<div id=\"fs-id1167793275008\" class=\"textbox\">\r\n<p id=\"fs-id1167793495090\">[latex]\\text{coth}(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794181011\" class=\"exercise\">\r\n<div id=\"fs-id1167794181013\" class=\"textbox\">\r\n<p id=\"fs-id1167794181016\">[latex]\\text{cosh}(x)+\\text{sinh}(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793361732\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793361732\"]\r\n<p id=\"fs-id1167793361732\">[latex]\\text{cosh}(x)+\\text{sinh}(x)+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1167793564129\" class=\"textbox\">\r\n<p id=\"fs-id1167793564131\">[latex]{(\\text{cosh}(x)+\\text{sinh}(x))}^{n}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793951599\">For the following exercises, find the derivatives for the functions.<\/p>\r\n\r\n<div id=\"fs-id1167793372386\" class=\"exercise\">\r\n<div id=\"fs-id1167793372388\" class=\"textbox\">\r\n<p id=\"fs-id1167793372390\">[latex]{\\text{tanh}}^{-1}(4x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793948851\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793948851\"]\r\n<p id=\"fs-id1167793948851\">[latex]\\frac{4}{1-16{x}^{2}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794206992\" class=\"exercise\">\r\n<div id=\"fs-id1167794206994\" class=\"textbox\">\r\n<p id=\"fs-id1167794206996\">[latex]{\\text{sinh}}^{-1}({x}^{2})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793936665\" class=\"exercise\">\r\n<div id=\"fs-id1167793936667\" class=\"textbox\">\r\n<p id=\"fs-id1167794329290\">[latex]{\\text{sinh}}^{-1}(\\text{cosh}(x))[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167794207012\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794207012\"]\r\n<p id=\"fs-id1167794207012\">[latex]\\frac{\\text{sinh}(x)}{\\sqrt{{\\text{cosh}}^{2}(x)+1}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793358453\" class=\"exercise\">\r\n<div id=\"fs-id1167793358455\" class=\"textbox\">\r\n<p id=\"fs-id1167793605572\">[latex]{\\text{cosh}}^{-1}({x}^{3})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793940311\" class=\"exercise\">\r\n<div id=\"fs-id1167793940313\" class=\"textbox\">\r\n<p id=\"fs-id1167793940316\">[latex]{\\text{tanh}}^{-1}( \\cos (x))[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793543556\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793543556\"]\r\n<p id=\"fs-id1167793543556\">[latex]\\text{\u2212} \\csc (x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793928572\" class=\"exercise\">\r\n<div id=\"fs-id1167793928574\" class=\"textbox\">\r\n<p id=\"fs-id1167793928576\">[latex]{e}^{{\\text{sinh}}^{-1}(x)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793290963\" class=\"exercise\">\r\n<div id=\"fs-id1167793290965\" class=\"textbox\">\r\n<p id=\"fs-id1167793290968\">[latex]\\text{ln}({\\text{tanh}}^{-1}(x))[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"solution\">\r\n<p id=\"fs-id1167793538362\">[latex]-\\frac{1}{({x}^{2}-1){\\text{tanh}}^{-1}(x)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793509961\">For the following exercises, find the antiderivatives for the functions.<\/p>\r\n\r\n<div class=\"exercise\">\r\n<div id=\"fs-id1167793579580\" class=\"textbox\">\r\n<p id=\"fs-id1167793579582\">[latex]\\int \\frac{dx}{4-{x}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794058014\" class=\"exercise\">\r\n<div id=\"fs-id1167794058016\" class=\"textbox\">\r\n<p id=\"fs-id1167794058019\">[latex]\\int \\frac{dx}{{a}^{2}-{x}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793399888\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793399888\"]\r\n<p id=\"fs-id1167793399888\">[latex]\\frac{1}{a}{\\text{tanh}}^{-1}(\\frac{x}{a})+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793420648\" class=\"exercise\">\r\n<div id=\"fs-id1167793420650\" class=\"textbox\">\r\n<p id=\"fs-id1167793421226\">[latex]\\int \\frac{dx}{\\sqrt{{x}^{2}+1}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793950155\" class=\"exercise\">\r\n<div id=\"fs-id1167793950157\" class=\"textbox\">\r\n<p id=\"fs-id1167793590468\">[latex]\\int \\frac{xdx}{\\sqrt{{x}^{2}+1}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793886745\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793886745\"]\r\n<p id=\"fs-id1167793886745\">[latex]\\sqrt{{x}^{2}+1}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793569521\" class=\"exercise\">\r\n<div id=\"fs-id1167793569523\" class=\"textbox\">\r\n\r\n[latex]\\int -\\frac{dx}{x\\sqrt{1-{x}^{2}}}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793510599\" class=\"exercise\">\r\n<div id=\"fs-id1167793510601\" class=\"textbox\">\r\n<p id=\"fs-id1167793510603\">[latex]\\int \\frac{{e}^{x}}{\\sqrt{{e}^{2x}-1}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1167793582489\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793582489\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793582489\"][latex]{\\text{cosh}}^{-1}({e}^{x})+C[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794058783\" class=\"exercise\">\r\n<div id=\"fs-id1167794058785\" class=\"textbox\">\r\n\r\n[latex]\\int -\\frac{2x}{{x}^{4}-1}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793384924\">For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation [latex]dv\\text{\/}dt=g-{v}^{2}.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1167793276964\" class=\"exercise\">\r\n<div id=\"fs-id1167793276967\" class=\"textbox\">\r\n<p id=\"fs-id1167793276969\">Show that [latex]v(t)=\\sqrt{g}\\text{tanh}(\\sqrt{gt})[\/latex] satisfies this equation.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793255883\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793255883\"]\r\n<p id=\"fs-id1167793255883\">Answers may vary<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793255888\" class=\"exercise\">\r\n<div id=\"fs-id1167793607690\" class=\"textbox\">\r\n<p id=\"fs-id1167793607692\">Derive the previous expression for [latex]v(t)[\/latex] by integrating [latex]\\frac{dv}{g-{v}^{2}}=dt.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793316091\" class=\"exercise\">\r\n<div id=\"fs-id1167793309335\" class=\"textbox\">\r\n<p id=\"fs-id1167793309337\"><strong>[T]<\/strong> Estimate how far a body has fallen in 12 seconds by finding the area underneath the curve of [latex]v(t).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1167794058934\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1167794058934\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794058934\"]37.30[\/hidden-answer]<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167794058944\">For the following exercises, use this scenario: A cable hanging under its own weight has a slope [latex]S=dy\\text{\/}dx[\/latex] that satisfies [latex]dS\\text{\/}dx=c\\sqrt{1+{S}^{2}}.[\/latex] The constant [latex]c[\/latex] is the ratio of cable density to tension.<\/p>\r\n\r\n<div id=\"fs-id1167793518527\" class=\"exercise\">\r\n<div id=\"fs-id1167793518530\" class=\"textbox\">\r\n<p id=\"fs-id1167793518532\">Show that [latex]S=\\text{sinh}(cx)[\/latex] satisfies this equation.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793619929\" class=\"exercise\">\r\n<div id=\"fs-id1167793619931\" class=\"textbox\">\r\n<p id=\"fs-id1167793619933\">Integrate [latex]dy\\text{\/}dx=\\text{sinh}(cx)[\/latex] to find the cable height [latex]y(x)[\/latex] if [latex]y(0)=1\\text{\/}c.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167794291509\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794291509\"]\r\n<p id=\"fs-id1167794291509\">[latex]y=\\frac{1}{c}\\text{cosh}(cx)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793932271\" class=\"exercise\">\r\n<div id=\"fs-id1167793932273\" class=\"textbox\">\r\n<p id=\"fs-id1167793932275\">Sketch the cable and determine how far down it sags at [latex]x=0.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793510618\">For the following exercises, solve each problem.<\/p>\r\n\r\n<div id=\"fs-id1167793510621\" class=\"exercise\">\r\n<div id=\"fs-id1167793510623\" class=\"textbox\">\r\n<p id=\"fs-id1167793510625\"><strong>[T]<\/strong> A chain hangs from two posts 2 m apart to form a catenary described by the equation [latex]y=2\\text{cosh}(x\\text{\/}2)-1.[\/latex] Find the slope of the catenary at the left fence post.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793463064\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793463064\"]\r\n<p id=\"fs-id1167793463064\">-0.521095<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793463073\" class=\"exercise\">\r\n<div id=\"fs-id1167793463075\" class=\"textbox\">\r\n<p id=\"fs-id1167793463077\"><strong>[T]<\/strong> A chain hangs from two posts four meters apart to form a catenary described by the equation [latex]y=4\\text{cosh}(x\\text{\/}4)-3.[\/latex] Find the total length of the catenary (arc length).<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793641978\" class=\"exercise\">\r\n<div id=\"fs-id1167793641980\" class=\"textbox\">\r\n<p id=\"fs-id1167793641982\"><strong>[T]<\/strong> A high-voltage power line is a catenary described by [latex]y=10\\text{cosh}(x\\text{\/}10).[\/latex] Find the ratio of the area under the catenary to its arc length. What do you notice?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1167793956533\" class=\"textbox shaded\">[reveal-answer q=\"fs-id1167793956533\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793956533\"]10[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n\r\nA telephone line is a catenary described by [latex]y=a\\text{cosh}(x\\text{\/}a).[\/latex] Find the ratio of the area under the catenary to its arc length. Does this confirm your answer for the previous question?\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793929410\" class=\"exercise\">\r\n<div class=\"textbox\">\r\n<p id=\"fs-id1167793929415\">Prove the formula for the derivative of [latex]y={\\text{sinh}}^{-1}(x)[\/latex] by differentiating [latex]x=\\text{sinh}(y).[\/latex] (<em>Hint:<\/em> Use hyperbolic trigonometric identities.)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793950965\" class=\"exercise\">\r\n<div id=\"fs-id1167793950967\" class=\"textbox\">\r\n<p id=\"fs-id1167793950969\">Prove the formula for the derivative of [latex]y={\\text{cosh}}^{-1}(x)[\/latex] by differentiating [latex]x=\\text{cosh}(y).[\/latex]<\/p>\r\n<p id=\"fs-id1167793395478\">(<em>Hint:<\/em> Use hyperbolic trigonometric identities.)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793395488\" class=\"exercise\">\r\n<div id=\"fs-id1167793395490\" class=\"textbox\">\r\n<p id=\"fs-id1167793395493\">Prove the formula for the derivative of [latex]y={\\text{sech}}^{-1}(x)[\/latex] by differentiating [latex]x=\\text{sech}(y).[\/latex] (<em>Hint:<\/em> Use hyperbolic trigonometric identities.)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167794210494\" class=\"exercise\">\r\n<div id=\"fs-id1167794210496\" class=\"textbox\">\r\n<p id=\"fs-id1167794210498\">Prove that [latex]{(\\text{cosh}(x)+\\text{sinh}(x))}^{n}=\\text{cosh}(nx)+\\text{sinh}(nx).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1167793379933\" class=\"exercise\">\r\n<div id=\"fs-id1167793379935\" class=\"textbox\">\r\n<p id=\"fs-id1167793379937\">Prove the expression for [latex]{\\text{sinh}}^{-1}(x).[\/latex] Multiply [latex]x=\\text{sinh}(y)=(1\\text{\/}2)({e}^{y}-{e}^{\\text{\u2212}y})[\/latex] by [latex]2{e}^{y}[\/latex] and solve for [latex]y.[\/latex] Does your expression match the textbook?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"exercise\">\r\n<div class=\"textbox\">\r\n<p id=\"fs-id1167793372779\">Prove the expression for [latex]{\\text{cosh}}^{-1}(x).[\/latex] Multiply [latex]x=\\text{cosh}(y)=(1\\text{\/}2)({e}^{y}-{e}^{\\text{\u2212}y})[\/latex] by [latex]2{e}^{y}[\/latex] and solve for [latex]y.[\/latex] Does your expression match the textbook?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"review-exercises\"><\/div>\r\n<div class=\"textbox shaded\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1167793432037\" class=\"definition\">\r\n \t<dt>catenary<\/dt>\r\n \t<dd id=\"fs-id1167793432042\">a curve in the shape of the function [latex]y=a\\text{cosh}(x\\text{\/}a)[\/latex] is a catenary; a cable of uniform density suspended between two supports assumes the shape of a catenary<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Apply the formulas for derivatives and integrals of the hyperbolic functions.<\/li>\n<li>Apply the formulas for the derivatives of the inverse hyperbolic functions and their associated integrals.<\/li>\n<li>Describe the common applied conditions of a catenary curve.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1167793385448\">We were introduced to hyperbolic functions in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/introduction\/\">Introduction to Functions and Graphs<\/a>, along with some of their basic properties. In this section, we look at differentiation and integration formulas for the hyperbolic functions and their inverses.<\/p>\n<div id=\"fs-id1167793586339\" class=\"bc-section section\">\n<h1>Derivatives and Integrals of the Hyperbolic Functions<\/h1>\n<p id=\"fs-id1167793637684\">Recall that the hyperbolic sine and hyperbolic cosine are defined as<\/p>\n<div id=\"fs-id1167793288534\" class=\"equation unnumbered\">[latex]\\text{sinh}x=\\frac{{e}^{x}-{e}^{\\text{\u2212}x}}{2}\\text{ and }\\text{cosh}x=\\frac{{e}^{x}+{e}^{\\text{\u2212}x}}{2}.[\/latex]<\/div>\n<p id=\"fs-id1167793233791\">The other hyperbolic functions are then defined in terms of [latex]\\text{sinh}x[\/latex] and [latex]\\text{cosh}x.[\/latex] The graphs of the hyperbolic functions are shown in the following figure.<\/p>\n<div id=\"CNX_Calc_Figure_06_09_001\" class=\"wp-caption aligncenter\">\n<div style=\"width: 968px\" 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\/11213342\/CNX_Calc_Figure_06_09_001.jpg\" alt=\"This figure has six graphs. The first graph labeled \u201ca\u201d is of the function y=sinh(x). It is an increasing function from the 3rd quadrant, through the origin to the first quadrant. The second graph is labeled \u201cb\u201d and is of the function y=cosh(x). It decreases in the second quadrant to the intercept y=1, then becomes an increasing function. The third graph labeled \u201cc\u201d is of the function y=tanh(x). It is an increasing function from the third quadrant, through the origin, to the first quadrant. The fourth graph is labeled \u201cd\u201d and is of the function y=coth(x). It has two pieces, one in the third quadrant and one in the first quadrant with a vertical asymptote at the y-axis. The fifth graph is labeled \u201ce\u201d and is of the function y=sech(x). It is a curve above the x-axis, increasing in the second quadrant, to the y-axis at y=1 and then decreases. The sixth graph is labeled \u201cf\u201d and is of the function y=csch(x). It has two pieces, one in the third quadrant and one in the first quadrant with a vertical asymptote at the y-axis.\" width=\"958\" height=\"749\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. Graphs of the hyperbolic functions.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793925306\">It is easy to develop differentiation formulas for the hyperbolic functions. For example, looking at [latex]\\text{sinh}x[\/latex] we have<\/p>\n<div id=\"fs-id1167794049237\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill \\frac{d}{dx}(\\text{sinh}x)& =\\frac{d}{dx}(\\frac{{e}^{x}-{e}^{\\text{\u2212}x}}{2})\\hfill \\\\ & =\\frac{1}{2}\\left[\\frac{d}{dx}({e}^{x})-\\frac{d}{dx}({e}^{\\text{\u2212}x})\\right]\\hfill \\\\ & =\\frac{1}{2}\\left[{e}^{x}+{e}^{\\text{\u2212}x}\\right]=\\text{cosh}x.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167793278441\">Similarly, [latex](d\\text{\/}dx)\\text{cosh}x=\\text{sinh}x.[\/latex] We summarize the differentiation formulas for the hyperbolic functions in the following table.<\/p>\n<table id=\"fs-id1167793363804\" summary=\"This is a table of two columns. The first column is labeled f(x). Its entries are the hyperbolic trigonometric functions. The second column is labeled d\/dx f(x) and is the corresponding derivatives of the hyperbolic trigonometric functions.\">\n<caption>Derivatives of the Hyperbolic Functions<\/caption>\n<thead>\n<tr valign=\"top\">\n<th>[latex]f(x)[\/latex]<\/th>\n<th>[latex]\\frac{d}{dx}f(x)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]\\text{sinh}x[\/latex]<\/td>\n<td>[latex]\\text{cosh}x[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\text{cosh}x[\/latex]<\/td>\n<td>[latex]\\text{sinh}x[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\text{tanh}x[\/latex]<\/td>\n<td>[latex]{\\text{sech}}^{2}x[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\text{coth}x[\/latex]<\/td>\n<td>[latex]\\text{\u2212}{\\text{csch}}^{2}x[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\text{sech}x[\/latex]<\/td>\n<td>[latex]\\text{\u2212}\\text{sech}x\\text{tanh}x[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\text{csch}x[\/latex]<\/td>\n<td>[latex]\\text{\u2212}\\text{csch}x\\text{coth}x[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167794139383\">Let\u2019s take a moment to compare the derivatives of the hyperbolic functions with the derivatives of the standard trigonometric functions. There are a lot of similarities, but differences as well. For example, the derivatives of the sine functions match: [latex](d\\text{\/}dx) \\sin x= \\cos x[\/latex] and [latex](d\\text{\/}dx)\\text{sinh}x=\\text{cosh}x.[\/latex] The derivatives of the cosine functions, however, differ in sign: [latex](d\\text{\/}dx) \\cos x=\\text{\u2212} \\sin x,[\/latex] but [latex](d\\text{\/}dx)\\text{cosh}x=\\text{sinh}x.[\/latex] As we continue our examination of the hyperbolic functions, we must be mindful of their similarities and differences to the standard trigonometric functions.<\/p>\n<p id=\"fs-id1167794334421\">These differentiation formulas for the hyperbolic functions lead directly to the following integral formulas.<\/p>\n<div id=\"fs-id1167793944458\" class=\"equation unnumbered\">[latex]\\begin{array}{cccccccc}\\hfill \\int \\text{sinh}udu& =\\hfill & \\text{cosh}u+C\\hfill & & & \\hfill \\int {\\text{csch}}^{2}udu& =\\hfill & \\text{\u2212}\\text{coth}u+C\\hfill \\\\ \\hfill \\int \\text{cosh}udu& =\\hfill & \\text{sinh}u+C\\hfill & & & \\hfill \\int \\text{sech}u\\text{tanh}udu& =\\hfill & \\text{\u2212}\\text{sech}u+C\\hfill \\\\ \\hfill \\int {\\text{sech}}^{2}udu& =\\hfill & \\text{tanh}u+C\\hfill & & & \\hfill \\int \\text{csch}u\\text{coth}udu& =\\hfill & \\text{\u2212}\\text{csch}u+C\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"fs-id1167793852238\" class=\"textbox examples\">\n<h3>Differentiating Hyperbolic Functions<\/h3>\n<div id=\"fs-id1167793936071\" class=\"exercise\">\n<div id=\"fs-id1167793876199\" class=\"textbox\">\n<p id=\"fs-id1167793278457\">Evaluate the following derivatives:<\/p>\n<ol id=\"fs-id1167794060382\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dx}(\\text{sinh}({x}^{2}))[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{(\\text{cosh}x)}^{2}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167794098810\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167794098810\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794098810\">Using the formulas in <a class=\"autogenerated-content\" href=\"#fs-id1167793363804\">(Figure)<\/a> and the chain rule, we get<\/p>\n<ol id=\"fs-id1167793500488\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dx}(\\text{sinh}({x}^{2}))=\\text{cosh}({x}^{2})\u00b72x[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{(\\text{cosh}x)}^{2}=2\\text{cosh}x\\text{sinh}x[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793970793\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1167794145876\" class=\"exercise\">\n<div id=\"fs-id1167794126285\" class=\"textbox\">\n<p id=\"fs-id1167794028505\">Evaluate the following derivatives:<\/p>\n<ol id=\"fs-id1167794043812\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dx}(\\text{tanh}({x}^{2}+3x))[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}(\\frac{1}{{(\\text{sinh}x)}^{2}})[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1167794187159\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167794187159\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167794187159\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dx}(\\text{tanh}({x}^{2}+3x))=({\\text{sech}}^{2}({x}^{2}+3x))(2x+3)[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}(\\frac{1}{{(\\text{sinh}x)}^{2}})=\\frac{d}{dx}{(\\text{sinh}x)}^{-2}=-2{(\\text{sinh}x)}^{-3}\\text{cosh}x[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1167793372558\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1167794072646\">Use the formulas in <a class=\"autogenerated-content\" href=\"#fs-id1167793363804\">(Figure)<\/a> and apply the chain rule as necessary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793279424\" class=\"textbox examples\">\n<h3>Integrals Involving Hyperbolic Functions<\/h3>\n<div id=\"fs-id1167794051555\" class=\"exercise\">\n<div id=\"fs-id1167794127457\" class=\"textbox\">\n<p id=\"fs-id1167793965336\">Evaluate the following integrals:<\/p>\n<ol id=\"fs-id1167794050806\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\int x\\text{cosh}({x}^{2})dx[\/latex]<\/li>\n<li>[latex]\\int \\text{tanh}xdx[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793254882\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793254882\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793254882\">We can use [latex]u[\/latex]-substitution in both cases.<\/p>\n<ol id=\"fs-id1167794042760\" style=\"list-style-type: lower-alpha\">\n<li>Let [latex]u={x}^{2}.[\/latex] Then, [latex]du=2xdx[\/latex] and\n<div id=\"fs-id1167794337482\" class=\"equation unnumbered\">[latex]\\int x\\text{cosh}({x}^{2})dx=\\int \\frac{1}{2}\\text{cosh}udu=\\frac{1}{2}\\text{sinh}u+C=\\frac{1}{2}\\text{sinh}({x}^{2})+C.[\/latex]<\/div>\n<\/li>\n<li>Let [latex]u=\\text{cosh}x.[\/latex] Then, [latex]du=\\text{sinh}xdx[\/latex] and\n<div id=\"fs-id1167794043265\" class=\"equation unnumbered\">[latex]\\int \\text{tanh}xdx=\\int \\frac{\\text{sinh}x}{\\text{cosh}x}dx=\\int \\frac{1}{u}du=\\text{ln}|u|+C=\\text{ln}|\\text{cosh}x|+C.[\/latex]<\/div>\n<p>Note that [latex]\\text{cosh}x>0[\/latex] for all [latex]x,[\/latex] so we can eliminate the absolute value signs and obtain<\/p>\n<div id=\"fs-id1167793358384\" class=\"equation unnumbered\">[latex]\\int \\text{tanh}xdx=\\text{ln}(\\text{cosh}x)+C.[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793504384\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1167794146347\" class=\"exercise\">\n<div id=\"fs-id1167794330193\" class=\"textbox\">\n<p id=\"fs-id1167793819956\">Evaluate the following integrals:<\/p>\n<ol id=\"fs-id1167793985861\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\int {\\text{sinh}}^{3}x\\text{cosh}xdx[\/latex]<\/li>\n<li>[latex]\\int {\\text{sech}}^{2}(3x)dx[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793373829\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793373829\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1167793373829\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\int {\\text{sinh}}^{3}x\\text{cosh}xdx=\\frac{{\\text{sinh}}^{4}x}{4}+C[\/latex]<\/li>\n<li>[latex]\\int {\\text{sech}}^{2}(3x)dx=\\frac{\\text{tanh}(3x)}{3}+C[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1167793245855\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1167794199986\">Use the formulas above and apply [latex]u[\/latex]-substitution as necessary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794077052\" class=\"bc-section section\">\n<h1>Calculus of Inverse Hyperbolic Functions<\/h1>\n<p>Looking at the graphs of the hyperbolic functions, we see that with appropriate range restrictions, they all have inverses. Most of the necessary range restrictions can be discerned by close examination of the graphs. The domains and ranges of the inverse hyperbolic functions are summarized in the following table.<\/p>\n<table id=\"fs-id1167793362111\" summary=\"This table has three columns. The first column is labeled function and has the inverse hyperbolic functions listed in the column. The second column is labeled domain and has the domains of the inverse hyperbolic functions. The third column is labeled range and has the ranges of the inverse hyperbolic functions.\">\n<caption>Domains and Ranges of the Inverse Hyperbolic Functions<\/caption>\n<thead>\n<tr valign=\"top\">\n<th>Function<\/th>\n<th>Domain<\/th>\n<th>Range<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]{\\text{sinh}}^{-1}x[\/latex]<\/td>\n<td>[latex](\\text{\u2212}\\infty ,\\infty )[\/latex]<\/td>\n<td>[latex](\\text{\u2212}\\infty ,\\infty )[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]{\\text{cosh}}^{-1}x[\/latex]<\/td>\n<td>[latex](1,\\infty )[\/latex]<\/td>\n<td>[latex]\\left[0,\\infty )[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]{\\text{tanh}}^{-1}x[\/latex]<\/td>\n<td>[latex](-1,1)[\/latex]<\/td>\n<td>[latex](\\text{\u2212}\\infty ,\\infty )[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]{\\text{coth}}^{-1}x[\/latex]<\/td>\n<td>[latex](\\text{\u2212}\\infty ,-1)\\cup (1,\\infty )[\/latex]<\/td>\n<td>[latex](\\text{\u2212}\\infty ,0)\\cup (0,\\infty )[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]{\\text{sech}}^{-1}x[\/latex]<\/td>\n<td>[latex](0\\text{, 1})[\/latex]<\/td>\n<td>[latex]\\left[0,\\infty )[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]{\\text{csch}}^{-1}x[\/latex]<\/td>\n<td>[latex](\\text{\u2212}\\infty ,0)\\cup (0,\\infty )[\/latex]<\/td>\n<td>[latex](\\text{\u2212}\\infty ,0)\\cup (0,\\infty )[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167793998256\">The graphs of the inverse hyperbolic functions are shown in the following figure.<\/p>\n<div id=\"CNX_Calc_Figure_06_09_002\" class=\"wp-caption aligncenter\"><span id=\"fs-id1167793245836\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213346\/CNX_Calc_Figure_06_09_002.jpg\" alt=\"This figure has six graphs. The first graph labeled \u201ca\u201d is of the function y=sinh^-1(x). It is an increasing function from the 3rd quadrant, through the origin to the first quadrant. The second graph is labeled \u201cb\u201d and is of the function y=cosh^-1(x). It is in the first quadrant, beginning on the x-axis at 2 and increasing. The third graph labeled \u201cc\u201d is of the function y=tanh^-1(x). It is an increasing function from the third quadrant, through the origin, to the first quadrant. The fourth graph is labeled \u201cd\u201d and is of the function y=coth^-1(x). It has two pieces, one in the third quadrant and one in the first quadrant with a vertical asymptote at the y-axis. The fifth graph is labeled \u201ce\u201d and is of the function y=sech^-1(x). It is a curve decreasing in the first quadrant and stopping on the x-axis at x=1. The sixth graph is labeled \u201cf\u201d and is of the function y=csch^-1(x). It has two pieces, one in the third quadrant and one in the first quadrant with a vertical asymptote at the y-axis.\" \/><\/span><\/div>\n<p id=\"fs-id1167793259261\">To find the derivatives of the inverse functions, we use implicit differentiation. We have<\/p>\n<div id=\"fs-id1167793959214\" class=\"equation unnumbered\">[latex]\\begin{array}{ccc}\\hfill y& =\\hfill & {\\text{sinh}}^{-1}x\\hfill \\\\ \\hfill \\text{sinh}y& =\\hfill & x\\hfill \\\\ \\hfill \\frac{d}{dx}\\text{sinh}y& =\\hfill & \\frac{d}{dx}x\\hfill \\\\ \\hfill \\text{cosh}y\\frac{dy}{dx}& =\\hfill & 1.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167793589391\">Recall that [latex]{\\text{cosh}}^{2}y-{\\text{sinh}}^{2}y=1,[\/latex] so [latex]\\text{cosh}y=\\sqrt{1+{\\text{sinh}}^{2}y}.[\/latex] Then,<\/p>\n<div id=\"fs-id1167794042534\" class=\"equation unnumbered\">[latex]\\frac{dy}{dx}=\\frac{1}{\\text{cosh}y}=\\frac{1}{\\sqrt{1+{\\text{sinh}}^{2}y}}=\\frac{1}{\\sqrt{1+{x}^{2}}}.[\/latex]<\/div>\n<p id=\"fs-id1167793638871\">We can derive differentiation formulas for the other inverse hyperbolic functions in a similar fashion. These differentiation formulas are summarized in the following table.<\/p>\n<table id=\"fs-id1167794020892\" summary=\"This table has two columns. The first column is labeled f(x) and has the inverse hyperbolic functions as entries. The second column is labeled d\/dx f(x) and is the derivatives of the inverse hyperbolic functions.\">\n<caption>Derivatives of the Inverse Hyperbolic Functions<\/caption>\n<thead>\n<tr valign=\"top\">\n<th>[latex]f(x)[\/latex]<\/th>\n<th>[latex]\\frac{d}{dx}f(x)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]{\\text{sinh}}^{-1}x[\/latex]<\/td>\n<td>[latex]\\frac{1}{\\sqrt{1+{x}^{2}}}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]{\\text{cosh}}^{-1}x[\/latex]<\/td>\n<td>[latex]\\frac{1}{\\sqrt{{x}^{2}-1}}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]{\\text{tanh}}^{-1}x[\/latex]<\/td>\n<td>[latex]\\frac{1}{1-{x}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]{\\text{coth}}^{-1}x[\/latex]<\/td>\n<td>[latex]\\frac{1}{1-{x}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]{\\text{sech}}^{-1}x[\/latex]<\/td>\n<td>[latex]\\frac{-1}{x\\sqrt{1-{x}^{2}}}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]{\\text{csch}}^{-1}x[\/latex]<\/td>\n<td>[latex]\\frac{-1}{|x|\\sqrt{1+{x}^{2}}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167794119173\">Note that the derivatives of [latex]{\\text{tanh}}^{-1}x[\/latex] and [latex]{\\text{coth}}^{-1}x[\/latex] are the same. Thus, when we integrate [latex]1\\text{\/}(1-{x}^{2}),[\/latex] we need to select the proper antiderivative based on the domain of the functions and the values of [latex]x.[\/latex] Integration formulas involving the inverse hyperbolic functions are summarized as follows.<\/p>\n<div id=\"fs-id1167793951927\" class=\"equation unnumbered\">[latex]\\begin{array}{cccccccc}\\hfill \\int \\frac{1}{\\sqrt{1+{u}^{2}}}du& =\\hfill & {\\text{sinh}}^{-1}u+C\\hfill & & & \\hfill \\int \\frac{1}{u\\sqrt{1-{u}^{2}}}du& =\\hfill & \\text{\u2212}{\\text{sech}}^{-1}|u|+C\\hfill \\\\ \\hfill \\int \\frac{1}{\\sqrt{{u}^{2}-1}}du& =\\hfill & {\\text{cosh}}^{-1}u+C\\hfill & & & \\hfill \\int \\frac{1}{u\\sqrt{1+{u}^{2}}}du& =\\hfill & \\text{\u2212}{\\text{csch}}^{-1}|u|+C\\hfill \\\\ \\hfill \\int \\frac{1}{1-{u}^{2}}du& =\\hfill & \\bigg\\{\\begin{array}{c}{\\text{tanh}}^{-1}u+C\\text{ if }|u|<1\\hfill \\\\ {\\text{coth}}^{-1}u+C\\text{ if }|u|>1\\hfill \\end{array}\\hfill & & & & & \\end{array}[\/latex]<\/div>\n<div id=\"fs-id1167793984979\" class=\"textbox examples\">\n<h3>Differentiating Inverse Hyperbolic Functions<\/h3>\n<div id=\"fs-id1167793984981\" class=\"exercise\">\n<div id=\"fs-id1167793829064\" class=\"textbox\">\n<p id=\"fs-id1167793372563\">Evaluate the following derivatives:<\/p>\n<ol id=\"fs-id1167794155431\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dx}({\\text{sinh}}^{-1}(\\frac{x}{3}))[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{({\\text{tanh}}^{-1}x)}^{2}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793949787\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793949787\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793949787\">Using the formulas in <a class=\"autogenerated-content\" href=\"#fs-id1167794020892\">(Figure)<\/a> and the chain rule, we obtain the following results:<\/p>\n<ol id=\"fs-id1167793298499\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dx}({\\text{sinh}}^{-1}(\\frac{x}{3}))=\\frac{1}{3\\sqrt{1+\\frac{{x}^{2}}{9}}}=\\frac{1}{\\sqrt{9+{x}^{2}}}[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{({\\text{tanh}}^{-1}x)}^{2}=\\frac{2({\\text{tanh}}^{-1}x)}{1-{x}^{2}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793626577\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1167793626580\" class=\"exercise\">\n<div id=\"fs-id1167794332393\" class=\"textbox\">\n<p id=\"fs-id1167794332395\">Evaluate the following derivatives:<\/p>\n<ol id=\"fs-id1167793590727\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dx}({\\text{cosh}}^{-1}(3x))[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{({\\text{coth}}^{-1}x)}^{3}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793498614\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793498614\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1167793498614\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\frac{d}{dx}({\\text{cosh}}^{-1}(3x))=\\frac{3}{\\sqrt{9{x}^{2}-1}}[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx}{({\\text{coth}}^{-1}x)}^{3}=\\frac{3{({\\text{coth}}^{-1}x)}^{2}}{1-{x}^{2}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1167794037907\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1167794137287\">Use the formulas in <a class=\"autogenerated-content\" href=\"#fs-id1167794020892\">(Figure)<\/a> and apply the chain rule as necessary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793285224\" class=\"textbox examples\">\n<h3>Integrals Involving Inverse Hyperbolic Functions<\/h3>\n<div id=\"fs-id1167793285226\" class=\"exercise\">\n<div id=\"fs-id1167793285228\" class=\"textbox\">\n<p id=\"fs-id1167793871641\">Evaluate the following integrals:<\/p>\n<ol id=\"fs-id1167793937415\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\int \\frac{1}{\\sqrt{4{x}^{2}-1}}dx[\/latex]<\/li>\n<li>[latex]\\int \\frac{1}{2x\\sqrt{1-9{x}^{2}}}dx[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793361818\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793361818\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793361818\">We can use [latex]u\\text{-substitution}[\/latex] in both cases.<\/p>\n<ol id=\"fs-id1167794296597\" style=\"list-style-type: lower-alpha\">\n<li>Let [latex]u=2x.[\/latex] Then, [latex]du=2dx[\/latex] and we have\n<div id=\"fs-id1167793883719\" class=\"equation unnumbered\">[latex]\\int \\frac{1}{\\sqrt{4{x}^{2}-1}}dx=\\int \\frac{1}{2\\sqrt{{u}^{2}-1}}du=\\frac{1}{2}{\\text{cosh}}^{-1}u+C=\\frac{1}{2}{\\text{cosh}}^{-1}(2x)+C.[\/latex]<\/div>\n<\/li>\n<li>Let [latex]u=3x.[\/latex] Then, [latex]du=3dx[\/latex] and we obtain\n<div id=\"fs-id1167793444519\" class=\"equation unnumbered\">[latex]\\int \\frac{1}{2x\\sqrt{1-9{x}^{2}}}dx=\\frac{1}{2}\\int \\frac{1}{u\\sqrt{1-{u}^{2}}}du=-\\frac{1}{2}{\\text{sech}}^{-1}|u|+C=-\\frac{1}{2}{\\text{sech}}^{-1}|3x|+C.[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793925131\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1167793969727\" class=\"exercise\">\n<div id=\"fs-id1167793969729\" class=\"textbox\">\n<p id=\"fs-id1167793265985\">Evaluate the following integrals:<\/p>\n<ol id=\"fs-id1167793265989\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\int \\frac{1}{\\sqrt{{x}^{2}-4}}dx,\\text{}x>2[\/latex]<\/li>\n<li>[latex]\\int \\frac{1}{\\sqrt{1-{e}^{2x}}}dx[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793371557\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793371557\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1167793371557\" style=\"list-style-type: lower-alpha\">\n<li>[latex]\\int \\frac{1}{\\sqrt{{x}^{2}-4}}dx={\\text{cosh}}^{-1}(\\frac{x}{2})+C[\/latex]<\/li>\n<li>[latex]\\int \\frac{1}{\\sqrt{1-{e}^{2x}}}dx=\\text{\u2212}{\\text{sech}}^{-1}({e}^{x})+C[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1167794333942\">Use the formulas above and apply [latex]u\\text{-substitution}[\/latex] as necessary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794046158\" class=\"bc-section section\">\n<h1>Applications<\/h1>\n<p id=\"fs-id1167793964561\">One physical application of hyperbolic functions involves <span class=\"no-emphasis\">hanging cables<\/span>. If a cable of uniform density is suspended between two supports without any load other than its own weight, the cable forms a curve called a <strong>catenary<\/strong>. High-voltage power lines, chains hanging between two posts, and strands of a spider\u2019s web all form catenaries. The following figure shows chains hanging from a row of posts.<\/p>\n<div id=\"CNX_Calc_Figure_06_09_003\" class=\"wp-caption aligncenter\">\n<div style=\"width: 985px\" 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\/11213354\/CNX_Calc_Figure_06_09_003.jpg\" alt=\"An image of chains hanging between posts that all take the shape of a catenary.\" width=\"975\" height=\"731\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. Chains between these posts take the shape of a catenary. (credit: modification of work by OKFoundryCompany, Flickr)<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793591346\">Hyperbolic functions can be used to model catenaries. Specifically, functions of the form [latex]y=a\\text{cosh}(x\\text{\/}a)[\/latex] are catenaries. <a class=\"autogenerated-content\" href=\"#CNX_Calc_Figure_06_09_004\">(Figure)<\/a> shows the graph of [latex]y=2\\text{cosh}(x\\text{\/}2).[\/latex]<\/p>\n<div id=\"CNX_Calc_Figure_06_09_004\" class=\"wp-caption aligncenter\">\n<div style=\"width: 352px\" 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\/11213358\/CNX_Calc_Figure_06_09_004.jpg\" alt=\"This figure is a graph. It is of the function f(x)=2cosh(x\/2). The curve decreases in the second quadrant to the y-axis. It intersects the y-axis at y=2. Then the curve becomes increasing.\" width=\"342\" height=\"347\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4. A hyperbolic cosine function forms the shape of a catenary.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793370895\" class=\"textbox examples\">\n<h3>Using a Catenary to Find the Length of a Cable<\/h3>\n<div id=\"fs-id1167793884153\" class=\"exercise\">\n<div id=\"fs-id1167793884155\" class=\"textbox\">\n<p id=\"fs-id1167793926505\">Assume a hanging cable has the shape [latex]10\\text{cosh}(x\\text{\/}10)[\/latex] for [latex]-15\\le x\\le 15,[\/latex] where [latex]x[\/latex] is measured in feet. Determine the length of the cable (in feet).<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793307481\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793307481\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793307481\">Recall from Section 6.4 that the formula for arc length is<\/p>\n<div id=\"fs-id1167793943925\" class=\"equation unnumbered\">[latex]\\text{Arc Length}={\\int }_{a}^{b}\\sqrt{1+{\\left[{f}^{\\prime }(x)\\right]}^{2}}dx.[\/latex]<\/div>\n<p id=\"fs-id1167794160035\">We have [latex]f(x)=10\\text{cosh}(x\\text{\/}10),[\/latex] so [latex]{f}^{\\prime }(x)=\\text{sinh}(x\\text{\/}10).[\/latex] Then<\/p>\n<div id=\"fs-id1167794337061\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill \\text{Arc Length}& ={\\int }_{a}^{b}\\sqrt{1+{\\left[{f}^{\\prime }(x)\\right]}^{2}}dx\\hfill \\\\ & ={\\int }_{-15}^{15}\\sqrt{1+{\\text{sinh}}^{2}(\\frac{x}{10})}dx.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167793964837\">Now recall that [latex]1+{\\text{sinh}}^{2}x={\\text{cosh}}^{2}x,[\/latex] so we have<\/p>\n<div id=\"fs-id1167794333245\" class=\"equation unnumbered\">[latex]\\begin{array}{cc}\\hfill \\text{Arc Length}& ={\\int }_{-15}^{15}\\sqrt{1+{\\text{sinh}}^{2}(\\frac{x}{10})}dx\\hfill \\\\ & ={\\int }_{-15}^{15}\\text{cosh}(\\frac{x}{10})dx\\hfill \\\\ & =10\\text{sinh}{(\\frac{x}{10})|}_{-15}^{15}=10\\left[\\text{sinh}(\\frac{3}{2})-\\text{sinh}(-\\frac{3}{2})\\right]=20\\text{sinh}(\\frac{3}{2})\\hfill \\\\ & \\approx 42.586\\text{ft}\\text{.}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794326134\" class=\"textbox exercises checkpoint\">\n<div id=\"fs-id1167793414872\" class=\"exercise\">\n<div id=\"fs-id1167793414874\" class=\"textbox\">\n<p id=\"fs-id1167794043358\">Assume a hanging cable has the shape [latex]15\\text{cosh}(x\\text{\/}15)[\/latex] for [latex]-20\\le x\\le 20.[\/latex] Determine the length of the cable (in feet).<\/p>\n<\/div>\n<div id=\"fs-id1167793292342\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793292342\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793292342\" class=\"hidden-answer\" style=\"display: none\">[latex]52.95\\text{ft}[\/latex]<\/div>\n<div id=\"fs-id1167793237822\" class=\"commentary\">\n<h4>Hint<\/h4>\n<p id=\"fs-id1167794022005\">Use the procedure from the previous example.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793269537\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1167793618899\">\n<li>Hyperbolic functions are defined in terms of exponential functions.<\/li>\n<li>Term-by-term differentiation yields differentiation formulas for the hyperbolic functions. These differentiation formulas give rise, in turn, to integration formulas.<\/li>\n<li>With appropriate range restrictions, the hyperbolic functions all have inverses.<\/li>\n<li>Implicit differentiation yields differentiation formulas for the inverse hyperbolic functions, which in turn give rise to integration formulas.<\/li>\n<li>The most common physical applications of hyperbolic functions are calculations involving catenaries.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1167794126430\" class=\"textbox exercises\">\n<div id=\"fs-id1167794126434\" class=\"exercise\">\n<div id=\"fs-id1167793940609\" class=\"textbox\">\n<p id=\"fs-id1167793940611\"><strong>[T]<\/strong> Find expressions for [latex]\\text{cosh}x+\\text{sinh}x[\/latex] and [latex]\\text{cosh}x-\\text{sinh}x.[\/latex] Use a calculator to graph these functions and ensure your expression is correct.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793552000\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793552000\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793552000\">[latex]{e}^{x}\\text{ and }{e}^{\\text{\u2212}x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794051638\" class=\"exercise\">\n<div id=\"fs-id1167794051640\" class=\"textbox\">\n<p id=\"fs-id1167793930753\">From the definitions of [latex]\\text{cosh}(x)[\/latex] and [latex]\\text{sinh}(x),[\/latex] find their antiderivatives.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793956239\" class=\"exercise\">\n<div id=\"fs-id1167793956241\" class=\"textbox\">\n<p id=\"fs-id1167793956243\">Show that [latex]\\text{cosh}(x)[\/latex] and [latex]\\text{sinh}(x)[\/latex] satisfy [latex]y\\text{\u2033}=y.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793416604\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793416604\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793416604\">Answers may vary<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793269055\" class=\"exercise\">\n<div id=\"fs-id1167793269058\" class=\"textbox\">\n<p>Use the quotient rule to verify that [latex]\\text{tanh}(x)\\prime ={\\text{sech}}^{2}(x).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793514512\" class=\"exercise\">\n<div id=\"fs-id1167793514514\" class=\"textbox\">\n<p id=\"fs-id1167793944608\">Derive [latex]{\\text{cosh}}^{2}(x)+{\\text{sinh}}^{2}(x)=\\text{cosh}(2x)[\/latex] from the definition.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793870399\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793870399\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793870399\">Answers may vary<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794037737\" class=\"exercise\">\n<div id=\"fs-id1167794037739\" class=\"textbox\">\n<p id=\"fs-id1167793219307\">Take the derivative of the previous expression to find an expression for [latex]\\text{sinh}(2x).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794141049\" class=\"exercise\">\n<div id=\"fs-id1167794141051\" class=\"textbox\">\n<p id=\"fs-id1167794141053\">Prove [latex]\\text{sinh}(x+y)=\\text{sinh}(x)\\text{cosh}(y)+\\text{cosh}(x)\\text{sinh}(y)[\/latex] by changing the expression to exponentials.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167794137118\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167794137118\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794137118\">Answers may vary<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794336126\" class=\"exercise\">\n<div id=\"fs-id1167794336128\" class=\"textbox\">\n<p id=\"fs-id1167794336130\">Take the derivative of the previous expression to find an expression for [latex]\\text{cosh}(x+y).[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793929899\">For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct.<\/p>\n<div id=\"fs-id1167793630189\" class=\"exercise\">\n<div id=\"fs-id1167793630191\" class=\"textbox\">\n<p id=\"fs-id1167793630193\"><strong>[T]<\/strong>[latex]\\text{cosh}(3x+1)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793318463\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793318463\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793318463\">[latex]3\\text{sinh}(3x+1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793382588\" class=\"exercise\">\n<div id=\"fs-id1167793382590\" class=\"textbox\">\n<p id=\"fs-id1167793382592\"><strong>[T]<\/strong>[latex]\\text{sinh}({x}^{2})[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793416764\" class=\"exercise\">\n<div id=\"fs-id1167794329964\" class=\"textbox\">\n<p id=\"fs-id1167794329966\"><strong>[T]<\/strong>[latex]\\frac{1}{\\text{cosh}(x)}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793268310\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793268310\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793268310\">[latex]\\text{\u2212}\\text{tanh}(x)\\text{sech}(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793985952\" class=\"exercise\">\n<div id=\"fs-id1167793985955\" class=\"textbox\">\n<p id=\"fs-id1167793985957\"><strong>[T]<\/strong>[latex]\\text{sinh}(\\text{ln}(x))[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793384557\" class=\"exercise\">\n<div id=\"fs-id1167793384559\" class=\"textbox\">\n<p id=\"fs-id1167793246794\"><strong>[T]<\/strong>[latex]{\\text{cosh}}^{2}(x)+{\\text{sinh}}^{2}(x)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793947964\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793947964\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793947964\">[latex]4\\text{cosh}(x)\\text{sinh}(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793301148\" class=\"exercise\">\n<div id=\"fs-id1167793301150\" class=\"textbox\">\n<p id=\"fs-id1167793301152\"><strong>[T]<\/strong>[latex]{\\text{cosh}}^{2}(x)-{\\text{sinh}}^{2}(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794284874\" class=\"exercise\">\n<div id=\"fs-id1167793455832\" class=\"textbox\">\n<p id=\"fs-id1167793455835\"><strong>[T]<\/strong>[latex]\\text{tanh}(\\sqrt{{x}^{2}+1})[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167794293324\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167794293324\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794293324\">[latex]\\frac{x{\\text{sech}}^{2}(\\sqrt{{x}^{2}+1})}{\\sqrt{{x}^{2}+1}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794199184\" class=\"exercise\">\n<div id=\"fs-id1167794199186\" class=\"textbox\">\n<p id=\"fs-id1167794199188\"><strong>[T]<\/strong>[latex]\\frac{1+\\text{tanh}(x)}{1-\\text{tanh}(x)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794043454\" class=\"exercise\">\n<div id=\"fs-id1167793883113\" class=\"textbox\">\n<p id=\"fs-id1167793883115\"><strong>[T]<\/strong>[latex]{\\text{sinh}}^{6}(x)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167794039862\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167794039862\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794039862\">[latex]6{\\text{sinh}}^{5}(x)\\text{cosh}(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793637691\" class=\"exercise\">\n<div id=\"fs-id1167793385422\" class=\"textbox\">\n<p id=\"fs-id1167793385424\"><strong>[T]<\/strong>[latex]\\text{ln}(\\text{sech}(x)+\\text{tanh}(x))[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793281078\">For the following exercises, find the antiderivatives for the given functions.<\/p>\n<div id=\"fs-id1167793281082\" class=\"exercise\">\n<div id=\"fs-id1167793281084\" class=\"textbox\">\n<p id=\"fs-id1167793638839\">[latex]\\text{cosh}(2x+1)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793372486\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793372486\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793372486\">[latex]\\frac{1}{2}\\text{sinh}(2x+1)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794178053\" class=\"exercise\">\n<div id=\"fs-id1167794178055\" class=\"textbox\">\n<p id=\"fs-id1167794178057\">[latex]\\text{tanh}(3x+2)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794336738\" class=\"exercise\">\n<div id=\"fs-id1167794336741\" class=\"textbox\">\n<p id=\"fs-id1167794336743\">[latex]x\\text{cosh}({x}^{2})[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793950952\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793950952\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793950952\">[latex]\\frac{1}{2}{\\text{sinh}}^{2}({x}^{2})+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793956996\" class=\"exercise\">\n<div id=\"fs-id1167793958193\" class=\"textbox\">\n<p id=\"fs-id1167793958195\">[latex]3{x}^{3}\\text{tanh}({x}^{4})[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793372582\" class=\"exercise\">\n<div id=\"fs-id1167793316107\" class=\"textbox\">\n<p id=\"fs-id1167793316109\">[latex]{\\text{cosh}}^{2}(x)\\text{sinh}(x)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793393695\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793393695\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793393695\">[latex]\\frac{1}{3}{\\text{cosh}}^{3}(x)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793265440\" class=\"exercise\">\n<div id=\"fs-id1167793265442\" class=\"textbox\">\n<p>[latex]{\\text{tanh}}^{2}(x){\\text{sech}}^{2}(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793450621\" class=\"exercise\">\n<div id=\"fs-id1167793450623\" class=\"textbox\">\n<p id=\"fs-id1167793450625\">[latex]\\frac{\\text{sinh}(x)}{1+\\text{cosh}(x)}[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p id=\"fs-id1167794118195\">[latex]\\text{ln}(1+\\text{cosh}(x))+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793275006\" class=\"exercise\">\n<div id=\"fs-id1167793275008\" class=\"textbox\">\n<p id=\"fs-id1167793495090\">[latex]\\text{coth}(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794181011\" class=\"exercise\">\n<div id=\"fs-id1167794181013\" class=\"textbox\">\n<p id=\"fs-id1167794181016\">[latex]\\text{cosh}(x)+\\text{sinh}(x)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793361732\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793361732\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793361732\">[latex]\\text{cosh}(x)+\\text{sinh}(x)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id1167793564129\" class=\"textbox\">\n<p id=\"fs-id1167793564131\">[latex]{(\\text{cosh}(x)+\\text{sinh}(x))}^{n}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793951599\">For the following exercises, find the derivatives for the functions.<\/p>\n<div id=\"fs-id1167793372386\" class=\"exercise\">\n<div id=\"fs-id1167793372388\" class=\"textbox\">\n<p id=\"fs-id1167793372390\">[latex]{\\text{tanh}}^{-1}(4x)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793948851\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793948851\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793948851\">[latex]\\frac{4}{1-16{x}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794206992\" class=\"exercise\">\n<div id=\"fs-id1167794206994\" class=\"textbox\">\n<p id=\"fs-id1167794206996\">[latex]{\\text{sinh}}^{-1}({x}^{2})[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793936665\" class=\"exercise\">\n<div id=\"fs-id1167793936667\" class=\"textbox\">\n<p id=\"fs-id1167794329290\">[latex]{\\text{sinh}}^{-1}(\\text{cosh}(x))[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167794207012\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167794207012\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794207012\">[latex]\\frac{\\text{sinh}(x)}{\\sqrt{{\\text{cosh}}^{2}(x)+1}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793358453\" class=\"exercise\">\n<div id=\"fs-id1167793358455\" class=\"textbox\">\n<p id=\"fs-id1167793605572\">[latex]{\\text{cosh}}^{-1}({x}^{3})[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793940311\" class=\"exercise\">\n<div id=\"fs-id1167793940313\" class=\"textbox\">\n<p id=\"fs-id1167793940316\">[latex]{\\text{tanh}}^{-1}( \\cos (x))[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793543556\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793543556\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793543556\">[latex]\\text{\u2212} \\csc (x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793928572\" class=\"exercise\">\n<div id=\"fs-id1167793928574\" class=\"textbox\">\n<p id=\"fs-id1167793928576\">[latex]{e}^{{\\text{sinh}}^{-1}(x)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793290963\" class=\"exercise\">\n<div id=\"fs-id1167793290965\" class=\"textbox\">\n<p id=\"fs-id1167793290968\">[latex]\\text{ln}({\\text{tanh}}^{-1}(x))[\/latex]<\/p>\n<\/div>\n<div class=\"solution\">\n<p id=\"fs-id1167793538362\">[latex]-\\frac{1}{({x}^{2}-1){\\text{tanh}}^{-1}(x)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793509961\">For the following exercises, find the antiderivatives for the functions.<\/p>\n<div class=\"exercise\">\n<div id=\"fs-id1167793579580\" class=\"textbox\">\n<p id=\"fs-id1167793579582\">[latex]\\int \\frac{dx}{4-{x}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794058014\" class=\"exercise\">\n<div id=\"fs-id1167794058016\" class=\"textbox\">\n<p id=\"fs-id1167794058019\">[latex]\\int \\frac{dx}{{a}^{2}-{x}^{2}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793399888\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793399888\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793399888\">[latex]\\frac{1}{a}{\\text{tanh}}^{-1}(\\frac{x}{a})+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793420648\" class=\"exercise\">\n<div id=\"fs-id1167793420650\" class=\"textbox\">\n<p id=\"fs-id1167793421226\">[latex]\\int \\frac{dx}{\\sqrt{{x}^{2}+1}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793950155\" class=\"exercise\">\n<div id=\"fs-id1167793950157\" class=\"textbox\">\n<p id=\"fs-id1167793590468\">[latex]\\int \\frac{xdx}{\\sqrt{{x}^{2}+1}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793886745\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793886745\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793886745\">[latex]\\sqrt{{x}^{2}+1}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793569521\" class=\"exercise\">\n<div id=\"fs-id1167793569523\" class=\"textbox\">\n<p>[latex]\\int -\\frac{dx}{x\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793510599\" class=\"exercise\">\n<div id=\"fs-id1167793510601\" class=\"textbox\">\n<p id=\"fs-id1167793510603\">[latex]\\int \\frac{{e}^{x}}{\\sqrt{{e}^{2x}-1}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167793582489\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793582489\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793582489\" class=\"hidden-answer\" style=\"display: none\">[latex]{\\text{cosh}}^{-1}({e}^{x})+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794058783\" class=\"exercise\">\n<div id=\"fs-id1167794058785\" class=\"textbox\">\n<p>[latex]\\int -\\frac{2x}{{x}^{4}-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793384924\">For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation [latex]dv\\text{\/}dt=g-{v}^{2}.[\/latex]<\/p>\n<div id=\"fs-id1167793276964\" class=\"exercise\">\n<div id=\"fs-id1167793276967\" class=\"textbox\">\n<p id=\"fs-id1167793276969\">Show that [latex]v(t)=\\sqrt{g}\\text{tanh}(\\sqrt{gt})[\/latex] satisfies this equation.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793255883\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793255883\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793255883\">Answers may vary<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793255888\" class=\"exercise\">\n<div id=\"fs-id1167793607690\" class=\"textbox\">\n<p id=\"fs-id1167793607692\">Derive the previous expression for [latex]v(t)[\/latex] by integrating [latex]\\frac{dv}{g-{v}^{2}}=dt.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793316091\" class=\"exercise\">\n<div id=\"fs-id1167793309335\" class=\"textbox\">\n<p id=\"fs-id1167793309337\"><strong>[T]<\/strong> Estimate how far a body has fallen in 12 seconds by finding the area underneath the curve of [latex]v(t).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167794058934\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167794058934\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167794058934\" class=\"hidden-answer\" style=\"display: none\">37.30<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167794058944\">For the following exercises, use this scenario: A cable hanging under its own weight has a slope [latex]S=dy\\text{\/}dx[\/latex] that satisfies [latex]dS\\text{\/}dx=c\\sqrt{1+{S}^{2}}.[\/latex] The constant [latex]c[\/latex] is the ratio of cable density to tension.<\/p>\n<div id=\"fs-id1167793518527\" class=\"exercise\">\n<div id=\"fs-id1167793518530\" class=\"textbox\">\n<p id=\"fs-id1167793518532\">Show that [latex]S=\\text{sinh}(cx)[\/latex] satisfies this equation.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793619929\" class=\"exercise\">\n<div id=\"fs-id1167793619931\" class=\"textbox\">\n<p id=\"fs-id1167793619933\">Integrate [latex]dy\\text{\/}dx=\\text{sinh}(cx)[\/latex] to find the cable height [latex]y(x)[\/latex] if [latex]y(0)=1\\text{\/}c.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167794291509\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167794291509\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794291509\">[latex]y=\\frac{1}{c}\\text{cosh}(cx)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793932271\" class=\"exercise\">\n<div id=\"fs-id1167793932273\" class=\"textbox\">\n<p id=\"fs-id1167793932275\">Sketch the cable and determine how far down it sags at [latex]x=0.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167793510618\">For the following exercises, solve each problem.<\/p>\n<div id=\"fs-id1167793510621\" class=\"exercise\">\n<div id=\"fs-id1167793510623\" class=\"textbox\">\n<p id=\"fs-id1167793510625\"><strong>[T]<\/strong> A chain hangs from two posts 2 m apart to form a catenary described by the equation [latex]y=2\\text{cosh}(x\\text{\/}2)-1.[\/latex] Find the slope of the catenary at the left fence post.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793463064\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793463064\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793463064\">-0.521095<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793463073\" class=\"exercise\">\n<div id=\"fs-id1167793463075\" class=\"textbox\">\n<p id=\"fs-id1167793463077\"><strong>[T]<\/strong> A chain hangs from two posts four meters apart to form a catenary described by the equation [latex]y=4\\text{cosh}(x\\text{\/}4)-3.[\/latex] Find the total length of the catenary (arc length).<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793641978\" class=\"exercise\">\n<div id=\"fs-id1167793641980\" class=\"textbox\">\n<p id=\"fs-id1167793641982\"><strong>[T]<\/strong> A high-voltage power line is a catenary described by [latex]y=10\\text{cosh}(x\\text{\/}10).[\/latex] Find the ratio of the area under the catenary to its arc length. What do you notice?<\/p>\n<\/div>\n<div id=\"fs-id1167793956533\" class=\"textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1167793956533\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1167793956533\" class=\"hidden-answer\" style=\"display: none\">10<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div class=\"textbox\">\n<p>A telephone line is a catenary described by [latex]y=a\\text{cosh}(x\\text{\/}a).[\/latex] Find the ratio of the area under the catenary to its arc length. Does this confirm your answer for the previous question?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793929410\" class=\"exercise\">\n<div class=\"textbox\">\n<p id=\"fs-id1167793929415\">Prove the formula for the derivative of [latex]y={\\text{sinh}}^{-1}(x)[\/latex] by differentiating [latex]x=\\text{sinh}(y).[\/latex] (<em>Hint:<\/em> Use hyperbolic trigonometric identities.)<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793950965\" class=\"exercise\">\n<div id=\"fs-id1167793950967\" class=\"textbox\">\n<p id=\"fs-id1167793950969\">Prove the formula for the derivative of [latex]y={\\text{cosh}}^{-1}(x)[\/latex] by differentiating [latex]x=\\text{cosh}(y).[\/latex]<\/p>\n<p id=\"fs-id1167793395478\">(<em>Hint:<\/em> Use hyperbolic trigonometric identities.)<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793395488\" class=\"exercise\">\n<div id=\"fs-id1167793395490\" class=\"textbox\">\n<p id=\"fs-id1167793395493\">Prove the formula for the derivative of [latex]y={\\text{sech}}^{-1}(x)[\/latex] by differentiating [latex]x=\\text{sech}(y).[\/latex] (<em>Hint:<\/em> Use hyperbolic trigonometric identities.)<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167794210494\" class=\"exercise\">\n<div id=\"fs-id1167794210496\" class=\"textbox\">\n<p id=\"fs-id1167794210498\">Prove that [latex]{(\\text{cosh}(x)+\\text{sinh}(x))}^{n}=\\text{cosh}(nx)+\\text{sinh}(nx).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167793379933\" class=\"exercise\">\n<div id=\"fs-id1167793379935\" class=\"textbox\">\n<p id=\"fs-id1167793379937\">Prove the expression for [latex]{\\text{sinh}}^{-1}(x).[\/latex] Multiply [latex]x=\\text{sinh}(y)=(1\\text{\/}2)({e}^{y}-{e}^{\\text{\u2212}y})[\/latex] by [latex]2{e}^{y}[\/latex] and solve for [latex]y.[\/latex] Does your expression match the textbook?<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div class=\"textbox\">\n<p id=\"fs-id1167793372779\">Prove the expression for [latex]{\\text{cosh}}^{-1}(x).[\/latex] Multiply [latex]x=\\text{cosh}(y)=(1\\text{\/}2)({e}^{y}-{e}^{\\text{\u2212}y})[\/latex] by [latex]2{e}^{y}[\/latex] and solve for [latex]y.[\/latex] Does your expression match the textbook?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"review-exercises\"><\/div>\n<div class=\"textbox shaded\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1167793432037\" class=\"definition\">\n<dt>catenary<\/dt>\n<dd id=\"fs-id1167793432042\">a curve in the shape of the function [latex]y=a\\text{cosh}(x\\text{\/}a)[\/latex] is a catenary; a cable of uniform density suspended between two supports assumes the shape of a catenary<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":311,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2205","chapter","type-chapter","status-publish","hentry"],"part":2032,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/2205","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/2205\/revisions"}],"predecessor-version":[{"id":2553,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/2205\/revisions\/2553"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/parts\/2032"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapters\/2205\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/media?parent=2205"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/chapter-type?post=2205"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/contributor?post=2205"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/license?post=2205"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}