{"id":1210,"date":"2021-06-30T17:02:11","date_gmt":"2021-06-30T17:02:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/problem-set-calculus-of-the-hyperbolic-functions\/"},"modified":"2021-11-17T02:20:27","modified_gmt":"2021-11-17T02:20:27","slug":"problem-set-calculus-of-the-hyperbolic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/problem-set-calculus-of-the-hyperbolic-functions\/","title":{"raw":"Problem Set: Calculus of the Hyperbolic Functions","rendered":"Problem Set: Calculus of the Hyperbolic Functions"},"content":{"raw":"
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1. [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[reveal-answer q=\"fs-id1167793552000\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793552000\"]\r\n

[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

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2.\u00a0<\/strong>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

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3.\u00a0<\/strong>Show that [latex]\\text{cosh}(x)[\/latex] and [latex]\\text{sinh}(x)[\/latex] satisfy [latex]y\\text{\u2033}=y.[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793416604\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793416604\"]\r\n

Answers may vary<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n4.<\/strong> Use 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
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5.\u00a0<\/strong>Derive [latex]{\\text{cosh}}^{2}(x)+{\\text{sinh}}^{2}(x)=\\text{cosh}(2x)[\/latex] from the definition.<\/p>\r\n[reveal-answer q=\"fs-id1167793870399\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793870399\"]\r\n

Answers may vary<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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6.\u00a0<\/strong>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

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7.\u00a0<\/strong>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[reveal-answer q=\"fs-id1167794137118\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794137118\"]\r\n

Answers may vary<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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8.\u00a0<\/strong>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

For the following exercises (9-18), find the derivatives of the given functions and graph along with the function to ensure your answer is correct.<\/p>\r\n\r\n

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9. [T]<\/strong> [latex]\\text{cosh}(3x+1)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793318463\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793318463\"]\r\n

[latex]3\\text{sinh}(3x+1)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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10. [T]<\/strong> [latex]\\text{sinh}({x}^{2})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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11. [T]<\/strong> [latex]\\frac{1}{\\text{cosh}(x)}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793268310\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793268310\"]\r\n

[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

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12. [T]<\/strong> [latex]\\text{sinh}(\\text{ln}(x))[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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13. [T]<\/strong> [latex]{\\text{cosh}}^{2}(x)+{\\text{sinh}}^{2}(x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793947964\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793947964\"]\r\n

[latex]4\\text{cosh}(x)\\text{sinh}(x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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14. [T]<\/strong> [latex]{\\text{cosh}}^{2}(x)-{\\text{sinh}}^{2}(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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15. [T]<\/strong> [latex]\\text{tanh}(\\sqrt{{x}^{2}+1})[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167794293324\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794293324\"]\r\n

[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

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16. [T]<\/strong> [latex]\\frac{1+\\text{tanh}(x)}{1-\\text{tanh}(x)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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17. [T]<\/strong> [latex]{\\text{sinh}}^{6}(x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167794039862\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794039862\"]\r\n

[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

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18. [T]<\/strong> [latex]\\text{ln}(\\text{sech}(x)+\\text{tanh}(x))[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (19-29), find the antiderivatives for the given functions.<\/p>\r\n\r\n

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19.\u00a0<\/strong>[latex]\\text{cosh}(2x+1)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793372486\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793372486\"]\r\n

[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

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20.\u00a0<\/strong>[latex]\\text{tanh}(3x+2)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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21.\u00a0<\/strong>[latex]x\\text{cosh}({x}^{2})[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793950952\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793950952\"]\r\n

[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

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23.\u00a0<\/strong>[latex]3{x}^{3}\\text{tanh}({x}^{4})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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24.\u00a0<\/strong>[latex]{\\text{cosh}}^{2}(x)\\text{sinh}(x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793393695\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793393695\"]\r\n

[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

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\r\n\r\n25.<\/strong> [latex]{\\text{tanh}}^{2}(x){\\text{sech}}^{2}(x)[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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26.\u00a0<\/strong>[latex]\\frac{\\text{sinh}(x)}{1+\\text{cosh}(x)}[\/latex]<\/p>\r\n[reveal-answer q=\"214220\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"214220\"][latex]\\text{ln}(1+\\text{cosh}(x))+C[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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27.\u00a0<\/strong>[latex]\\text{coth}(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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28.\u00a0<\/strong>[latex]\\text{cosh}(x)+\\text{sinh}(x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793361732\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793361732\"]\r\n

[latex]\\text{cosh}(x)+\\text{sinh}(x)+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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29.\u00a0<\/strong>[latex]{(\\text{cosh}(x)+\\text{sinh}(x))}^{n}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (30-36), find the derivatives for the functions.<\/p>\r\n\r\n

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30.<\/strong> [latex]{\\text{tanh}}^{-1}(4x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793948851\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793948851\"]\r\n

[latex]\\frac{4}{1-16{x}^{2}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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31.\u00a0<\/strong>[latex]{\\text{sinh}}^{-1}({x}^{2})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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32.\u00a0<\/strong>[latex]{\\text{sinh}}^{-1}(\\text{cosh}(x))[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167794207012\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794207012\"]\r\n

[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

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33.\u00a0<\/strong>[latex]{\\text{cosh}}^{-1}({x}^{3})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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34.\u00a0<\/strong>[latex]{\\text{tanh}}^{-1}( \\cos (x))[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793543556\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793543556\"]\r\n

[latex]\\text{\u2212} \\csc (x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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35.\u00a0<\/strong>[latex]{e}^{{\\text{sinh}}^{-1}(x)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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36.\u00a0<\/strong>[latex]\\text{ln}({\\text{tanh}}^{-1}(x))[\/latex]<\/p>\r\n[reveal-answer q=\"509394\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"509394\"][latex]-\\frac{1}{({x}^{2}-1){\\text{tanh}}^{-1}(x)}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (37-43), find the antiderivatives for the functions.<\/p>\r\n\r\n

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37.<\/strong> [latex]\\displaystyle\\int \\frac{dx}{4-{x}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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38.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{a}^{2}-{x}^{2}}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793399888\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793399888\"]\r\n

[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

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39.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}+1}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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40.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{xdx}{\\sqrt{{x}^{2}+1}}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793886745\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793886745\"]\r\n

[latex]\\sqrt{{x}^{2}+1}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n41.\u00a0<\/strong>[latex]\\displaystyle\\int -\\frac{dx}{x\\sqrt{1-{x}^{2}}}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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42.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{e}^{x}}{\\sqrt{{e}^{2x}-1}}[\/latex]<\/p>\r\n[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]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n43.\u00a0<\/strong>[latex]\\displaystyle\\int -\\frac{2x}{{x}^{4}-1}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (44-46), 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

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44.<\/strong> Show that [latex]v(t)=\\sqrt{g}\\text{tanh}(\\sqrt{gt})[\/latex] satisfies this equation.<\/p>\r\n[reveal-answer q=\"fs-id1167793255883\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793255883\"]\r\n

Answers may vary<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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45.<\/strong> 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

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46. [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[reveal-answer q=\"fs-id1167794058934\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794058934\"]37.30[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (47-49), 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

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47.\u00a0<\/strong>Show that [latex]S=\\text{sinh}(cx)[\/latex] satisfies this equation.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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48.\u00a0<\/strong>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[reveal-answer q=\"fs-id1167794291509\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794291509\"]\r\n

[latex]y=\\frac{1}{c}\\text{cosh}(cx)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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49.<\/strong> 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

For the following exercises (50-53), solve each problem.<\/p>\r\n\r\n

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50. [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[reveal-answer q=\"fs-id1167793463064\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793463064\"]\r\n

-0.521095<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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51. [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

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52. [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[reveal-answer q=\"fs-id1167793956533\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793956533\"]10[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n53.\u00a0<\/strong>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?\r\n\r\n<\/div>\r\n<\/div>\r\n
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54.\u00a0<\/strong>Prove the formula for the derivative of [latex]y={\\text{sinh}}^{-1}(x)[\/latex] by differentiating [latex]x=\\text{sinh}(y).[\/latex] (Hint:<\/em> Use hyperbolic trigonometric identities.)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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55.\u00a0<\/strong>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

(Hint:<\/em> Use hyperbolic trigonometric identities.)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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56.\u00a0<\/strong>Prove the formula for the derivative of [latex]y={\\text{sech}}^{-1}(x)[\/latex] by differentiating [latex]x=\\text{sech}(y).[\/latex] (Hint:<\/em> Use hyperbolic trigonometric identities.)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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57.\u00a0<\/strong>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

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58.\u00a0<\/strong>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

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59.\u00a0<\/strong>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>","rendered":"

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1. [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

Show Solution<\/span><\/p>\n
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[latex]{e}^{x}\\text{ and }{e}^{\\text{\u2212}x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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2.\u00a0<\/strong>From the definitions of [latex]\\text{cosh}(x)[\/latex] and [latex]\\text{sinh}(x),[\/latex] find their antiderivatives.<\/p>\n<\/div>\n<\/div>\n

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3.\u00a0<\/strong>Show that [latex]\\text{cosh}(x)[\/latex] and [latex]\\text{sinh}(x)[\/latex] satisfy [latex]y\\text{\u2033}=y.[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Answers may vary<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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4.<\/strong> Use the quotient rule to verify that [latex]\\text{tanh}(x)\\prime ={\\text{sech}}^{2}(x).[\/latex]<\/p>\n<\/div>\n<\/div>\n

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5.\u00a0<\/strong>Derive [latex]{\\text{cosh}}^{2}(x)+{\\text{sinh}}^{2}(x)=\\text{cosh}(2x)[\/latex] from the definition.<\/p>\n

Show Solution<\/span><\/p>\n
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Answers may vary<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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6.\u00a0<\/strong>Take the derivative of the previous expression to find an expression for [latex]\\text{sinh}(2x).[\/latex]<\/p>\n<\/div>\n<\/div>\n

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7.\u00a0<\/strong>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

Show Solution<\/span><\/p>\n
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Answers may vary<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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8.\u00a0<\/strong>Take the derivative of the previous expression to find an expression for [latex]\\text{cosh}(x+y).[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises (9-18), find the derivatives of the given functions and graph along with the function to ensure your answer is correct.<\/p>\n

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9. [T]<\/strong> [latex]\\text{cosh}(3x+1)[\/latex]<\/p>\n

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[latex]3\\text{sinh}(3x+1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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10. [T]<\/strong> [latex]\\text{sinh}({x}^{2})[\/latex]<\/p>\n<\/div>\n<\/div>\n

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11. [T]<\/strong> [latex]\\frac{1}{\\text{cosh}(x)}[\/latex]<\/p>\n

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[latex]\\text{\u2212}\\text{tanh}(x)\\text{sech}(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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12. [T]<\/strong> [latex]\\text{sinh}(\\text{ln}(x))[\/latex]<\/p>\n<\/div>\n<\/div>\n

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13. [T]<\/strong> [latex]{\\text{cosh}}^{2}(x)+{\\text{sinh}}^{2}(x)[\/latex]<\/p>\n

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[latex]4\\text{cosh}(x)\\text{sinh}(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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14. [T]<\/strong> [latex]{\\text{cosh}}^{2}(x)-{\\text{sinh}}^{2}(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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15. [T]<\/strong> [latex]\\text{tanh}(\\sqrt{{x}^{2}+1})[\/latex]<\/p>\n

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[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

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16. [T]<\/strong> [latex]\\frac{1+\\text{tanh}(x)}{1-\\text{tanh}(x)}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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17. [T]<\/strong> [latex]{\\text{sinh}}^{6}(x)[\/latex]<\/p>\n

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[latex]6{\\text{sinh}}^{5}(x)\\text{cosh}(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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18. [T]<\/strong> [latex]\\text{ln}(\\text{sech}(x)+\\text{tanh}(x))[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises (19-29), find the antiderivatives for the given functions.<\/p>\n

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19.\u00a0<\/strong>[latex]\\text{cosh}(2x+1)[\/latex]<\/p>\n

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[latex]\\frac{1}{2}\\text{sinh}(2x+1)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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20.\u00a0<\/strong>[latex]\\text{tanh}(3x+2)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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21.\u00a0<\/strong>[latex]x\\text{cosh}({x}^{2})[\/latex]<\/p>\n

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[latex]\\frac{1}{2}{\\text{sinh}}^{2}({x}^{2})+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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23.\u00a0<\/strong>[latex]3{x}^{3}\\text{tanh}({x}^{4})[\/latex]<\/p>\n<\/div>\n<\/div>\n

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24.\u00a0<\/strong>[latex]{\\text{cosh}}^{2}(x)\\text{sinh}(x)[\/latex]<\/p>\n

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[latex]\\frac{1}{3}{\\text{cosh}}^{3}(x)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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25.<\/strong> [latex]{\\text{tanh}}^{2}(x){\\text{sech}}^{2}(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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26.\u00a0<\/strong>[latex]\\frac{\\text{sinh}(x)}{1+\\text{cosh}(x)}[\/latex]<\/p>\n

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[latex]\\text{ln}(1+\\text{cosh}(x))+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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27.\u00a0<\/strong>[latex]\\text{coth}(x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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28.\u00a0<\/strong>[latex]\\text{cosh}(x)+\\text{sinh}(x)[\/latex]<\/p>\n

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[latex]\\text{cosh}(x)+\\text{sinh}(x)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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29.\u00a0<\/strong>[latex]{(\\text{cosh}(x)+\\text{sinh}(x))}^{n}[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises (30-36), find the derivatives for the functions.<\/p>\n

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30.<\/strong> [latex]{\\text{tanh}}^{-1}(4x)[\/latex]<\/p>\n

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[latex]\\frac{4}{1-16{x}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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31.\u00a0<\/strong>[latex]{\\text{sinh}}^{-1}({x}^{2})[\/latex]<\/p>\n<\/div>\n<\/div>\n

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32.\u00a0<\/strong>[latex]{\\text{sinh}}^{-1}(\\text{cosh}(x))[\/latex]<\/p>\n

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[latex]\\frac{\\text{sinh}(x)}{\\sqrt{{\\text{cosh}}^{2}(x)+1}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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33.\u00a0<\/strong>[latex]{\\text{cosh}}^{-1}({x}^{3})[\/latex]<\/p>\n<\/div>\n<\/div>\n

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34.\u00a0<\/strong>[latex]{\\text{tanh}}^{-1}( \\cos (x))[\/latex]<\/p>\n

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[latex]\\text{\u2212} \\csc (x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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35.\u00a0<\/strong>[latex]{e}^{{\\text{sinh}}^{-1}(x)}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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36.\u00a0<\/strong>[latex]\\text{ln}({\\text{tanh}}^{-1}(x))[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
[latex]-\\frac{1}{({x}^{2}-1){\\text{tanh}}^{-1}(x)}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (37-43), find the antiderivatives for the functions.<\/p>\n

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37.<\/strong> [latex]\\displaystyle\\int \\frac{dx}{4-{x}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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38.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{a}^{2}-{x}^{2}}[\/latex]<\/p>\n

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[latex]\\frac{1}{a}{\\text{tanh}}^{-1}(\\frac{x}{a})+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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39.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}+1}}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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40.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{xdx}{\\sqrt{{x}^{2}+1}}[\/latex]<\/p>\n

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[latex]\\sqrt{{x}^{2}+1}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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41.\u00a0<\/strong>[latex]\\displaystyle\\int -\\frac{dx}{x\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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42.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{e}^{x}}{\\sqrt{{e}^{2x}-1}}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
[latex]{\\text{cosh}}^{-1}({e}^{x})+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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43.\u00a0<\/strong>[latex]\\displaystyle\\int -\\frac{2x}{{x}^{4}-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises (44-46), 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

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44.<\/strong> Show that [latex]v(t)=\\sqrt{g}\\text{tanh}(\\sqrt{gt})[\/latex] satisfies this equation.<\/p>\n

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Answers may vary<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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45.<\/strong> 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

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46. [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

Show Solution<\/span><\/p>\n
37.30<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (47-49), 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

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47.\u00a0<\/strong>Show that [latex]S=\\text{sinh}(cx)[\/latex] satisfies this equation.<\/p>\n<\/div>\n<\/div>\n

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48.\u00a0<\/strong>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

Show Solution<\/span><\/p>\n
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[latex]y=\\frac{1}{c}\\text{cosh}(cx)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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49.<\/strong> Sketch the cable and determine how far down it sags at [latex]x=0.[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises (50-53), solve each problem.<\/p>\n

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50. [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

Show Solution<\/span><\/p>\n
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-0.521095<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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51. [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

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52. [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

Show Solution<\/span><\/p>\n
10<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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53.\u00a0<\/strong>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

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54.\u00a0<\/strong>Prove the formula for the derivative of [latex]y={\\text{sinh}}^{-1}(x)[\/latex] by differentiating [latex]x=\\text{sinh}(y).[\/latex] (Hint:<\/em> Use hyperbolic trigonometric identities.)<\/p>\n<\/div>\n<\/div>\n

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55.\u00a0<\/strong>Prove the formula for the derivative of [latex]y={\\text{cosh}}^{-1}(x)[\/latex] by differentiating [latex]x=\\text{cosh}(y).[\/latex]<\/p>\n

(Hint:<\/em> Use hyperbolic trigonometric identities.)<\/p>\n<\/div>\n<\/div>\n

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56.\u00a0<\/strong>Prove the formula for the derivative of [latex]y={\\text{sech}}^{-1}(x)[\/latex] by differentiating [latex]x=\\text{sech}(y).[\/latex] (Hint:<\/em> Use hyperbolic trigonometric identities.)<\/p>\n<\/div>\n<\/div>\n

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57.\u00a0<\/strong>Prove that [latex]{(\\text{cosh}(x)+\\text{sinh}(x))}^{n}=\\text{cosh}(nx)+\\text{sinh}(nx).[\/latex]<\/p>\n<\/div>\n<\/div>\n

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58.\u00a0<\/strong>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

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59.\u00a0<\/strong>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\n\t\t\t

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