{"id":498,"date":"2025-02-13T19:45:29","date_gmt":"2025-02-13T19:45:29","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/integration-of-exponential-logarithmic-and-hyperbolic-functions-get-stronger\/"},"modified":"2025-02-13T19:45:29","modified_gmt":"2025-02-13T19:45:29","slug":"integration-of-exponential-logarithmic-and-hyperbolic-functions-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/integration-of-exponential-logarithmic-and-hyperbolic-functions-get-stronger\/","title":{"raw":"Integration of Exponential, Logarithmic, and Hyperbolic Functions: Get Stronger","rendered":"Integration of Exponential, Logarithmic, and Hyperbolic Functions: Get Stronger"},"content":{"raw":"\n

Integrals, Exponential Functions, and Logarithms<\/h2>\n

For the following exercises (1-2), find the derivative [latex]\\frac{dy}{dx}.[\/latex]<\/strong><\/p>\n

    \n\t
  1. [latex]y=\\text{ln}(2x)[\/latex]<\/li>\n\t
  2. [latex]y=\\dfrac{1}{\\text{ln}x}[\/latex]<\/li>\n<\/ol>\n

    For the following exercise, find the indefinite integral.<\/strong><\/p>\n

      \n\t
    1. [latex]\\displaystyle\\int \\frac{dx}{1+x}[\/latex]<\/li>\n<\/ol>\n

      For the following exercises (4-8), find the derivative [latex]dy\\text{\/}dx.[\/latex] (You can use a calculator to plot the function and the derivative to confirm that it is correct.)<\/strong><\/p>\n

        \n\t
      1. [latex]y=x\\text{ln}(x)[\/latex]<\/li>\n\t
      2. [latex]y=\\text{ln}( \\sin x)[\/latex]<\/li>\n\t
      3. [latex]y=7\\text{ln}(4x)[\/latex]<\/li>\n\t
      4. [latex]y=\\text{ln}( \\tan x)[\/latex]<\/li>\n\t
      5. [latex]y=\\text{ln}({ \\cos }^{2}x)[\/latex]<\/li>\n<\/ol>\n

        For the following exercises (9-13), find the definite or indefinite integral.<\/strong><\/p>\n

          \n\t
        1. [latex]{\\displaystyle\\int }_{0}^{1}\\dfrac{dt}{3+2t}[\/latex]<\/li>\n\t
        2. [latex]{\\displaystyle\\int }_{0}^{2}\\dfrac{{x}^{3}dx}{{x}^{2}+1}[\/latex]<\/li>\n\t
        3. [latex]{\\displaystyle\\int }_{2}^{e}\\dfrac{dx}{{(x\\text{ln}(x))}^{2}}[\/latex]<\/li>\n\t
        4. [latex]{\\displaystyle\\int }_{0}^{\\pi \\text{\/}4} \\tan xdx[\/latex]<\/li>\n\t
        5. [latex]\\displaystyle\\int \\frac{{(\\text{ln}x)}^{2}dx}{x}[\/latex]<\/li>\n<\/ol>\n

          For the following exercises (14-18), compute [latex]dy\\text{\/}dx[\/latex] by differentiating [latex]\\text{ln}y.[\/latex]<\/strong><\/p>\n

            \n\t
          1. [latex]y=\\sqrt{{x}^{2}+1}\\sqrt{{x}^{2}-1}[\/latex]<\/li>\n\t
          2. [latex]y={x}^{-1\\text{\/}x}[\/latex]<\/li>\n\t
          3. [latex]y={x}^{e}[\/latex]<\/li>\n\t
          4. [latex]y=\\sqrt{x}\\sqrt[3]{x}\\sqrt[6]{x}[\/latex]<\/li>\n\t
          5. [latex]y={e}^{\\text{\u2212}\\text{ln}x}[\/latex]<\/li>\n<\/ol>\n

            For the following exercises (19-20), evaluate by any method.<\/strong><\/p>\n

              \n\t
            1. [latex]{\\displaystyle\\int }_{1}^{{e}^{\\pi }}\\frac{dx}{x}+{\\displaystyle\\int }_{-2}^{-1}\\frac{dx}{x}[\/latex]<\/li>\n\t
            2. [latex]\\frac{d}{dx}{\\displaystyle\\int }_{x}^{{x}^{2}}\\frac{dt}{t}[\/latex]<\/li>\n<\/ol>\n

              For the following exercises (21-22), use the function [latex]\\text{ln}x.[\/latex] If you are unable to find intersection points analytically, use a calculator.<\/strong><\/p>\n

                \n\t
              1. Find the area of the region enclosed by [latex]x=1[\/latex] and [latex]y=5[\/latex] above [latex]y=\\text{ln}x.[\/latex]<\/li>\n\t
              2. Find the area between [latex]\\text{ln}x[\/latex] and the [latex]x[\/latex]-axis from [latex]x=1\\text{ to }x=2.[\/latex]<\/li>\n<\/ol>\n

                For the following exercises (23-27), verify the derivatives and antiderivatives.<\/strong><\/p>\n

                  \n\t
                1. [latex]\\frac{d}{dx}\\text{ln}(x+\\sqrt{{x}^{2}+1})=\\dfrac{1}{\\sqrt{1+{x}^{2}}}[\/latex]<\/li>\n\t
                2. [latex]\\frac{d}{dx}\\text{ln}\\left(\\dfrac{x-a}{x+a}\\right)=\\dfrac{2a}{({x}^{2}-{a}^{2})}[\/latex]<\/li>\n\t
                3. [latex]\\frac{d}{dx}\\text{ln}\\left(\\dfrac{1+\\sqrt{1-{x}^{2}}}{x}\\right)=-\\dfrac{1}{x\\sqrt{1-{x}^{2}}}[\/latex]<\/li>\n\t
                4. [latex]\\frac{d}{dx}\\text{ln}(x+\\sqrt{{x}^{2}-{a}^{2}})=\\dfrac{1}{\\sqrt{{x}^{2}-{a}^{2}}}[\/latex]<\/li>\n\t
                5. [latex]\\displaystyle\\int \\frac{dx}{x\\text{ln}(x)\\text{ln}(\\text{ln}x)}=\\text{ln}(\\text{ln}(\\text{ln}x))+C[\/latex]<\/li>\n<\/ol>\n

                  Exponential Growth and Decay<\/h2>\n

                  True or False<\/em>? If true, prove it. If false, find the true answer (1-2).<\/strong><\/p>\n

                    \n\t
                  1. If you invest [latex]$500,[\/latex] an annual rate of interest of [latex]3\\%[\/latex] yields more money in the first year than a [latex]2.5\\%[\/latex] continuous rate of interest.<\/li>\n\t
                  2. If given a half-life of [latex]t[\/latex] years, the constant [latex]k[\/latex] for [latex]y={e}^{kt}[\/latex] is calculated by [latex]k=\\text{ln}(1\\text{\/}2)\\text{\/}t.[\/latex]<\/li>\n<\/ol>\n

                    For the following exercises (3-11), use [latex]y={y}_{0}{e}^{kt}.[\/latex]<\/strong><\/p>\n

                      \n\t
                    1. If bacteria increase by a factor of [latex]10[\/latex] in [latex]10[\/latex] hours, how many hours does it take to increase by [latex]100?[\/latex]<\/li>\n\t
                    2. If a relic contains [latex]90\\%[\/latex] as much radiocarbon as new material, can it have come from the time of Christ (approximately 2000 years ago)? Note that the half-life of radiocarbon is [latex]5730[\/latex] years.<\/li>\n\t
                    3. The populations of New York and Los Angeles are growing at [latex]1\\%[\/latex] and [latex]1.4\\%[\/latex] a year, respectively. Starting from [latex]8[\/latex] million (New York) and [latex]6[\/latex] million (Los Angeles), when are the populations equal?<\/li>\n\t
                    4. The effect of advertising decays exponentially. If [latex]40\\%[\/latex] of the population remembers a new product after [latex]3[\/latex] days, how long will [latex]20\\%[\/latex] remember it?<\/li>\n\t
                    5. If [latex]y=100[\/latex] at [latex]t=4[\/latex] and [latex]y=10[\/latex] at [latex]t=8,[\/latex] when does [latex]y=1?[\/latex]<\/li>\n\t
                    6. What continuous interest rate has the same yield as an annual rate of [latex]9\\text{%}?[\/latex]<\/li>\n\t
                    7. You are trying to save [latex]$50,000[\/latex] in [latex]20[\/latex] years for college tuition for your child. If interest is a continuous [latex]10\\text{%},[\/latex] how much do you need to invest initially?<\/li>\n\t
                    8. You are trying to thaw some vegetables that are at a temperature of [latex]1\\text{\u00b0}\\text{F}\\text{.}[\/latex] To thaw vegetables safely, you must put them in the refrigerator, which has an ambient temperature of [latex]44\\text{\u00b0}\\text{F}.[\/latex] You check on your vegetables [latex]2[\/latex] hours after putting them in the refrigerator to find that they are now [latex]12\\text{\u00b0}\\text{F}\\text{.}[\/latex] Plot the resulting temperature curve and use it to determine when the vegetables reach [latex]33\\text{\u00b0}\\text{F}\\text{.}[\/latex]<\/li>\n\t
                    9. The spent fuel of a nuclear reactor contains plutonium-239, which has a half-life of [latex]24,000[\/latex] years. If [latex]1[\/latex] barrel containing [latex]10[\/latex]kg of plutonium-239 is sealed, how many years must pass until only [latex]10g[\/latex] of plutonium-239 is left?<\/li>\n<\/ol>\n

                      For the next set of exercises (12-15), use the following table, which features the world population by decade.<\/strong><\/p>\n\n\n\n\n\n\n\n\n\n\n\n
                      Source<\/em>: http:\/\/www.factmonster.com\/ipka\/A0762181.html.<\/caption>\n
                      Years since 1950<\/th>\nPopulation (millions)<\/th>\n<\/tr>\n<\/thead>\n
                      [latex]0[\/latex]<\/td>\n[latex]2,556[\/latex]<\/td>\n<\/tr>\n
                      [latex]10[\/latex]<\/td>\n[latex]3,039[\/latex]<\/td>\n<\/tr>\n
                      [latex]20[\/latex]<\/td>\n[latex]3,706[\/latex]<\/td>\n<\/tr>\n
                      [latex]30[\/latex]<\/td>\n[latex]4,453[\/latex]<\/td>\n<\/tr>\n
                      [latex]40[\/latex]<\/td>\n[latex]5,279[\/latex]<\/td>\n<\/tr>\n
                      [latex]50[\/latex]<\/td>\n[latex]6,083[\/latex]<\/td>\n<\/tr>\n
                      [latex]60[\/latex]<\/td>\n[latex]6,849[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                        \n\t
                      1. The best-fit exponential curve to the data of the form [latex]P(t)=a{e}^{bt}[\/latex] is given by [latex]P(t)=2686{e}^{0.01604t}.[\/latex] Use a graphing calculator to graph the data and the exponential curve together.<\/li>\n\t
                      2. Find and graph the derivative [latex]{y}^{\\prime }[\/latex] of your equation. Where is it increasing and what is the meaning of this increase?<\/li>\n\t
                      3. Find and graph the second derivative of your equation. Where is it increasing and what is the meaning of this increase?<\/li>\n\t
                      4. Find the predicted date when the population reaches [latex]10[\/latex] billion. Using your previous answers about the first and second derivatives, explain why exponential growth is unsuccessful in predicting the future.<\/li>\n<\/ol>\n

                        For the next set of exercises (16-18), use the following table, which shows the population of San Francisco during the 19th century.<\/strong><\/p>\n\n\n\n\n\n\n\n\n
                        Source<\/em>: http:\/\/www.sfgenealogy.com\/sf\/history\/hgpop.htm.<\/caption>\n
                        Years since 1850<\/strong><\/th>\nPopulation (thousands)<\/strong><\/th>\n<\/tr>\n<\/thead>\n
                        [latex]0[\/latex]<\/td>\n[latex]21.00[\/latex]<\/td>\n<\/tr>\n
                        [latex]10[\/latex]<\/td>\n[latex]56.80[\/latex]<\/td>\n<\/tr>\n
                        [latex]20[\/latex]<\/td>\n[latex]149.5[\/latex]<\/td>\n<\/tr>\n
                        [latex]30[\/latex]<\/td>\n[latex]234.0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                          \n\t
                        1. The best-fit exponential curve to the data of the form [latex]P(t)=a{e}^{bt}[\/latex] is given by [latex]P(t)=35.26{e}^{0.06407t}.[\/latex] Use a graphing calculator to graph the data and the exponential curve together.<\/li>\n\t
                        2. Find and graph the derivative [latex]{y}^{\\prime }[\/latex] of your equation. Where is it increasing? What is the meaning of this increase? Is there a value where the increase is maximal?<\/li>\n\t
                        3. Find and graph the second derivative of your equation. Where is it increasing? What is the meaning of this increase?<\/li>\n<\/ol>\n

                          Calculus of the Hyperbolic Functions<\/h2>\n
                            \n\t
                          1. 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.<\/li>\n\t
                          2. Show that [latex]\\text{cosh}(x)[\/latex] and [latex]\\text{sinh}(x)[\/latex] satisfy [latex]y\\text{\u2033}=y.[\/latex]<\/li>\n\t
                          3. Derive [latex]{\\text{cosh}}^{2}(x)+{\\text{sinh}}^{2}(x)=\\text{cosh}(2x)[\/latex] from the definition.<\/li>\n\t
                          4. 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.<\/li>\n<\/ol>\n

                            For the following exercises (5-9), find the derivatives of the given functions and graph along with the function to ensure your answer is correct.<\/strong><\/p>\n

                              \n\t
                            1. [latex]\\text{cosh}(3x+1)[\/latex]<\/li>\n\t
                            2. [latex]\\frac{1}{\\text{cosh}(x)}[\/latex]<\/li>\n\t
                            3. [latex]{\\text{cosh}}^{2}(x)+{\\text{sinh}}^{2}(x)[\/latex]<\/li>\n\t
                            4. [latex]\\text{tanh}(\\sqrt{{x}^{2}+1})[\/latex]<\/li>\n\t
                            5. [latex]{\\text{sinh}}^{6}(x)[\/latex]<\/li>\n<\/ol>\n

                              For the following exercises (10-14), find the antiderivatives for the given functions.<\/strong><\/p>\n

                                \n\t
                              1. [latex]\\text{cosh}(2x+1)[\/latex]<\/li>\n\t
                              2. [latex]x\\text{cosh}({x}^{2})[\/latex]<\/li>\n\t
                              3. [latex]{\\text{cosh}}^{2}(x)\\text{sinh}(x)[\/latex]<\/li>\n\t
                              4. [latex]\\frac{\\text{sinh}(x)}{1+\\text{cosh}(x)}[\/latex]<\/li>\n\t
                              5. [latex]\\text{cosh}(x)+\\text{sinh}(x)[\/latex]<\/li>\n<\/ol>\n

                                For the following exercises (15-18), find the derivatives for the functions.<\/strong><\/p>\n

                                  \n\t
                                1. [latex]{\\text{tanh}}^{-1}(4x)[\/latex]<\/li>\n\t
                                2. [latex]{\\text{sinh}}^{-1}(\\text{cosh}(x))[\/latex]<\/li>\n\t
                                3. [latex]{\\text{tanh}}^{-1}( \\cos (x))[\/latex]<\/li>\n\t
                                4. [latex]\\text{ln}({\\text{tanh}}^{-1}(x))[\/latex]<\/li>\n<\/ol>\n

                                  For the following exercises (19-21), find the antiderivatives for the functions.<\/strong><\/p>\n

                                    \n\t
                                  1. [latex]\\displaystyle\\int \\frac{dx}{{a}^{2}-{x}^{2}}[\/latex]<\/li>\n\t
                                  2. [latex]\\displaystyle\\int \\frac{xdx}{\\sqrt{{x}^{2}+1}}[\/latex]<\/li>\n\t
                                  3. [latex]\\displaystyle\\int \\frac{{e}^{x}}{\\sqrt{{e}^{2x}-1}}[\/latex]<\/li>\n<\/ol>\n

                                    For the following exercises (22-26), solve each problem.<\/strong><\/p>\n

                                      \n\t
                                    1. A chain hangs from two posts [latex]2[\/latex] 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.<\/li>\n\t
                                    2. 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?<\/li>\n\t
                                    3. 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.)<\/li>\n\t
                                    4. 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.)<\/li>\n\t
                                    5. 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 text?<\/li>\n<\/ol>\n","rendered":"

                                      Integrals, Exponential Functions, and Logarithms<\/h2>\n

                                      For the following exercises (1-2), find the derivative [latex]\\frac{dy}{dx}.[\/latex]<\/strong><\/p>\n

                                        \n
                                      1. [latex]y=\\text{ln}(2x)[\/latex]<\/li>\n
                                      2. [latex]y=\\dfrac{1}{\\text{ln}x}[\/latex]<\/li>\n<\/ol>\n

                                        For the following exercise, find the indefinite integral.<\/strong><\/p>\n

                                          \n
                                        1. [latex]\\displaystyle\\int \\frac{dx}{1+x}[\/latex]<\/li>\n<\/ol>\n

                                          For the following exercises (4-8), find the derivative [latex]dy\\text{\/}dx.[\/latex] (You can use a calculator to plot the function and the derivative to confirm that it is correct.)<\/strong><\/p>\n

                                            \n
                                          1. [latex]y=x\\text{ln}(x)[\/latex]<\/li>\n
                                          2. [latex]y=\\text{ln}( \\sin x)[\/latex]<\/li>\n
                                          3. [latex]y=7\\text{ln}(4x)[\/latex]<\/li>\n
                                          4. [latex]y=\\text{ln}( \\tan x)[\/latex]<\/li>\n
                                          5. [latex]y=\\text{ln}({ \\cos }^{2}x)[\/latex]<\/li>\n<\/ol>\n

                                            For the following exercises (9-13), find the definite or indefinite integral.<\/strong><\/p>\n

                                              \n
                                            1. [latex]{\\displaystyle\\int }_{0}^{1}\\dfrac{dt}{3+2t}[\/latex]<\/li>\n
                                            2. [latex]{\\displaystyle\\int }_{0}^{2}\\dfrac{{x}^{3}dx}{{x}^{2}+1}[\/latex]<\/li>\n
                                            3. [latex]{\\displaystyle\\int }_{2}^{e}\\dfrac{dx}{{(x\\text{ln}(x))}^{2}}[\/latex]<\/li>\n
                                            4. [latex]{\\displaystyle\\int }_{0}^{\\pi \\text{\/}4} \\tan xdx[\/latex]<\/li>\n
                                            5. [latex]\\displaystyle\\int \\frac{{(\\text{ln}x)}^{2}dx}{x}[\/latex]<\/li>\n<\/ol>\n

                                              For the following exercises (14-18), compute [latex]dy\\text{\/}dx[\/latex] by differentiating [latex]\\text{ln}y.[\/latex]<\/strong><\/p>\n

                                                \n
                                              1. [latex]y=\\sqrt{{x}^{2}+1}\\sqrt{{x}^{2}-1}[\/latex]<\/li>\n
                                              2. [latex]y={x}^{-1\\text{\/}x}[\/latex]<\/li>\n
                                              3. [latex]y={x}^{e}[\/latex]<\/li>\n
                                              4. [latex]y=\\sqrt{x}\\sqrt[3]{x}\\sqrt[6]{x}[\/latex]<\/li>\n
                                              5. [latex]y={e}^{\\text{\u2212}\\text{ln}x}[\/latex]<\/li>\n<\/ol>\n

                                                For the following exercises (19-20), evaluate by any method.<\/strong><\/p>\n

                                                  \n
                                                1. [latex]{\\displaystyle\\int }_{1}^{{e}^{\\pi }}\\frac{dx}{x}+{\\displaystyle\\int }_{-2}^{-1}\\frac{dx}{x}[\/latex]<\/li>\n
                                                2. [latex]\\frac{d}{dx}{\\displaystyle\\int }_{x}^{{x}^{2}}\\frac{dt}{t}[\/latex]<\/li>\n<\/ol>\n

                                                  For the following exercises (21-22), use the function [latex]\\text{ln}x.[\/latex] If you are unable to find intersection points analytically, use a calculator.<\/strong><\/p>\n

                                                    \n
                                                  1. Find the area of the region enclosed by [latex]x=1[\/latex] and [latex]y=5[\/latex] above [latex]y=\\text{ln}x.[\/latex]<\/li>\n
                                                  2. Find the area between [latex]\\text{ln}x[\/latex] and the [latex]x[\/latex]-axis from [latex]x=1\\text{ to }x=2.[\/latex]<\/li>\n<\/ol>\n

                                                    For the following exercises (23-27), verify the derivatives and antiderivatives.<\/strong><\/p>\n

                                                      \n
                                                    1. [latex]\\frac{d}{dx}\\text{ln}(x+\\sqrt{{x}^{2}+1})=\\dfrac{1}{\\sqrt{1+{x}^{2}}}[\/latex]<\/li>\n
                                                    2. [latex]\\frac{d}{dx}\\text{ln}\\left(\\dfrac{x-a}{x+a}\\right)=\\dfrac{2a}{({x}^{2}-{a}^{2})}[\/latex]<\/li>\n
                                                    3. [latex]\\frac{d}{dx}\\text{ln}\\left(\\dfrac{1+\\sqrt{1-{x}^{2}}}{x}\\right)=-\\dfrac{1}{x\\sqrt{1-{x}^{2}}}[\/latex]<\/li>\n
                                                    4. [latex]\\frac{d}{dx}\\text{ln}(x+\\sqrt{{x}^{2}-{a}^{2}})=\\dfrac{1}{\\sqrt{{x}^{2}-{a}^{2}}}[\/latex]<\/li>\n
                                                    5. [latex]\\displaystyle\\int \\frac{dx}{x\\text{ln}(x)\\text{ln}(\\text{ln}x)}=\\text{ln}(\\text{ln}(\\text{ln}x))+C[\/latex]<\/li>\n<\/ol>\n

                                                      Exponential Growth and Decay<\/h2>\n

                                                      True or False<\/em>? If true, prove it. If false, find the true answer (1-2).<\/strong><\/p>\n

                                                        \n
                                                      1. If you invest [latex]$500,[\/latex] an annual rate of interest of [latex]3\\%[\/latex] yields more money in the first year than a [latex]2.5\\%[\/latex] continuous rate of interest.<\/li>\n
                                                      2. If given a half-life of [latex]t[\/latex] years, the constant [latex]k[\/latex] for [latex]y={e}^{kt}[\/latex] is calculated by [latex]k=\\text{ln}(1\\text{\/}2)\\text{\/}t.[\/latex]<\/li>\n<\/ol>\n

                                                        For the following exercises (3-11), use [latex]y={y}_{0}{e}^{kt}.[\/latex]<\/strong><\/p>\n

                                                          \n
                                                        1. If bacteria increase by a factor of [latex]10[\/latex] in [latex]10[\/latex] hours, how many hours does it take to increase by [latex]100?[\/latex]<\/li>\n
                                                        2. If a relic contains [latex]90\\%[\/latex] as much radiocarbon as new material, can it have come from the time of Christ (approximately 2000 years ago)? Note that the half-life of radiocarbon is [latex]5730[\/latex] years.<\/li>\n
                                                        3. The populations of New York and Los Angeles are growing at [latex]1\\%[\/latex] and [latex]1.4\\%[\/latex] a year, respectively. Starting from [latex]8[\/latex] million (New York) and [latex]6[\/latex] million (Los Angeles), when are the populations equal?<\/li>\n
                                                        4. The effect of advertising decays exponentially. If [latex]40\\%[\/latex] of the population remembers a new product after [latex]3[\/latex] days, how long will [latex]20\\%[\/latex] remember it?<\/li>\n
                                                        5. If [latex]y=100[\/latex] at [latex]t=4[\/latex] and [latex]y=10[\/latex] at [latex]t=8,[\/latex] when does [latex]y=1?[\/latex]<\/li>\n
                                                        6. What continuous interest rate has the same yield as an annual rate of [latex]9\\text{%}?[\/latex]<\/li>\n
                                                        7. You are trying to save [latex]$50,000[\/latex] in [latex]20[\/latex] years for college tuition for your child. If interest is a continuous [latex]10\\text{%},[\/latex] how much do you need to invest initially?<\/li>\n
                                                        8. You are trying to thaw some vegetables that are at a temperature of [latex]1\\text{\u00b0}\\text{F}\\text{.}[\/latex] To thaw vegetables safely, you must put them in the refrigerator, which has an ambient temperature of [latex]44\\text{\u00b0}\\text{F}.[\/latex] You check on your vegetables [latex]2[\/latex] hours after putting them in the refrigerator to find that they are now [latex]12\\text{\u00b0}\\text{F}\\text{.}[\/latex] Plot the resulting temperature curve and use it to determine when the vegetables reach [latex]33\\text{\u00b0}\\text{F}\\text{.}[\/latex]<\/li>\n
                                                        9. The spent fuel of a nuclear reactor contains plutonium-239, which has a half-life of [latex]24,000[\/latex] years. If [latex]1[\/latex] barrel containing [latex]10[\/latex]kg of plutonium-239 is sealed, how many years must pass until only [latex]10g[\/latex] of plutonium-239 is left?<\/li>\n<\/ol>\n

                                                          For the next set of exercises (12-15), use the following table, which features the world population by decade.<\/strong><\/p>\n\n\n\n\n\n\n\n\n\n\n\n
                                                          Source<\/em>: http:\/\/www.factmonster.com\/ipka\/A0762181.html.<\/caption>\n
                                                          Years since 1950<\/th>\nPopulation (millions)<\/th>\n<\/tr>\n<\/thead>\n
                                                          [latex]0[\/latex]<\/td>\n[latex]2,556[\/latex]<\/td>\n<\/tr>\n
                                                          [latex]10[\/latex]<\/td>\n[latex]3,039[\/latex]<\/td>\n<\/tr>\n
                                                          [latex]20[\/latex]<\/td>\n[latex]3,706[\/latex]<\/td>\n<\/tr>\n
                                                          [latex]30[\/latex]<\/td>\n[latex]4,453[\/latex]<\/td>\n<\/tr>\n
                                                          [latex]40[\/latex]<\/td>\n[latex]5,279[\/latex]<\/td>\n<\/tr>\n
                                                          [latex]50[\/latex]<\/td>\n[latex]6,083[\/latex]<\/td>\n<\/tr>\n
                                                          [latex]60[\/latex]<\/td>\n[latex]6,849[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                                                            \n
                                                          1. The best-fit exponential curve to the data of the form [latex]P(t)=a{e}^{bt}[\/latex] is given by [latex]P(t)=2686{e}^{0.01604t}.[\/latex] Use a graphing calculator to graph the data and the exponential curve together.<\/li>\n
                                                          2. Find and graph the derivative [latex]{y}^{\\prime }[\/latex] of your equation. Where is it increasing and what is the meaning of this increase?<\/li>\n
                                                          3. Find and graph the second derivative of your equation. Where is it increasing and what is the meaning of this increase?<\/li>\n
                                                          4. Find the predicted date when the population reaches [latex]10[\/latex] billion. Using your previous answers about the first and second derivatives, explain why exponential growth is unsuccessful in predicting the future.<\/li>\n<\/ol>\n

                                                            For the next set of exercises (16-18), use the following table, which shows the population of San Francisco during the 19th century.<\/strong><\/p>\n\n\n\n\n\n\n\n\n
                                                            Source<\/em>: http:\/\/www.sfgenealogy.com\/sf\/history\/hgpop.htm.<\/caption>\n
                                                            Years since 1850<\/strong><\/th>\nPopulation (thousands)<\/strong><\/th>\n<\/tr>\n<\/thead>\n
                                                            [latex]0[\/latex]<\/td>\n[latex]21.00[\/latex]<\/td>\n<\/tr>\n
                                                            [latex]10[\/latex]<\/td>\n[latex]56.80[\/latex]<\/td>\n<\/tr>\n
                                                            [latex]20[\/latex]<\/td>\n[latex]149.5[\/latex]<\/td>\n<\/tr>\n
                                                            [latex]30[\/latex]<\/td>\n[latex]234.0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                                                              \n
                                                            1. The best-fit exponential curve to the data of the form [latex]P(t)=a{e}^{bt}[\/latex] is given by [latex]P(t)=35.26{e}^{0.06407t}.[\/latex] Use a graphing calculator to graph the data and the exponential curve together.<\/li>\n
                                                            2. Find and graph the derivative [latex]{y}^{\\prime }[\/latex] of your equation. Where is it increasing? What is the meaning of this increase? Is there a value where the increase is maximal?<\/li>\n
                                                            3. Find and graph the second derivative of your equation. Where is it increasing? What is the meaning of this increase?<\/li>\n<\/ol>\n

                                                              Calculus of the Hyperbolic Functions<\/h2>\n
                                                                \n
                                                              1. 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.<\/li>\n
                                                              2. Show that [latex]\\text{cosh}(x)[\/latex] and [latex]\\text{sinh}(x)[\/latex] satisfy [latex]y\\text{\u2033}=y.[\/latex]<\/li>\n
                                                              3. Derive [latex]{\\text{cosh}}^{2}(x)+{\\text{sinh}}^{2}(x)=\\text{cosh}(2x)[\/latex] from the definition.<\/li>\n
                                                              4. 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.<\/li>\n<\/ol>\n

                                                                For the following exercises (5-9), find the derivatives of the given functions and graph along with the function to ensure your answer is correct.<\/strong><\/p>\n

                                                                  \n
                                                                1. [latex]\\text{cosh}(3x+1)[\/latex]<\/li>\n
                                                                2. [latex]\\frac{1}{\\text{cosh}(x)}[\/latex]<\/li>\n
                                                                3. [latex]{\\text{cosh}}^{2}(x)+{\\text{sinh}}^{2}(x)[\/latex]<\/li>\n
                                                                4. [latex]\\text{tanh}(\\sqrt{{x}^{2}+1})[\/latex]<\/li>\n
                                                                5. [latex]{\\text{sinh}}^{6}(x)[\/latex]<\/li>\n<\/ol>\n

                                                                  For the following exercises (10-14), find the antiderivatives for the given functions.<\/strong><\/p>\n

                                                                    \n
                                                                  1. [latex]\\text{cosh}(2x+1)[\/latex]<\/li>\n
                                                                  2. [latex]x\\text{cosh}({x}^{2})[\/latex]<\/li>\n
                                                                  3. [latex]{\\text{cosh}}^{2}(x)\\text{sinh}(x)[\/latex]<\/li>\n
                                                                  4. [latex]\\frac{\\text{sinh}(x)}{1+\\text{cosh}(x)}[\/latex]<\/li>\n
                                                                  5. [latex]\\text{cosh}(x)+\\text{sinh}(x)[\/latex]<\/li>\n<\/ol>\n

                                                                    For the following exercises (15-18), find the derivatives for the functions.<\/strong><\/p>\n

                                                                      \n
                                                                    1. [latex]{\\text{tanh}}^{-1}(4x)[\/latex]<\/li>\n
                                                                    2. [latex]{\\text{sinh}}^{-1}(\\text{cosh}(x))[\/latex]<\/li>\n
                                                                    3. [latex]{\\text{tanh}}^{-1}( \\cos (x))[\/latex]<\/li>\n
                                                                    4. [latex]\\text{ln}({\\text{tanh}}^{-1}(x))[\/latex]<\/li>\n<\/ol>\n

                                                                      For the following exercises (19-21), find the antiderivatives for the functions.<\/strong><\/p>\n

                                                                        \n
                                                                      1. [latex]\\displaystyle\\int \\frac{dx}{{a}^{2}-{x}^{2}}[\/latex]<\/li>\n
                                                                      2. [latex]\\displaystyle\\int \\frac{xdx}{\\sqrt{{x}^{2}+1}}[\/latex]<\/li>\n
                                                                      3. [latex]\\displaystyle\\int \\frac{{e}^{x}}{\\sqrt{{e}^{2x}-1}}[\/latex]<\/li>\n<\/ol>\n

                                                                        For the following exercises (22-26), solve each problem.<\/strong><\/p>\n

                                                                          \n
                                                                        1. A chain hangs from two posts [latex]2[\/latex] 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.<\/li>\n
                                                                        2. 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?<\/li>\n
                                                                        3. 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.)<\/li>\n
                                                                        4. 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.)<\/li>\n
                                                                        5. 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 text?<\/li>\n<\/ol>\n","protected":false},"author":6,"menu_order":19,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":479,"module-header":"","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/498"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/498\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/479"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/498\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=498"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=498"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=498"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=498"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}