{"id":1005,"date":"2025-06-20T17:27:51","date_gmt":"2025-06-20T17:27:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&p=1005"},"modified":"2025-08-14T17:32:59","modified_gmt":"2025-08-14T17:32:59","slug":"parametric-curves-and-their-applications-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/parametric-curves-and-their-applications-get-stronger\/","title":{"raw":"Parametric Curves and Their Applications: Get Stronger","rendered":"Parametric Curves and Their Applications: Get Stronger"},"content":{"raw":"

Fundamentals of Parametric Equations<\/span><\/h2>\r\n

For the following exercises (1-2), sketch the curves below by eliminating the parameter [latex]t[\/latex]. Give the orientation of the curve.<\/strong><\/p>\r\n\r\n

    \r\n \t
  1. [latex]x={t}^{2}+2t[\/latex], [latex]y=t+1[\/latex]<\/li>\r\n \t
  2. [latex]x=2t+4,y=t - 1[\/latex]<\/li>\r\n<\/ol>\r\n

    For the following exercise, eliminate the parameter and sketch the graphs.<\/strong><\/p>\r\n\r\n

      \r\n \t
    1. [latex]x=2{t}^{2},y={t}^{4}+1[\/latex]<\/li>\r\n<\/ol>\r\n

      For the following exercises (4-5), use technology (CAS or calculator) to sketch the parametric equations.<\/strong><\/p>\r\n\r\n

        \r\n \t
      1. [latex]\\begin{array}{cc}x={e}^{\\text{-}t},\\hfill & y={e}^{2t}-1\\hfill \\end{array}[\/latex]<\/li>\r\n \t
      2. [latex]\\begin{array}{cc}x=\\sec{t},\\hfill & y=\\cos{t}\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n

        For the following exercises (6-10), sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph.<\/strong><\/p>\r\n\r\n

          \r\n \t
        1. [latex]x=6\\sin\\left(2\\theta \\right),y=4\\cos\\left(2\\theta \\right)[\/latex]<\/li>\r\n \t
        2. [latex]\\begin{array}{cc}x=3 - 2\\cos\\theta ,\\hfill & y=-5+3\\sin\\theta \\hfill \\end{array}[\/latex]<\/li>\r\n \t
        3. [latex]\\begin{array}{cc}x=\\sec{t},\\hfill & y=\\tan{t}\\hfill \\end{array}[\/latex]<\/li>\r\n \t
        4. [latex]\\begin{array}{cc}x={e}^{t},\\hfill & y={e}^{2t}\\hfill \\end{array}[\/latex]<\/li>\r\n \t
        5. [latex]\\begin{array}{cc}x={t}^{3},\\hfill & y=3\\text{ln}t\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n

          For the following exercises (11-19), convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form.<\/strong><\/p>\r\n\r\n

            \r\n \t
          1. [latex]\\begin{array}{cc}x={t}^{2}-1,\\hfill & y=\\dfrac{t}{2}\\hfill \\end{array}[\/latex]<\/li>\r\n \t
          2. [latex]x=4\\cos\\theta ,y=3\\sin\\theta ,t\\in \\left(0,2\\pi \\right][\/latex]<\/li>\r\n \t
          3. [latex]\\begin{array}{cc}x=2t - 3,\\hfill & y=6t - 7\\hfill \\end{array}[\/latex]<\/li>\r\n \t
          4. [latex]\\begin{array}{cc}x=1+\\cos{t},\\hfill & y=3-\\sin{t}\\hfill \\end{array}[\/latex]<\/li>\r\n \t
          5. [latex]\\begin{array}{cc}x=\\sec{t},\\hfill & y=\\tan{t},\\pi \\le t<\\dfrac{3\\pi }{2}\\hfill \\end{array}[\/latex]<\/li>\r\n \t
          6. [latex]\\begin{array}{cc}x=\\cos\\left(2t\\right),\\hfill & y=\\sin{t}\\hfill \\end{array}[\/latex]<\/li>\r\n \t
          7. [latex]\\begin{array}{cc}x={t}^{2},\\hfill & y=2\\text{ln}t,t\\ge 1\\hfill \\end{array}[\/latex]<\/li>\r\n \t
          8. [latex]\\begin{array}{cc}x={t}^{n},\\hfill & y=n\\text{ln}t,t\\ge 1,\\hfill \\end{array}[\/latex] where n is a natural number<\/li>\r\n \t
          9. [latex]\\begin{array}{c}x=2\\sin\\left(8t\\right)\\hfill \\ y=2\\cos\\left(8t\\right)\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n

            For the following exercises (20-24), the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents.<\/strong><\/p>\r\n\r\n

              \r\n \t
            1. [latex]\\begin{array}{c}x=3t+4\\hfill \\ y=5t - 2\\hfill \\end{array}[\/latex]<\/li>\r\n \t
            2. [latex]\\begin{array}{c}x=2t+1\\hfill \\ y={t}^{2}-3\\hfill \\end{array}[\/latex]<\/li>\r\n \t
            3. [latex]\\begin{array}{c}x=2\\cos\\left(3t\\right)\\hfill \\ y=2\\sin\\left(3t\\right)\\hfill \\end{array}[\/latex]<\/li>\r\n \t
            4. [latex]\\begin{array}{c}x=3\\cos{t}\\hfill \\ y=4\\sin{t}\\hfill \\end{array}[\/latex]<\/li>\r\n \t
            5. [latex]\\begin{array}{c}x=3\\text{cosh}\\left(4t\\right)\\hfill \\ y=4\\text{sinh}\\left(4t\\right)\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n

              For the following exercises (25-26), use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation.<\/strong><\/p>\r\n\r\n

                \r\n \t
              1. [latex]\\begin{array}{c}x=\\theta +\\sin\\theta \\hfill \\ y=1-\\cos\\theta \\hfill \\end{array}[\/latex]<\/li>\r\n \t
              2. [latex]\\begin{array}{c}x=t - 0.5\\sin{t}\\hfill \\ y=1 - 1.5\\cos{t}\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n

                Calculus with Parametric Curves<\/span><\/h2>\r\n

                For the following exercises (1-2), each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.<\/strong><\/p>\r\n\r\n

                  \r\n \t
                1. [latex]\\begin{array}{cc}x=8+2t,\\hfill & y=1\\hfill \\end{array}[\/latex]<\/li>\r\n \t
                2. [latex]\\begin{array}{cc}x=-5t+7,\\hfill & y=3t - 1\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n

                  For the following exercises (3-4), determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter.<\/strong><\/p>\r\n\r\n

                    \r\n \t
                  1. [latex]\\begin{array}{cc}x=\\cos{t},\\hfill & y=8\\sin{t},\\hfill \\end{array}t=\\dfrac{\\pi }{2}[\/latex]<\/li>\r\n \t
                  2. [latex]\\begin{array}{cc}x=t+\\dfrac{1}{t},\\hfill & y=t-\\dfrac{1}{t},t=1\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n

                    For the following exercises (5-6), find all points on the curve that have the given slope.<\/strong><\/p>\r\n\r\n

                      \r\n \t
                    1. [latex]\\begin{array}{cc}x=4\\cos{t},\\hfill & y=4\\sin{t},\\hfill \\end{array}[\/latex] slope = [latex]0.5[\/latex]<\/li>\r\n \t
                    2. [latex]\\begin{array}{cc}x=t+\\dfrac{1}{t},\\hfill & y=t-\\dfrac{1}{t},\\text{slope}=1\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n

                      For the following exercises (7-8), write the equation of the tangent line in Cartesian coordinates for the given parameter t.<\/strong><\/p>\r\n\r\n

                        \r\n \t
                      1. [latex]\\begin{array}{cc}x={e}^{\\sqrt{t}},\\hfill & y=1-\\text{ln}{t}^{2},t=1\\hfill \\end{array}[\/latex]<\/li>\r\n \t
                      2. [latex]\\begin{array}{cc}x={e}^{t},\\hfill & y={\\left(t - 1\\right)}^{2},\\text{at}\\left(1,1\\right)\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises (9-13), solve each problem.<\/strong>\r\n
                          \r\n \t
                        1. For [latex]x=\\sin\\left(2t\\right),y=2\\sin{t}[\/latex] where [latex]0\\le t<2\\pi [\/latex]. Find all values of [latex]t[\/latex] at which a vertical tangent line exists.<\/li>\r\n \t
                        2. Find [latex]\\dfrac{dy}{dx}[\/latex] for [latex]x=\\sin\\left(t\\right),y=\\cos\\left(t\\right)[\/latex].<\/li>\r\n \t
                        3. For the curve [latex]x=4t,y=3t - 2[\/latex], find the slope and concavity of the curve at [latex]t=3[\/latex].<\/li>\r\n \t
                        4. Find the slope and concavity for the curve whose equation is [latex]x=2+\\sec\\theta ,y=1+2\\tan\\theta [\/latex] at [latex]\\theta =\\dfrac{\\pi }{6}[\/latex].<\/li>\r\n \t
                        5. Find all points on the curve [latex]x=\\sec\\theta ,y=\\tan\\theta [\/latex] at which horizontal and vertical tangents exist.<\/li>\r\n<\/ol>\r\n

                          For the following exercise, find [latex]\\dfrac{{d}^{2}y}{d{x}^{2}}[\/latex].<\/strong><\/p>\r\n\r\n

                            \r\n \t
                          1. [latex]\\begin{array}{cc}x=\\sin\\left(\\pi t\\right),\\hfill & y=\\cos\\left(\\pi t\\right)\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n

                            For the following exercise, find points on the curve at which tangent line is horizontal or vertical.<\/strong><\/p>\r\n\r\n

                              \r\n \t
                            1. [latex]\\begin{array}{cc}x=t\\left({t}^{2}-3\\right),\\hfill & y=3\\left({t}^{2}-3\\right)\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n

                              For the following exercises (16-17), find [latex]\\dfrac{dy}{dx}[\/latex] at the value of the parameter.<\/strong><\/p>\r\n\r\n

                                \r\n \t
                              1. [latex]\\begin{array}{cc}x=\\cos{t},\\hfill & y=\\sin{t},t=\\dfrac{3\\pi }{4}\\hfill \\end{array}[\/latex]<\/li>\r\n \t
                              2. [latex]\\begin{array}{cc}x=4\\cos{(2\\pi s)},\\hfill & y=3\\sin{(2\\pi s)}\\hfill \\end{array} ,s=-\\dfrac{1}{4}[\/latex]<\/li>\r\n<\/ol>\r\n

                                For the following exercise, find [latex]\\dfrac{{d}^{2}y}{d{x}^{2}}[\/latex] at the given point without eliminating the parameter.<\/strong><\/p>\r\n\r\n

                                  \r\n \t
                                1. [latex]x=\\sqrt{t},y=2t+4,t=1[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises (19-21), solve each problem.<\/strong>\r\n
                                    \r\n \t
                                  1. Determine the concavity of the curve [latex]x=2t+\\text{ln}t,y=2t-\\text{ln}t[\/latex].<\/li>\r\n \t
                                  2. Find the area bounded by the curve [latex]x=\\cos{t},y={e}^{t},0\\le t\\le \\dfrac{\\pi }{2}[\/latex] and the lines [latex]y=1[\/latex] and [latex]x=0[\/latex].<\/li>\r\n \t
                                  3. Find the area of the region bounded by [latex]x=2{\\sin}^{2}\\theta ,y=2{\\sin}^{2}\\theta \\tan\\theta [\/latex], for [latex]0\\le \\theta \\le \\dfrac{\\pi }{2}[\/latex].<\/li>\r\n<\/ol>\r\n

                                    For the following exercises (22-23), find the area of the regions bounded by the parametric curves and the indicated values of the parameter.<\/strong><\/p>\r\n\r\n

                                      \r\n \t
                                    1. \u00a0[latex]x=2a\\cos{t}-a\\cos\\left(2t\\right),y=2a\\sin{t}-a\\sin\\left(2t\\right),0\\le t<2\\pi [\/latex]<\/li>\r\n \t
                                    2. [latex]x=2a\\cos{t}-a\\sin\\left(2t\\right),y=b\\sin{t},0\\le t<2\\pi [\/latex] (the \"teardrop\")<\/li>\r\n<\/ol>\r\n

                                      For the following exercises (24-26), find the arc length of the curve on the indicated interval of the parameter.<\/strong><\/p>\r\n\r\n

                                        \r\n \t
                                      1. [latex]\\begin{array}{ccc}x=\\dfrac{1}{3}{t}^{3},\\hfill & y=\\dfrac{1}{2}{t}^{2},\\hfill & 0\\le t\\le 1\\hfill \\end{array}[\/latex]<\/li>\r\n \t
                                      2. [latex]\\begin{array}{ccc}x=1+{t}^{2},\\hfill & y={\\left(1+t\\right)}^{3},\\hfill & 0\\le t\\le 1\\hfill \\end{array}[\/latex]<\/li>\r\n \t
                                      3. [latex]x=a{\\cos}^{3}\\theta ,y=a{\\sin}^{3}\\theta [\/latex] on the interval [latex]\\left[0,2\\pi \\right)[\/latex] (the hypocycloid)<\/li>\r\n<\/ol>\r\n

                                        For the following exercises (27-29), find the area of the surface obtained by rotating the given curve about the [latex]x[\/latex]-axis.<\/strong><\/p>\r\n\r\n

                                          \r\n \t
                                        1. [latex]\\begin{array}{ccc}x={t}^{3},\\hfill & y={t}^{2},\\hfill & 0\\le t\\le 1\\hfill \\end{array}[\/latex]<\/li>\r\n \t
                                        2. Use a CAS to find the area of the surface generated by rotating [latex]x=t+{t}^{3},y=t-\\dfrac{1}{{t}^{2}},1\\le t\\le 2[\/latex] about the [latex]x[\/latex]-axis. (Answer to three decimal places.)<\/li>\r\n \t
                                        3. Find the area of the surface generated by revolving [latex]x={t}^{2},y=2t,0\\le t\\le 4[\/latex] about the [latex]x[\/latex]-axis.<\/li>\r\n<\/ol>","rendered":"

                                          Fundamentals of Parametric Equations<\/span><\/h2>\n

                                          For the following exercises (1-2), sketch the curves below by eliminating the parameter [latex]t[\/latex]. Give the orientation of the curve.<\/strong><\/p>\n

                                            \n
                                          1. [latex]x={t}^{2}+2t[\/latex], [latex]y=t+1[\/latex]<\/li>\n
                                          2. [latex]x=2t+4,y=t - 1[\/latex]<\/li>\n<\/ol>\n

                                            For the following exercise, eliminate the parameter and sketch the graphs.<\/strong><\/p>\n

                                              \n
                                            1. [latex]x=2{t}^{2},y={t}^{4}+1[\/latex]<\/li>\n<\/ol>\n

                                              For the following exercises (4-5), use technology (CAS or calculator) to sketch the parametric equations.<\/strong><\/p>\n

                                                \n
                                              1. [latex]\\begin{array}{cc}x={e}^{\\text{-}t},\\hfill & y={e}^{2t}-1\\hfill \\end{array}[\/latex]<\/li>\n
                                              2. [latex]\\begin{array}{cc}x=\\sec{t},\\hfill & y=\\cos{t}\\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n

                                                For the following exercises (6-10), sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph.<\/strong><\/p>\n

                                                  \n
                                                1. [latex]x=6\\sin\\left(2\\theta \\right),y=4\\cos\\left(2\\theta \\right)[\/latex]<\/li>\n
                                                2. [latex]\\begin{array}{cc}x=3 - 2\\cos\\theta ,\\hfill & y=-5+3\\sin\\theta \\hfill \\end{array}[\/latex]<\/li>\n
                                                3. [latex]\\begin{array}{cc}x=\\sec{t},\\hfill & y=\\tan{t}\\hfill \\end{array}[\/latex]<\/li>\n
                                                4. [latex]\\begin{array}{cc}x={e}^{t},\\hfill & y={e}^{2t}\\hfill \\end{array}[\/latex]<\/li>\n
                                                5. [latex]\\begin{array}{cc}x={t}^{3},\\hfill & y=3\\text{ln}t\\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n

                                                  For the following exercises (11-19), convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form.<\/strong><\/p>\n

                                                    \n
                                                  1. [latex]\\begin{array}{cc}x={t}^{2}-1,\\hfill & y=\\dfrac{t}{2}\\hfill \\end{array}[\/latex]<\/li>\n
                                                  2. [latex]x=4\\cos\\theta ,y=3\\sin\\theta ,t\\in \\left(0,2\\pi \\right][\/latex]<\/li>\n
                                                  3. [latex]\\begin{array}{cc}x=2t - 3,\\hfill & y=6t - 7\\hfill \\end{array}[\/latex]<\/li>\n
                                                  4. [latex]\\begin{array}{cc}x=1+\\cos{t},\\hfill & y=3-\\sin{t}\\hfill \\end{array}[\/latex]<\/li>\n
                                                  5. [latex]\\begin{array}{cc}x=\\sec{t},\\hfill & y=\\tan{t},\\pi \\le t<\\dfrac{3\\pi }{2}\\hfill \\end{array}[\/latex]<\/li>\n
                                                  6. [latex]\\begin{array}{cc}x=\\cos\\left(2t\\right),\\hfill & y=\\sin{t}\\hfill \\end{array}[\/latex]<\/li>\n
                                                  7. [latex]\\begin{array}{cc}x={t}^{2},\\hfill & y=2\\text{ln}t,t\\ge 1\\hfill \\end{array}[\/latex]<\/li>\n
                                                  8. [latex]\\begin{array}{cc}x={t}^{n},\\hfill & y=n\\text{ln}t,t\\ge 1,\\hfill \\end{array}[\/latex] where n is a natural number<\/li>\n
                                                  9. [latex]\\begin{array}{c}x=2\\sin\\left(8t\\right)\\hfill \\ y=2\\cos\\left(8t\\right)\\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n

                                                    For the following exercises (20-24), the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents.<\/strong><\/p>\n

                                                      \n
                                                    1. [latex]\\begin{array}{c}x=3t+4\\hfill \\ y=5t - 2\\hfill \\end{array}[\/latex]<\/li>\n
                                                    2. [latex]\\begin{array}{c}x=2t+1\\hfill \\ y={t}^{2}-3\\hfill \\end{array}[\/latex]<\/li>\n
                                                    3. [latex]\\begin{array}{c}x=2\\cos\\left(3t\\right)\\hfill \\ y=2\\sin\\left(3t\\right)\\hfill \\end{array}[\/latex]<\/li>\n
                                                    4. [latex]\\begin{array}{c}x=3\\cos{t}\\hfill \\ y=4\\sin{t}\\hfill \\end{array}[\/latex]<\/li>\n
                                                    5. [latex]\\begin{array}{c}x=3\\text{cosh}\\left(4t\\right)\\hfill \\ y=4\\text{sinh}\\left(4t\\right)\\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n

                                                      For the following exercises (25-26), use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation.<\/strong><\/p>\n

                                                        \n
                                                      1. [latex]\\begin{array}{c}x=\\theta +\\sin\\theta \\hfill \\ y=1-\\cos\\theta \\hfill \\end{array}[\/latex]<\/li>\n
                                                      2. [latex]\\begin{array}{c}x=t - 0.5\\sin{t}\\hfill \\ y=1 - 1.5\\cos{t}\\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n

                                                        Calculus with Parametric Curves<\/span><\/h2>\n

                                                        For the following exercises (1-2), each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.<\/strong><\/p>\n

                                                          \n
                                                        1. [latex]\\begin{array}{cc}x=8+2t,\\hfill & y=1\\hfill \\end{array}[\/latex]<\/li>\n
                                                        2. [latex]\\begin{array}{cc}x=-5t+7,\\hfill & y=3t - 1\\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n

                                                          For the following exercises (3-4), determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter.<\/strong><\/p>\n

                                                            \n
                                                          1. [latex]\\begin{array}{cc}x=\\cos{t},\\hfill & y=8\\sin{t},\\hfill \\end{array}t=\\dfrac{\\pi }{2}[\/latex]<\/li>\n
                                                          2. [latex]\\begin{array}{cc}x=t+\\dfrac{1}{t},\\hfill & y=t-\\dfrac{1}{t},t=1\\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n

                                                            For the following exercises (5-6), find all points on the curve that have the given slope.<\/strong><\/p>\n

                                                              \n
                                                            1. [latex]\\begin{array}{cc}x=4\\cos{t},\\hfill & y=4\\sin{t},\\hfill \\end{array}[\/latex] slope = [latex]0.5[\/latex]<\/li>\n
                                                            2. [latex]\\begin{array}{cc}x=t+\\dfrac{1}{t},\\hfill & y=t-\\dfrac{1}{t},\\text{slope}=1\\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n

                                                              For the following exercises (7-8), write the equation of the tangent line in Cartesian coordinates for the given parameter t.<\/strong><\/p>\n

                                                                \n
                                                              1. [latex]\\begin{array}{cc}x={e}^{\\sqrt{t}},\\hfill & y=1-\\text{ln}{t}^{2},t=1\\hfill \\end{array}[\/latex]<\/li>\n
                                                              2. [latex]\\begin{array}{cc}x={e}^{t},\\hfill & y={\\left(t - 1\\right)}^{2},\\text{at}\\left(1,1\\right)\\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n

                                                                For the following exercises (9-13), solve each problem.<\/strong><\/p>\n

                                                                  \n
                                                                1. For [latex]x=\\sin\\left(2t\\right),y=2\\sin{t}[\/latex] where [latex]0\\le t<2\\pi[\/latex]. Find all values of [latex]t[\/latex] at which a vertical tangent line exists.<\/li>\n
                                                                2. Find [latex]\\dfrac{dy}{dx}[\/latex] for [latex]x=\\sin\\left(t\\right),y=\\cos\\left(t\\right)[\/latex].<\/li>\n
                                                                3. For the curve [latex]x=4t,y=3t - 2[\/latex], find the slope and concavity of the curve at [latex]t=3[\/latex].<\/li>\n
                                                                4. Find the slope and concavity for the curve whose equation is [latex]x=2+\\sec\\theta ,y=1+2\\tan\\theta[\/latex] at [latex]\\theta =\\dfrac{\\pi }{6}[\/latex].<\/li>\n
                                                                5. Find all points on the curve [latex]x=\\sec\\theta ,y=\\tan\\theta[\/latex] at which horizontal and vertical tangents exist.<\/li>\n<\/ol>\n

                                                                  For the following exercise, find [latex]\\dfrac{{d}^{2}y}{d{x}^{2}}[\/latex].<\/strong><\/p>\n

                                                                    \n
                                                                  1. [latex]\\begin{array}{cc}x=\\sin\\left(\\pi t\\right),\\hfill & y=\\cos\\left(\\pi t\\right)\\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n

                                                                    For the following exercise, find points on the curve at which tangent line is horizontal or vertical.<\/strong><\/p>\n

                                                                      \n
                                                                    1. [latex]\\begin{array}{cc}x=t\\left({t}^{2}-3\\right),\\hfill & y=3\\left({t}^{2}-3\\right)\\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n

                                                                      For the following exercises (16-17), find [latex]\\dfrac{dy}{dx}[\/latex] at the value of the parameter.<\/strong><\/p>\n

                                                                        \n
                                                                      1. [latex]\\begin{array}{cc}x=\\cos{t},\\hfill & y=\\sin{t},t=\\dfrac{3\\pi }{4}\\hfill \\end{array}[\/latex]<\/li>\n
                                                                      2. [latex]\\begin{array}{cc}x=4\\cos{(2\\pi s)},\\hfill & y=3\\sin{(2\\pi s)}\\hfill \\end{array} ,s=-\\dfrac{1}{4}[\/latex]<\/li>\n<\/ol>\n

                                                                        For the following exercise, find [latex]\\dfrac{{d}^{2}y}{d{x}^{2}}[\/latex] at the given point without eliminating the parameter.<\/strong><\/p>\n

                                                                          \n
                                                                        1. [latex]x=\\sqrt{t},y=2t+4,t=1[\/latex]<\/li>\n<\/ol>\n

                                                                          For the following exercises (19-21), solve each problem.<\/strong><\/p>\n

                                                                            \n
                                                                          1. Determine the concavity of the curve [latex]x=2t+\\text{ln}t,y=2t-\\text{ln}t[\/latex].<\/li>\n
                                                                          2. Find the area bounded by the curve [latex]x=\\cos{t},y={e}^{t},0\\le t\\le \\dfrac{\\pi }{2}[\/latex] and the lines [latex]y=1[\/latex] and [latex]x=0[\/latex].<\/li>\n
                                                                          3. Find the area of the region bounded by [latex]x=2{\\sin}^{2}\\theta ,y=2{\\sin}^{2}\\theta \\tan\\theta[\/latex], for [latex]0\\le \\theta \\le \\dfrac{\\pi }{2}[\/latex].<\/li>\n<\/ol>\n

                                                                            For the following exercises (22-23), find the area of the regions bounded by the parametric curves and the indicated values of the parameter.<\/strong><\/p>\n

                                                                              \n
                                                                            1. \u00a0[latex]x=2a\\cos{t}-a\\cos\\left(2t\\right),y=2a\\sin{t}-a\\sin\\left(2t\\right),0\\le t<2\\pi[\/latex]<\/li>\n
                                                                            2. [latex]x=2a\\cos{t}-a\\sin\\left(2t\\right),y=b\\sin{t},0\\le t<2\\pi[\/latex] (the “teardrop”)<\/li>\n<\/ol>\n

                                                                              For the following exercises (24-26), find the arc length of the curve on the indicated interval of the parameter.<\/strong><\/p>\n

                                                                                \n
                                                                              1. [latex]\\begin{array}{ccc}x=\\dfrac{1}{3}{t}^{3},\\hfill & y=\\dfrac{1}{2}{t}^{2},\\hfill & 0\\le t\\le 1\\hfill \\end{array}[\/latex]<\/li>\n
                                                                              2. [latex]\\begin{array}{ccc}x=1+{t}^{2},\\hfill & y={\\left(1+t\\right)}^{3},\\hfill & 0\\le t\\le 1\\hfill \\end{array}[\/latex]<\/li>\n
                                                                              3. [latex]x=a{\\cos}^{3}\\theta ,y=a{\\sin}^{3}\\theta[\/latex] on the interval [latex]\\left[0,2\\pi \\right)[\/latex] (the hypocycloid)<\/li>\n<\/ol>\n

                                                                                For the following exercises (27-29), find the area of the surface obtained by rotating the given curve about the [latex]x[\/latex]-axis.<\/strong><\/p>\n

                                                                                  \n
                                                                                1. [latex]\\begin{array}{ccc}x={t}^{3},\\hfill & y={t}^{2},\\hfill & 0\\le t\\le 1\\hfill \\end{array}[\/latex]<\/li>\n
                                                                                2. Use a CAS to find the area of the surface generated by rotating [latex]x=t+{t}^{3},y=t-\\dfrac{1}{{t}^{2}},1\\le t\\le 2[\/latex] about the [latex]x[\/latex]-axis. (Answer to three decimal places.)<\/li>\n
                                                                                3. Find the area of the surface generated by revolving [latex]x={t}^{2},y=2t,0\\le t\\le 4[\/latex] about the [latex]x[\/latex]-axis.<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":17,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":675,"module-header":"- Select 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\/1005"}],"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\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1005\/revisions"}],"predecessor-version":[{"id":1818,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1005\/revisions\/1818"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/675"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1005\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1005"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1005"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1005"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1005"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}