{"id":1170,"date":"2025-06-30T16:42:50","date_gmt":"2025-06-30T16:42:50","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&p=1170"},"modified":"2025-08-14T17:47:13","modified_gmt":"2025-08-14T17:47:13","slug":"polar-coordinates-and-conic-sections-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/polar-coordinates-and-conic-sections-get-stronger\/","title":{"raw":"Polar Coordinates and Conic Sections: Get Stronger","rendered":"Polar Coordinates and Conic Sections: Get Stronger"},"content":{"raw":"

Understanding Polar Coordinates<\/span><\/h2>\r\n

In the following exercises (1-4), plot the point whose polar coordinates are given by first constructing the angle [latex]\\theta [\/latex] and then marking off the distance [latex]r[\/latex] along the ray.<\/strong><\/p>\r\n\r\n

    \r\n \t
  1. [latex]\\left(3,\\dfrac{\\pi }{6}\\right)[\/latex]<\/li>\r\n \t
  2. [latex]\\left(0,\\dfrac{7\\pi }{6}\\right)[\/latex]<\/li>\r\n \t
  3. [latex]\\left(1,\\dfrac{\\pi }{4}\\right)[\/latex]<\/li>\r\n \t
  4. [latex]\\left(1,\\dfrac{\\pi }{2}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n

    For the following exercises (5-6), consider the polar graph below. Give two sets of polar coordinates for each point.\r\n\"The<\/span>\r\n<\/strong><\/p>\r\n\r\n

      \r\n \t
    1. Coordinates of point B.<\/li>\r\n \t
    2. Coordinates of point D.<\/li>\r\n<\/ol>\r\n

      For the following exercises (7-9), the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in [latex]\\left(0,2\\pi \\right][\/latex]. Round to three decimal places.<\/strong><\/p>\r\n\r\n

        \r\n \t
      1. [latex]\\left(3,-4\\right)[\/latex]<\/li>\r\n \t
      2. [latex]\\left(-6,8\\right)[\/latex]<\/li>\r\n \t
      3. [latex]\\left(3,\\text{-}\\sqrt{3}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n

        For the following exercises (10-12), find rectangular coordinates for the given point in polar coordinates.<\/strong><\/p>\r\n\r\n

          \r\n \t
        1. [latex]\\left(-2,\\dfrac{\\pi }{6}\\right)[\/latex]<\/li>\r\n \t
        2. [latex]\\left(1,\\dfrac{7\\pi }{6}\\right)[\/latex]<\/li>\r\n \t
        3. [latex]\\left(0,\\dfrac{\\pi }{2}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n

          For the following exercises (13-15), determine whether the graphs of the polar equation are symmetric with respect to the [latex]x[\/latex] -axis, the [latex]y[\/latex] -axis, or the origin.<\/strong><\/p>\r\n\r\n

            \r\n \t
          1. [latex]r=3\\sin\\left(2\\theta \\right)[\/latex]<\/li>\r\n \t
          2. [latex]r=\\cos\\left(\\dfrac{\\theta }{5}\\right)[\/latex]<\/li>\r\n \t
          3. [latex]r=1+\\cos\\theta [\/latex]<\/li>\r\n<\/ol>\r\n

            For the following exercises (16-17), describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.<\/strong><\/p>\r\n\r\n

              \r\n \t
            1. [latex]\\theta =\\dfrac{\\pi }{4}[\/latex]<\/li>\r\n \t
            2. [latex]r=\\csc\\theta [\/latex]<\/li>\r\n<\/ol>\r\n

              For the following exercise, convert the rectangular equation to polar form and sketch its graph.<\/strong><\/p>\r\n\r\n

                \r\n \t
              1. [latex]{x}^{2}-{y}^{2}=16[\/latex]<\/li>\r\n<\/ol>\r\n

                For the following exercise, convert the rectangular equation to polar form and sketch its graph.<\/strong><\/p>\r\n\r\n

                  \r\n \t
                1. [latex]3x-y=2[\/latex]<\/li>\r\n<\/ol>\r\n

                  For the following exercises (20-21), convert the polar equation to rectangular form and sketch its graph.<\/strong><\/p>\r\n\r\n

                    \r\n \t
                  1. [latex]r=4\\sin\\theta [\/latex]<\/li>\r\n \t
                  2. [latex]r=\\theta [\/latex]<\/li>\r\n<\/ol>\r\n

                    For the following exercises (22-27), sketch a graph of the polar equation and identify any symmetry.<\/strong><\/p>\r\n\r\n

                      \r\n \t
                    1. [latex]r=1+\\sin\\theta [\/latex]<\/li>\r\n \t
                    2. [latex]r=2 - 2\\sin\\theta [\/latex]<\/li>\r\n \t
                    3. [latex]r=3\\cos\\left(2\\theta \\right)[\/latex]<\/li>\r\n \t
                    4. [latex]r=2\\cos\\left(3\\theta \\right)[\/latex]<\/li>\r\n \t
                    5. [latex]{r}^{2}=4\\cos\\left(2\\theta \\right)[\/latex]<\/li>\r\n \t
                    6. [latex]r=2\\theta [\/latex]<\/li>\r\n<\/ol>\r\n

                      Area and Arc Length in Polar Coordinates<\/span><\/h2>\r\n

                      For the following exercises (1-6), determine a definite integral that represents the area.<\/strong><\/p>\r\n\r\n

                        \r\n \t
                      1. Region enclosed by [latex]r=3\\sin\\theta [\/latex]<\/li>\r\n \t
                      2. Region enclosed by one petal of [latex]r=8\\sin\\left(2\\theta \\right)[\/latex]<\/li>\r\n \t
                      3. Region below the polar axis and enclosed by [latex]r=1-\\sin\\theta [\/latex]<\/li>\r\n \t
                      4. Region enclosed by the inner loop of [latex]r=2 - 3\\sin\\theta [\/latex]<\/li>\r\n \t
                      5. Region enclosed by [latex]r=1 - 2\\cos\\theta [\/latex] and outside the inner loop<\/li>\r\n \t
                      6. Region common to [latex]r=2\\text{ and }r=4\\cos\\theta [\/latex]<\/li>\r\n<\/ol>\r\n

                        For the following exercises (7-13), find the area of the described region.<\/strong><\/p>\r\n\r\n

                          \r\n \t
                        1. Enclosed by [latex]r=6\\sin\\theta [\/latex]<\/li>\r\n \t
                        2. Below the polar axis and enclosed by [latex]r=2-\\cos\\theta [\/latex]<\/li>\r\n \t
                        3. Enclosed by one petal of [latex]r=3\\cos\\left(2\\theta \\right)[\/latex]<\/li>\r\n \t
                        4. Enclosed by the inner loop of [latex]r=3+6\\cos\\theta [\/latex]<\/li>\r\n \t
                        5. Common interior of [latex]r=4\\sin\\left(2\\theta \\right)\\text{and }r=2[\/latex]<\/li>\r\n \t
                        6. Common interior of [latex]r=6\\sin\\theta \\text{ and }r=3[\/latex]<\/li>\r\n \t
                        7. Common interior of [latex]r=2+2\\cos\\theta \\text{ and }r=2\\sin\\theta [\/latex]<\/li>\r\n<\/ol>\r\n

                          For the following exercises (14-15), find a definite integral that represents the arc length.<\/strong><\/p>\r\n\r\n

                            \r\n \t
                          1. [latex]r=1+\\sin\\theta [\/latex] on the interval [latex]0\\le \\theta \\le 2\\pi [\/latex]<\/li>\r\n \t
                          2. [latex]r={e}^{\\theta }\\text{ on the interval }0\\le \\theta \\le 1[\/latex]<\/li>\r\n<\/ol>\r\n

                            For the following exercises (16-17), find the length of the curve over the given interval.<\/strong><\/p>\r\n\r\n

                              \r\n \t
                            1. [latex]r={e}^{3\\theta }\\text{ on the interval }0\\le \\theta \\le 2[\/latex]<\/li>\r\n \t
                            2. [latex]r=8+8\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\pi [\/latex]<\/li>\r\n<\/ol>\r\n

                              For the following exercises (18-20), use the integration capabilities of a calculator to approximate the length of the curve.<\/strong><\/p>\r\n\r\n

                                \r\n \t
                              1. [latex]r=3\\theta \\text{ on the interval }0\\le \\theta \\le \\dfrac{\\pi }{2}[\/latex]<\/li>\r\n \t
                              2. [latex]r={\\sin}^{2}\\left(\\dfrac{\\theta }{2}\\right)\\text{ on the interval }0\\le \\theta \\le \\pi [\/latex]<\/li>\r\n \t
                              3. [latex]r=\\sin\\left(3\\cos\\theta \\right)\\text{ on the interval }0\\le \\theta \\le \\pi [\/latex]<\/li>\r\n<\/ol>\r\n

                                For the following exercise, use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral.<\/strong><\/p>\r\n\r\n

                                  \r\n \t
                                1. [latex]r=\\sin\\theta +\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\pi [\/latex]<\/li>\r\n<\/ol>\r\n

                                  For the following exercises (22-23), use the familiar formula from geometry to find the length of the curve and then confirm using the definite integral.<\/strong><\/p>\r\n\r\n

                                    \r\n \t
                                  1. [latex]r=3\\sin\\theta \\text{ on the interval }0\\le \\theta \\le \\pi [\/latex]<\/li>\r\n \t
                                  2. [latex]r=6\\sin\\theta +8\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\pi [\/latex]<\/li>\r\n<\/ol>\r\n

                                    For the following exercises (24-29), find the slope of a tangent line to a polar curve [latex]r=f\\left(\\theta \\right)[\/latex]. Let [latex]x=r\\cos\\theta =f\\left(\\theta \\right)\\cos\\theta [\/latex] and [latex]y=r\\sin\\theta =f\\left(\\theta \\right)\\sin\\theta [\/latex], so the polar equation [latex]r=f\\left(\\theta \\right)[\/latex] is now written in parametric form.<\/strong><\/p>\r\n\r\n

                                      \r\n \t
                                    1. Use the definition of the derivative [latex]\\dfrac{dy}{dx}=\\dfrac{\\dfrac{dy}{d}\\theta }{\\dfrac{dx}{d}\\theta }[\/latex] and the product rule to derive the derivative of a polar equation.<\/li>\r\n \t
                                    2. [latex]r=4\\cos\\theta [\/latex]; [latex]\\left(2,\\dfrac{\\pi }{3}\\right)[\/latex]<\/li>\r\n \t
                                    3. [latex]r=4+\\sin\\theta [\/latex]; [latex]\\left(3,\\dfrac{3\\pi }{2}\\right)[\/latex]<\/li>\r\n \t
                                    4. [latex]r=4\\cos\\left(2\\theta \\right)[\/latex]; tips of the leaves<\/li>\r\n \t
                                    5. [latex]r=2\\theta [\/latex]; [latex]\\left(\\dfrac{\\pi }{2},\\dfrac{\\pi }{4}\\right)[\/latex]<\/li>\r\n \t
                                    6. For the cardioid [latex]r=1+\\sin\\theta [\/latex], find the slope of the tangent line when [latex]\\theta =\\dfrac{\\pi }{3}[\/latex].<\/li>\r\n<\/ol>\r\n

                                      For the following exercises (30-31), find the slope of the tangent line to the given polar curve at the point given by the value of [latex]\\theta [\/latex].<\/strong><\/p>\r\n\r\n

                                        \r\n \t
                                      1. [latex]r=\\theta [\/latex], [latex]\\theta =\\dfrac{\\pi }{2}[\/latex]<\/li>\r\n \t
                                      2. Use technology: [latex]r=2+4\\cos\\theta [\/latex] at [latex]\\theta =\\dfrac{\\pi }{6}[\/latex]<\/li>\r\n<\/ol>\r\n

                                        For the following exercises (32-33), find the points at which the following polar curves have a horizontal or vertical tangent line.<\/strong><\/p>\r\n\r\n

                                          \r\n \t
                                        1. [latex]{r}^{2}=4\\cos\\left(2\\theta \\right)[\/latex]<\/li>\r\n \t
                                        2. The cardioid [latex]r=1+\\sin\\theta [\/latex]<\/li>\r\n<\/ol>\r\n

                                          Conic Sections<\/span><\/h2>\r\n

                                          For the following exercises (1-4), determine the equation of the parabola using the information given.<\/strong><\/p>\r\n\r\n

                                            \r\n \t
                                          1. Focus [latex]\\left(4,0\\right)[\/latex] and directrix [latex]x=-4[\/latex]<\/li>\r\n \t
                                          2. Focus [latex]\\left(0,0.5\\right)[\/latex] and directrix [latex]y=-0.5[\/latex]<\/li>\r\n \t
                                          3. Focus [latex]\\left(0,2\\right)[\/latex] and directrix [latex]y=4[\/latex]<\/li>\r\n \t
                                          4. Focus [latex]\\left(-3,5\\right)[\/latex] and directrix [latex]y=1[\/latex]<\/li>\r\n<\/ol>\r\n

                                            For the following exercises (5-8), determine the equation of the ellipse using the information given.<\/strong><\/p>\r\n\r\n

                                              \r\n \t
                                            1. Endpoints of major axis at [latex]\\left(4,0\\right),\\left(-4,0\\right)[\/latex] and foci located at [latex]\\left(2,0\\right),\\left(-2,0\\right)[\/latex]<\/li>\r\n \t
                                            2. Endpoints of major axis at [latex]\\left(0,2\\right),\\left(0,-2\\right)[\/latex] and foci located at [latex]\\left(3,0\\right),\\left(-3,0\\right)[\/latex]<\/li>\r\n \t
                                            3. Endpoints of major axis at [latex]\\left(-3,5\\right),\\left(-3,-3\\right)[\/latex] and foci located at [latex]\\left(-3,3\\right),\\left(-3,-1\\right)[\/latex]<\/li>\r\n \t
                                            4. Foci located at [latex]\\left(2,0\\right),\\left(-2,0\\right)[\/latex] and eccentricity of [latex]\\dfrac{1}{2}[\/latex]<\/li>\r\n<\/ol>\r\n

                                              For the following exercises (9-12), determine the equation of the hyperbola using the information given.<\/strong><\/p>\r\n\r\n

                                                \r\n \t
                                              1. Vertices located at [latex]\\left(5,0\\right),\\left(-5,0\\right)[\/latex] and foci located at [latex]\\left(6,0\\right),\\left(-6,0\\right)[\/latex]<\/li>\r\n \t
                                              2. Endpoints of the conjugate axis located at [latex]\\left(0,3\\right),\\left(0,-3\\right)[\/latex] and foci located [latex]\\left(4,0\\right),\\left(-4,0\\right)[\/latex]<\/li>\r\n \t
                                              3. Vertices located at [latex]\\left(-2,0\\right),\\left(-2,-4\\right)[\/latex] and focus located at [latex]\\left(-2,-8\\right)[\/latex]<\/li>\r\n \t
                                              4. Foci located at [latex]\\left(6,-0\\right),\\left(6,0\\right)[\/latex] and eccentricity of [latex]3[\/latex]<\/li>\r\n<\/ol>\r\n

                                                For the following exercises (13-15), consider the following polar equations of conics. Determine the eccentricity and identify the conic.<\/strong><\/p>\r\n\r\n

                                                  \r\n \t
                                                1. [latex]r=\\dfrac{-1}{1+\\cos\\theta }[\/latex]<\/li>\r\n \t
                                                2. [latex]r=\\dfrac{5}{2+\\sin\\theta }[\/latex]<\/li>\r\n \t
                                                3. [latex]r=\\dfrac{3}{2 - 6\\sin\\theta }[\/latex]<\/li>\r\n<\/ol>\r\n

                                                  For the following exercises (16-17), find a polar equation of the conic with focus at the origin and eccentricity and directrix as given.<\/strong><\/p>\r\n\r\n

                                                    \r\n \t
                                                  1. [latex]\\text{Directrix:}x=4;e=\\dfrac{1}{5}[\/latex]<\/li>\r\n \t
                                                  2. [latex]\\text{Directrix: y}=2;e=2[\/latex]<\/li>\r\n<\/ol>\r\n

                                                    For the following equations (18-20), determine which of the conic sections is described.<\/strong><\/p>\r\n\r\n

                                                      \r\n \t
                                                    1. [latex]{x}^{2}+4xy - 2{y}^{2}-6=0[\/latex]<\/li>\r\n \t
                                                    2. [latex]{x}^{2}-xy+{y}^{2}-2=0[\/latex]<\/li>\r\n \t
                                                    3. [latex]52{x}^{2}-72xy+73{y}^{2}+40x+30y - 75=0[\/latex]<\/li>\r\n<\/ol>","rendered":"

                                                      Understanding Polar Coordinates<\/span><\/h2>\n

                                                      In the following exercises (1-4), plot the point whose polar coordinates are given by first constructing the angle [latex]\\theta[\/latex] and then marking off the distance [latex]r[\/latex] along the ray.<\/strong><\/p>\n

                                                        \n
                                                      1. [latex]\\left(3,\\dfrac{\\pi }{6}\\right)[\/latex]<\/li>\n
                                                      2. [latex]\\left(0,\\dfrac{7\\pi }{6}\\right)[\/latex]<\/li>\n
                                                      3. [latex]\\left(1,\\dfrac{\\pi }{4}\\right)[\/latex]<\/li>\n
                                                      4. [latex]\\left(1,\\dfrac{\\pi }{2}\\right)[\/latex]<\/li>\n<\/ol>\n

                                                        For the following exercises (5-6), consider the polar graph below. Give two sets of polar coordinates for each point.
                                                        \n\"The<\/span>
                                                        \n<\/strong><\/p>\n

                                                          \n
                                                        1. Coordinates of point B.<\/li>\n
                                                        2. Coordinates of point D.<\/li>\n<\/ol>\n

                                                          For the following exercises (7-9), the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in [latex]\\left(0,2\\pi \\right][\/latex]. Round to three decimal places.<\/strong><\/p>\n

                                                            \n
                                                          1. [latex]\\left(3,-4\\right)[\/latex]<\/li>\n
                                                          2. [latex]\\left(-6,8\\right)[\/latex]<\/li>\n
                                                          3. [latex]\\left(3,\\text{-}\\sqrt{3}\\right)[\/latex]<\/li>\n<\/ol>\n

                                                            For the following exercises (10-12), find rectangular coordinates for the given point in polar coordinates.<\/strong><\/p>\n

                                                              \n
                                                            1. [latex]\\left(-2,\\dfrac{\\pi }{6}\\right)[\/latex]<\/li>\n
                                                            2. [latex]\\left(1,\\dfrac{7\\pi }{6}\\right)[\/latex]<\/li>\n
                                                            3. [latex]\\left(0,\\dfrac{\\pi }{2}\\right)[\/latex]<\/li>\n<\/ol>\n

                                                              For the following exercises (13-15), determine whether the graphs of the polar equation are symmetric with respect to the [latex]x[\/latex] -axis, the [latex]y[\/latex] -axis, or the origin.<\/strong><\/p>\n

                                                                \n
                                                              1. [latex]r=3\\sin\\left(2\\theta \\right)[\/latex]<\/li>\n
                                                              2. [latex]r=\\cos\\left(\\dfrac{\\theta }{5}\\right)[\/latex]<\/li>\n
                                                              3. [latex]r=1+\\cos\\theta[\/latex]<\/li>\n<\/ol>\n

                                                                For the following exercises (16-17), describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.<\/strong><\/p>\n

                                                                  \n
                                                                1. [latex]\\theta =\\dfrac{\\pi }{4}[\/latex]<\/li>\n
                                                                2. [latex]r=\\csc\\theta[\/latex]<\/li>\n<\/ol>\n

                                                                  For the following exercise, convert the rectangular equation to polar form and sketch its graph.<\/strong><\/p>\n

                                                                    \n
                                                                  1. [latex]{x}^{2}-{y}^{2}=16[\/latex]<\/li>\n<\/ol>\n

                                                                    For the following exercise, convert the rectangular equation to polar form and sketch its graph.<\/strong><\/p>\n

                                                                      \n
                                                                    1. [latex]3x-y=2[\/latex]<\/li>\n<\/ol>\n

                                                                      For the following exercises (20-21), convert the polar equation to rectangular form and sketch its graph.<\/strong><\/p>\n

                                                                        \n
                                                                      1. [latex]r=4\\sin\\theta[\/latex]<\/li>\n
                                                                      2. [latex]r=\\theta[\/latex]<\/li>\n<\/ol>\n

                                                                        For the following exercises (22-27), sketch a graph of the polar equation and identify any symmetry.<\/strong><\/p>\n

                                                                          \n
                                                                        1. [latex]r=1+\\sin\\theta[\/latex]<\/li>\n
                                                                        2. [latex]r=2 - 2\\sin\\theta[\/latex]<\/li>\n
                                                                        3. [latex]r=3\\cos\\left(2\\theta \\right)[\/latex]<\/li>\n
                                                                        4. [latex]r=2\\cos\\left(3\\theta \\right)[\/latex]<\/li>\n
                                                                        5. [latex]{r}^{2}=4\\cos\\left(2\\theta \\right)[\/latex]<\/li>\n
                                                                        6. [latex]r=2\\theta[\/latex]<\/li>\n<\/ol>\n

                                                                          Area and Arc Length in Polar Coordinates<\/span><\/h2>\n

                                                                          For the following exercises (1-6), determine a definite integral that represents the area.<\/strong><\/p>\n

                                                                            \n
                                                                          1. Region enclosed by [latex]r=3\\sin\\theta[\/latex]<\/li>\n
                                                                          2. Region enclosed by one petal of [latex]r=8\\sin\\left(2\\theta \\right)[\/latex]<\/li>\n
                                                                          3. Region below the polar axis and enclosed by [latex]r=1-\\sin\\theta[\/latex]<\/li>\n
                                                                          4. Region enclosed by the inner loop of [latex]r=2 - 3\\sin\\theta[\/latex]<\/li>\n
                                                                          5. Region enclosed by [latex]r=1 - 2\\cos\\theta[\/latex] and outside the inner loop<\/li>\n
                                                                          6. Region common to [latex]r=2\\text{ and }r=4\\cos\\theta[\/latex]<\/li>\n<\/ol>\n

                                                                            For the following exercises (7-13), find the area of the described region.<\/strong><\/p>\n

                                                                              \n
                                                                            1. Enclosed by [latex]r=6\\sin\\theta[\/latex]<\/li>\n
                                                                            2. Below the polar axis and enclosed by [latex]r=2-\\cos\\theta[\/latex]<\/li>\n
                                                                            3. Enclosed by one petal of [latex]r=3\\cos\\left(2\\theta \\right)[\/latex]<\/li>\n
                                                                            4. Enclosed by the inner loop of [latex]r=3+6\\cos\\theta[\/latex]<\/li>\n
                                                                            5. Common interior of [latex]r=4\\sin\\left(2\\theta \\right)\\text{and }r=2[\/latex]<\/li>\n
                                                                            6. Common interior of [latex]r=6\\sin\\theta \\text{ and }r=3[\/latex]<\/li>\n
                                                                            7. Common interior of [latex]r=2+2\\cos\\theta \\text{ and }r=2\\sin\\theta[\/latex]<\/li>\n<\/ol>\n

                                                                              For the following exercises (14-15), find a definite integral that represents the arc length.<\/strong><\/p>\n

                                                                                \n
                                                                              1. [latex]r=1+\\sin\\theta[\/latex] on the interval [latex]0\\le \\theta \\le 2\\pi[\/latex]<\/li>\n
                                                                              2. [latex]r={e}^{\\theta }\\text{ on the interval }0\\le \\theta \\le 1[\/latex]<\/li>\n<\/ol>\n

                                                                                For the following exercises (16-17), find the length of the curve over the given interval.<\/strong><\/p>\n

                                                                                  \n
                                                                                1. [latex]r={e}^{3\\theta }\\text{ on the interval }0\\le \\theta \\le 2[\/latex]<\/li>\n
                                                                                2. [latex]r=8+8\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\pi[\/latex]<\/li>\n<\/ol>\n

                                                                                  For the following exercises (18-20), use the integration capabilities of a calculator to approximate the length of the curve.<\/strong><\/p>\n

                                                                                    \n
                                                                                  1. [latex]r=3\\theta \\text{ on the interval }0\\le \\theta \\le \\dfrac{\\pi }{2}[\/latex]<\/li>\n
                                                                                  2. [latex]r={\\sin}^{2}\\left(\\dfrac{\\theta }{2}\\right)\\text{ on the interval }0\\le \\theta \\le \\pi[\/latex]<\/li>\n
                                                                                  3. [latex]r=\\sin\\left(3\\cos\\theta \\right)\\text{ on the interval }0\\le \\theta \\le \\pi[\/latex]<\/li>\n<\/ol>\n

                                                                                    For the following exercise, use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral.<\/strong><\/p>\n

                                                                                      \n
                                                                                    1. [latex]r=\\sin\\theta +\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\pi[\/latex]<\/li>\n<\/ol>\n

                                                                                      For the following exercises (22-23), use the familiar formula from geometry to find the length of the curve and then confirm using the definite integral.<\/strong><\/p>\n

                                                                                        \n
                                                                                      1. [latex]r=3\\sin\\theta \\text{ on the interval }0\\le \\theta \\le \\pi[\/latex]<\/li>\n
                                                                                      2. [latex]r=6\\sin\\theta +8\\cos\\theta \\text{ on the interval }0\\le \\theta \\le \\pi[\/latex]<\/li>\n<\/ol>\n

                                                                                        For the following exercises (24-29), find the slope of a tangent line to a polar curve [latex]r=f\\left(\\theta \\right)[\/latex]. Let [latex]x=r\\cos\\theta =f\\left(\\theta \\right)\\cos\\theta[\/latex] and [latex]y=r\\sin\\theta =f\\left(\\theta \\right)\\sin\\theta[\/latex], so the polar equation [latex]r=f\\left(\\theta \\right)[\/latex] is now written in parametric form.<\/strong><\/p>\n

                                                                                          \n
                                                                                        1. Use the definition of the derivative [latex]\\dfrac{dy}{dx}=\\dfrac{\\dfrac{dy}{d}\\theta }{\\dfrac{dx}{d}\\theta }[\/latex] and the product rule to derive the derivative of a polar equation.<\/li>\n
                                                                                        2. [latex]r=4\\cos\\theta[\/latex]; [latex]\\left(2,\\dfrac{\\pi }{3}\\right)[\/latex]<\/li>\n
                                                                                        3. [latex]r=4+\\sin\\theta[\/latex]; [latex]\\left(3,\\dfrac{3\\pi }{2}\\right)[\/latex]<\/li>\n
                                                                                        4. [latex]r=4\\cos\\left(2\\theta \\right)[\/latex]; tips of the leaves<\/li>\n
                                                                                        5. [latex]r=2\\theta[\/latex]; [latex]\\left(\\dfrac{\\pi }{2},\\dfrac{\\pi }{4}\\right)[\/latex]<\/li>\n
                                                                                        6. For the cardioid [latex]r=1+\\sin\\theta[\/latex], find the slope of the tangent line when [latex]\\theta =\\dfrac{\\pi }{3}[\/latex].<\/li>\n<\/ol>\n

                                                                                          For the following exercises (30-31), find the slope of the tangent line to the given polar curve at the point given by the value of [latex]\\theta[\/latex].<\/strong><\/p>\n

                                                                                            \n
                                                                                          1. [latex]r=\\theta[\/latex], [latex]\\theta =\\dfrac{\\pi }{2}[\/latex]<\/li>\n
                                                                                          2. Use technology: [latex]r=2+4\\cos\\theta[\/latex] at [latex]\\theta =\\dfrac{\\pi }{6}[\/latex]<\/li>\n<\/ol>\n

                                                                                            For the following exercises (32-33), find the points at which the following polar curves have a horizontal or vertical tangent line.<\/strong><\/p>\n

                                                                                              \n
                                                                                            1. [latex]{r}^{2}=4\\cos\\left(2\\theta \\right)[\/latex]<\/li>\n
                                                                                            2. The cardioid [latex]r=1+\\sin\\theta[\/latex]<\/li>\n<\/ol>\n

                                                                                              Conic Sections<\/span><\/h2>\n

                                                                                              For the following exercises (1-4), determine the equation of the parabola using the information given.<\/strong><\/p>\n

                                                                                                \n
                                                                                              1. Focus [latex]\\left(4,0\\right)[\/latex] and directrix [latex]x=-4[\/latex]<\/li>\n
                                                                                              2. Focus [latex]\\left(0,0.5\\right)[\/latex] and directrix [latex]y=-0.5[\/latex]<\/li>\n
                                                                                              3. Focus [latex]\\left(0,2\\right)[\/latex] and directrix [latex]y=4[\/latex]<\/li>\n
                                                                                              4. Focus [latex]\\left(-3,5\\right)[\/latex] and directrix [latex]y=1[\/latex]<\/li>\n<\/ol>\n

                                                                                                For the following exercises (5-8), determine the equation of the ellipse using the information given.<\/strong><\/p>\n

                                                                                                  \n
                                                                                                1. Endpoints of major axis at [latex]\\left(4,0\\right),\\left(-4,0\\right)[\/latex] and foci located at [latex]\\left(2,0\\right),\\left(-2,0\\right)[\/latex]<\/li>\n
                                                                                                2. Endpoints of major axis at [latex]\\left(0,2\\right),\\left(0,-2\\right)[\/latex] and foci located at [latex]\\left(3,0\\right),\\left(-3,0\\right)[\/latex]<\/li>\n
                                                                                                3. Endpoints of major axis at [latex]\\left(-3,5\\right),\\left(-3,-3\\right)[\/latex] and foci located at [latex]\\left(-3,3\\right),\\left(-3,-1\\right)[\/latex]<\/li>\n
                                                                                                4. Foci located at [latex]\\left(2,0\\right),\\left(-2,0\\right)[\/latex] and eccentricity of [latex]\\dfrac{1}{2}[\/latex]<\/li>\n<\/ol>\n

                                                                                                  For the following exercises (9-12), determine the equation of the hyperbola using the information given.<\/strong><\/p>\n

                                                                                                    \n
                                                                                                  1. Vertices located at [latex]\\left(5,0\\right),\\left(-5,0\\right)[\/latex] and foci located at [latex]\\left(6,0\\right),\\left(-6,0\\right)[\/latex]<\/li>\n
                                                                                                  2. Endpoints of the conjugate axis located at [latex]\\left(0,3\\right),\\left(0,-3\\right)[\/latex] and foci located [latex]\\left(4,0\\right),\\left(-4,0\\right)[\/latex]<\/li>\n
                                                                                                  3. Vertices located at [latex]\\left(-2,0\\right),\\left(-2,-4\\right)[\/latex] and focus located at [latex]\\left(-2,-8\\right)[\/latex]<\/li>\n
                                                                                                  4. Foci located at [latex]\\left(6,-0\\right),\\left(6,0\\right)[\/latex] and eccentricity of [latex]3[\/latex]<\/li>\n<\/ol>\n

                                                                                                    For the following exercises (13-15), consider the following polar equations of conics. Determine the eccentricity and identify the conic.<\/strong><\/p>\n

                                                                                                      \n
                                                                                                    1. [latex]r=\\dfrac{-1}{1+\\cos\\theta }[\/latex]<\/li>\n
                                                                                                    2. [latex]r=\\dfrac{5}{2+\\sin\\theta }[\/latex]<\/li>\n
                                                                                                    3. [latex]r=\\dfrac{3}{2 - 6\\sin\\theta }[\/latex]<\/li>\n<\/ol>\n

                                                                                                      For the following exercises (16-17), find a polar equation of the conic with focus at the origin and eccentricity and directrix as given.<\/strong><\/p>\n

                                                                                                        \n
                                                                                                      1. [latex]\\text{Directrix:}x=4;e=\\dfrac{1}{5}[\/latex]<\/li>\n
                                                                                                      2. [latex]\\text{Directrix: y}=2;e=2[\/latex]<\/li>\n<\/ol>\n

                                                                                                        For the following equations (18-20), determine which of the conic sections is described.<\/strong><\/p>\n

                                                                                                          \n
                                                                                                        1. [latex]{x}^{2}+4xy - 2{y}^{2}-6=0[\/latex]<\/li>\n
                                                                                                        2. [latex]{x}^{2}-xy+{y}^{2}-2=0[\/latex]<\/li>\n
                                                                                                        3. [latex]52{x}^{2}-72xy+73{y}^{2}+40x+30y - 75=0[\/latex]<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":23,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":677,"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\/1170"}],"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":7,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1170\/revisions"}],"predecessor-version":[{"id":1830,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1170\/revisions\/1830"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/677"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1170\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1170"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1170"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1170"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1170"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}