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All Textbook Solutions for Calculus Volume 3

Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However. perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the "versoria," a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, "versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a circle of radius a so that the points (0,0) and (0,2a) are points on the circle (Figure 1.12). Let O denote the origin. Choose any other point A on the (tilde, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through (0,2a) . The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution, In this project you will parameterize these curves. Figure 1.12 As the point A moves around the circle, the point P traces out the witch of Agnesi curve for the given circle. 1. On the figure, label the following points, lengths, and angle: a. C is the point on the x-axis with the same x-coordinate as A. b. x is the x-coordinate of P, and y is the y-coordinate of P. c. E is the point (0,a) . d. F is the point on the line segment OA such that the line segment EF is perpendicular to the line segment OA. e. b is the distance from O to F. f. c is the distance from F to A. g. d is the distance from O to B. h. is the measure of angle COA . The goal of this project is to parameterize the witch using as a parameter. To do this, write equations for x and y in terms of only .Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However. perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the "versoria," a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, "versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a circle of radius a so that the points (0,0) and (0,2a) are points on the circle (Figure 1.12). Let O denote the origin. Choose any other point A on the (tilde, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through (0,2a) . The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution, In this project you will parameterize these curves. Figure 1.12 As the point A moves around the circle, the point P traces out the witch of Agnesi curve for the given circle. Show that d=2asin.Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However. perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the "versoria," a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, "versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a circle of radius a so that the points (0,0) and (0,2a) are points on the circle (Figure 1.12). Let O denote the origin. Choose any other point A on the (tilde, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through (0,2a) . The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution, In this project you will parameterize these curves. Figure 1.12 As the point A moves around the circle, the point P traces out the witch of Agnesi curve for the given circle. 3. Note that x=dcos . Show that x=2acot . When you do this, you will have parameterized the x-coordinate of the curve with respect to . If you can get a similar equation for y, you will have parameterized the curve.Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However. perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the "versoria," a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, "versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a circle of radius a so that the points (0,0) and (0,2a) are points on the circle (Figure 1.12). Let O denote the origin. Choose any other point A on the (tilde, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through (0,2a) . The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution, In this project you will parameterize these curves. Figure 1.12 As the point A moves around the circle, the point P traces out the witch of Agnesi curve for the given circle. 4. In terms of , what is the angle EOA ?Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However. perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the "versoria," a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, "versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a circle of radius a so that the points (0,0) and (0,2a) are points on the circle (Figure 1.12). Let O denote the origin. Choose any other point A on the (tilde, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through (0,2a) . The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution, In this project you will parameterize these curves. Figure 1.12 As the point A moves around the circle, the point P traces out the witch of Agnesi curve for the given circle. 5. Show that b+c=2acos(2) .Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However. perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the "versoria," a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, "versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a circle of radius a so that the points (0,0) and (0,2a) are points on the circle (Figure 1.12). Let O denote the origin. Choose any other point A on the (tilde, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through (0,2a) . The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution, In this project you will parameterize these curves. Figure 1.12 As the point A moves around the circle, the point P traces out the witch of Agnesi curve for the given circle. 6. Show that y=2acos(2) .Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However. perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the "versoria," a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, "versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a circle of radius a so that the points (0,0) and (0,2a) are points on the circle (Figure 1.12). Let O denote the origin. Choose any other point A on the (tilde, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through (0,2a) . The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution, In this project you will parameterize these curves. Figure 1.12 As the point A moves around the circle, the point P traces out the witch of Agnesi curve for the given circle. 7. Show that y=2asin2 . You have now parameterized the y-coordinate of the curve with respect to .Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However. perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the "versoria," a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, "versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a circle of radius a so that the points (0,0) and (0,2a) are points on the circle (Figure 1.12). Let O denote the origin. Choose any other point A on the (tilde, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through (0,2a) . The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution, In this project you will parameterize these curves. Figure 1.12 As the point A moves around the circle, the point P traces out the witch of Agnesi curve for the given circle. 8. Conclude that a parameterization of the given witch curve is x=2acot,y=2asin2, .Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However. perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the "versoria," a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, "versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a circle of radius a so that the points (0,0) and (0,2a) are points on the circle (Figure 1.12). Let O denote the origin. Choose any other point A on the (tilde, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through (0,2a) . The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution, In this project you will parameterize these curves. Figure 1.12 As the point A moves around the circle, the point P traces out the witch of Agnesi curve for the given circle. 9. Use yum parameterization to show that the given witch curve is the graph of the function f(x)=8a3x2+4a2 .Earlier in this section, we looked at the parametric equations for a cycloid, which is the path a point on the edge of a wheel traces as the wheel rolls along a straight path. In this project we look at two different variations of the cycloid. called the curtate and prolate cycloids. First. let’s revisit the derivation of the parametric equations for a cycloid. Recall that we considered a tenacious ant trying to get home by hanging onto the edge of a bicycle tire. We have assumed the ant climbed onto the tire at the very edge, where the tire touches the ground. As the wheel rolls, the ant moves with the edge of the tire (Figure 1.13). As we have discussed, we have a lot of flexibility when parameterizing a curve. In this case we let our parameter c represent the angle the tire has rotated through. Looking at Figure 1.13, we see that after the tire has rotated through an angle of t, the Position of the center of the wheel, C=(xc,yc) , is given by Furthermore, letting A=(xA,yA) denote the position of the ant, we note that xCxA=asint and yCyA=acost . Then xA=xcasint=atasint=a(tsint)yA=yCacost=aacost=a(1cost) . Figure 1.13 (a) The ant clings to die edge of the bicycle tile as the tire rolls along the ground. (b) Using geometry to determine the position of the ant after the tire has rotated through an angle of t. Note that these are the same parametric representations we had before, but we have now assigned a physical meaning to the parametric variable t. After a while the am is getting dizzy from going round and round on the edge of the tire. So he climbs up one of the spokes toward the center of the wheel. By climbing toward the center of the wheel. the ant has changed his path of motion. The new path has less up-and-down motion and is called a mutate cycloid (Figure 1.14). As shown in the figure. we let b denote the distance along the spoke flow the center of the wheel to the ant. As before, we let t represent the angle the fire has rotated through. Additionally, we let C=(xC,yC) represent the position of the center of the wheel and A=(xA,yA) represent the position of the ant. Figure 1.14 (a) The ant climbs up one of the spokes toward the center of the wheel. (b) The ant's path of motion after he climbs closer to the center of the wheel. This is called a mutate cycloid. (c) The new Setup, now that the ant has moved closer to the center of the wheel. 1. What is the position of the center of the wheel after the fire has rotated through an angle of t?Earlier in this section, we looked at the parametric equations for a cycloid, which is the path a point on the edge of a wheel traces as the wheel rolls along a straight path. In this project we look at two different variations of the cycloid. called the curtate and prolate cycloids. First. let’s revisit the derivation of the parametric equations for a cycloid. Recall that we considered a tenacious ant trying to get home by hanging onto the edge of a bicycle tire. We have assumed the ant climbed onto the tire at the very edge, where the tire touches the ground. As the wheel rolls, the ant moves with the edge of the tire (Figure 1.13). As we have discussed, we have a lot of flexibility when parameterizing a curve. In this case we let our parameter c represent the angle the tire has rotated through. Looking at Figure 1.13, we see that after the tire has rotated through an angle of t, the Position of the center of the wheel, C=(xc,yc) , is given by Furthermore, letting A=(xA,yA) denote the position of the ant, we note that xCxA=asint and yCyA=acost . Then xA=xcasint=atasint=a(tsint)yA=yCacost=aacost=a(1cost) . Figure 1.13 (a) The ant clings to die edge of the bicycle tile as the tire rolls along the ground. (b) Using geometry to determine the position of the ant after the tire has rotated through an angle of t. Note that these are the same parametric representations we had before, but we have now assigned a physical meaning to the parametric variable t. After a while the am is getting dizzy from going round and round on the edge of the tire. So he climbs up one of the spokes toward the center of the wheel. By climbing toward the center of the wheel. the ant has changed his path of motion. The new path has less up-and-down motion and is called a mutate cycloid (Figure 1.14). As shown in the figure. we let b denote the distance along the spoke flow the center of the wheel to the ant. As before, we let t represent the angle the fire has rotated through. Additionally, we let C=(xC,yC) represent the position of the center of the wheel and A=(xA,yA) represent the position of the ant. Figure 1.14 (a) The ant climbs up one of the spokes toward the center of the wheel. (b) The ant's path of motion after he climbs closer to the center of the wheel. This is called a mutate cycloid. (c) The new Setup, now that the ant has moved closer to the center of the wheel. 2. Use geometry to find expressions for xCxA and for yCyA .Earlier in this section, we looked at the parametric equations for a cycloid, which is the path a point on the edge of a wheel traces as the wheel rolls along a straight path. In this project we look at two different variations of the cycloid. called the curtate and prolate cycloids. First. let’s revisit the derivation of the parametric equations for a cycloid. Recall that we considered a tenacious ant trying to get home by hanging onto the edge of a bicycle tire. We have assumed the ant climbed onto the tire at the very edge, where the tire touches the ground. As the wheel rolls, the ant moves with the edge of the tire (Figure 1.13). As we have discussed, we have a lot of flexibility when parameterizing a curve. In this case we let our parameter c represent the angle the tire has rotated through. Looking at Figure 1.13, we see that after the tire has rotated through an angle of t, the Position of the center of the wheel, C=(xc,yc) , is given by Furthermore, letting A=(xA,yA) denote the position of the ant, we note that xCxA=asint and yCyA=acost . Then xA=xcasint=atasint=a(tsint)yA=yCacost=aacost=a(1cost) . Figure 1.13 (a) The ant clings to die edge of the bicycle tile as the tire rolls along the ground. (b) Using geometry to determine the position of the ant after the tire has rotated through an angle of t. Note that these are the same parametric representations we had before, but we have now assigned a physical meaning to the parametric variable t. After a while the am is getting dizzy from going round and round on the edge of the tire. So he climbs up one of the spokes toward the center of the wheel. By climbing toward the center of the wheel. the ant has changed his path of motion. The new path has less up-and-down motion and is called a mutate cycloid (Figure 1.14). As shown in the figure. we let b denote the distance along the spoke flow the center of the wheel to the ant. As before, we let t represent the angle the fire has rotated through. Additionally, we let C=(xC,yC) represent the position of the center of the wheel and A=(xA,yA) represent the position of the ant. Figure 1.14 (a) The ant climbs up one of the spokes toward the center of the wheel. (b) The ant's path of motion after he climbs closer to the center of the wheel. This is called a mutate cycloid. (c) The new Setup, now that the ant has moved closer to the center of the wheel. 3. On the basis of your answer: to pans 1 and 2, what are the parametric equations representing the curtate cycloid? Once the ant’s head clears, he realizes that the bicyclist has made a turn, and is now traveling away from his home. So he drops off the bicycle tile and looks around. Fortunately, there is a set of train tracks nearby, headed back in the right direction. So the ant heads over to the train tracks to wait. After a while, a train goes by, heading in the right direction, and he manages to jump up and just catch the edge of the train wheel (without getting squished!). The ant is still worried about getting dizzy, but the train wheel is slippery and has no spokes to climb, so he decides to just hang on to the edge of the wheel and hope for the best. Now, train wheels have a ?ange to keep the wheel running on the tracks. So, in this case, since the ant is hanging on to the very edge of the ?ange, the distance from the center of the wheel to the ant is actually greater than the radius of the wheel (Figure 1.15). The setup here is essentially the same as when the ant Climbed up the spoke on the bicycle wheel. We let b denote the distance from the center of the wheel to the ant, and we let t represent the angle the tire has rotated through. Additionally, we let C=(xC,yC) represent the position of the center of the wheel and A=(xA,yA) represent the position of the ant (Figure 1.15). When the distance from the center of the wheel to the ant is greater than the radius of the wheel, his path of motion is called a prolate cycloid. A graph of a prolate cycloid is shown in the figure. Figure 1.15 (a) The am is hanging onto the ?ange of the train wheel. (b) The new setup, now that the am has jumped onto the train wheel. {C} The ant travels along a prolate cycloid.Earlier in this section, we looked at the parametric equations for a cycloid, which is the path a point on the edge of a wheel traces as the wheel rolls along a straight path. In this project we look at two different variations of the cycloid. called the curtate and prolate cycloids. First. let’s revisit the derivation of the parametric equations for a cycloid. Recall that we considered a tenacious ant trying to get home by hanging onto the edge of a bicycle tire. We have assumed the ant climbed onto the tire at the very edge, where the tire touches the ground. As the wheel rolls, the ant moves with the edge of the tire (Figure 1.13). As we have discussed, we have a lot of flexibility when parameterizing a curve. In this case we let our parameter c represent the angle the tire has rotated through. Looking at Figure 1.13, we see that after the tire has rotated through an angle of t, the Position of the center of the wheel, C=(xc,yc) , is given by Furthermore, letting A=(xA,yA) denote the position of the ant, we note that xCxA=asint and yCyA=acost . Then xA=xcasint=atasint=a(tsint)yA=yCacost=aacost=a(1cost) . Figure 1.13 (a) The ant clings to die edge of the bicycle tile as the tire rolls along the ground. (b) Using geometry to determine the position of the ant after the tire has rotated through an angle of t. Note that these are the same parametric representations we had before, but we have now assigned a physical meaning to the parametric variable t. After a while the am is getting dizzy from going round and round on the edge of the tire. So he climbs up one of the spokes toward the center of the wheel. By climbing toward the center of the wheel. the ant has changed his path of motion. The new path has less up-and-down motion and is called a mutate cycloid (Figure 1.14). As shown in the figure. we let b denote the distance along the spoke flow the center of the wheel to the ant. As before, we let t represent the angle the fire has rotated through. Additionally, we let C=(xC,yC) represent the position of the center of the wheel and A=(xA,yA) represent the position of the ant. Figure 1.14 (a) The ant climbs up one of the spokes toward the center of the wheel. (b) The ant's path of motion after he climbs closer to the center of the wheel. This is called a mutate cycloid. (c) The new Setup, now that the ant has moved closer to the center of the wheel. 4. Using the same approach you used in parts 1- 3. find the parametric equations for the path of motion of the ant.Earlier in this section, we looked at the parametric equations for a cycloid, which is the path a point on the edge of a wheel traces as the wheel rolls along a straight path. In this project we look at two different variations of the cycloid. called the curtate and prolate cycloids. First. let’s revisit the derivation of the parametric equations for a cycloid. Recall that we considered a tenacious ant trying to get home by hanging onto the edge of a bicycle tire. We have assumed the ant climbed onto the tire at the very edge, where the tire touches the ground. As the wheel rolls, the ant moves with the edge of the tire (Figure 1.13). As we have discussed, we have a lot of flexibility when parameterizing a curve. In this case we let our parameter c represent the angle the tire has rotated through. Looking at Figure 1.13, we see that after the tire has rotated through an angle of t, the Position of the center of the wheel, C=(xc,yc) , is given by Furthermore, letting A=(xA,yA) denote the position of the ant, we note that xCxA=asint and yCyA=acost . Then xA=xcasint=atasint=a(tsint)yA=yCacost=aacost=a(1cost) . Figure 1.13 (a) The ant clings to die edge of the bicycle tile as the tire rolls along the ground. (b) Using geometry to determine the position of the ant after the tire has rotated through an angle of t. Note that these are the same parametric representations we had before, but we have now assigned a physical meaning to the parametric variable t. After a while the am is getting dizzy from going round and round on the edge of the tire. So he climbs up one of the spokes toward the center of the wheel. By climbing toward the center of the wheel. the ant has changed his path of motion. The new path has less up-and-down motion and is called a mutate cycloid (Figure 1.14). As shown in the figure. we let b denote the distance along the spoke flow the center of the wheel to the ant. As before, we let t represent the angle the fire has rotated through. Additionally, we let C=(xC,yC) represent the position of the center of the wheel and A=(xA,yA) represent the position of the ant. Figure 1.14 (a) The ant climbs up one of the spokes toward the center of the wheel. (b) The ant's path of motion after he climbs closer to the center of the wheel. This is called a mutate cycloid. (c) The new Setup, now that the ant has moved closer to the center of the wheel. 5. What do you notice about your answer to pan 3 and your answer to part 4? Notice that the ant is actually traveling backward at times (the “loops” in the graph), even though the train continues to move forward. He is probably going to be really dizzy by the time he gets home!For the following exercises, sketch the curves below by eliminating the parameter t. Give the orientation of the curve. x=t2+2t,y=t+1 For the following exercises, sketch the curves below by eliminating the parameter t. Give the orientation of the curve. x=cos(t),y=sin(t),(0,2]For the following exercises, sketch the curves below by eliminating the parameter t. Give the orientation of the curve. 3.x=2t+4,y=t1 For the following exercises, sketch the curves below by eliminating the parameter t. Give the orientation of the curve. 4. x=3t,y=2t3,1.5t3 For the following exercises, eliminate the parameter and sketch the graphs. 5. x=2t2,y=t4+1 For the following exercises, use technology (CAS or calculator) to sketch the parametric equations. 6. [T]x=t2+t,y=t21 For the following exercises, use technology (CAS or calculator) to sketch the parametric equations. 7. [T]x=et,y=e2t1 For the following exercises, use technology (CAS or calculator) to sketch the parametric equations. 8. [T]x=3cost,y=4sint For the following exercises, use technology (CAS or calculator) to sketch the parametric equations. 9. [T]x=sect,y=cost For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 10. x=et,y=e2t+1 For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 11. x=6sin(2),y=4cos(2)For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 12. x=cos,y=2sin(2)For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 13. x=32cos,y=5+3sin For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 14. x=4+2cos,y=1+sin For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 15. x=sect,y=tant For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 16. x=ln(2t),y=t2 For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 17. x=et,y=e2t For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 18. x=e2t,y=e3t For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 19. x=t3,y=3lnt For the following exercises, sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. 20. x=4sec,y=3tan For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 21. x=t21,y=t2 For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 22. x=1t+1,y=t1+t,t1 For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 23. x=4cos,y=3sin,t(0,2]For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 24. x=cosht,y=sinht For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 25. x=2t3,y=6t7 For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 26. x=t2,y=t3 For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 27. x=1+cost,y=3sint For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 28. x=t,y=2t+4 For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 29. x=sect,y=tant,t32 For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 30. x=2cosht,y=4sinht For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 31. x=cos(2t),y=sint For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 32. x=4t+3,y=16t29 For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 33. x=t2,y=2lnt,t1 For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 34. x=t3,y=3lnt,t1 For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 35. x=tn,y=nlnt,t1, where n is a natural number For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 36. x=ln(5t)y=ln(t2) where 1te For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 37. x=2sin(8t)y=2cos(8t)For the following exercises, convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. 38. x=tanty=sec2t1 For the following exercises, the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 39. x=3t+4y=5t2 For the following exercises, the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 40. x4=5ty+2=t For the following exercises, the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 41. x=2t+1y=t23 For the following exercises, the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 42. x=3costy=3sint For the following exercises, the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 43. x=2cos(3t)y=2sin(3t)For the following exercises, the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 44. x=coshty=sinht For the following exercises, the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 45. x=3costy=4sint For the following exercises, the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 46. x=2cos(3t)y=5sin(3t)For the following exercises, the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 48. x=2coshty=2sinht For the following exercises, the pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. 48. x=2coshty=2sinht Show that x=h+rcos , y=k+rsin represents the equation of a circle.Use the equations in the preceding problem to find a set of parametric equations for a circle whose radius is 5 and whose center is (2,3) .For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. 51.[T]y=1cosx=+sin For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. 52.[T]y=22costx=2t2sint For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. 53. [T]y=11.5costx=t0.5sint For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. 54. An airplane traveling horizontally at 100m/s over flat ground at an elevation of 4000 meters must drop an emergency package on a target on the ground. The trajectory of the package is given by x=100t,y=4.9t2+4000,t0 where the origin is the point on the ground directly beneath the plane at the moment of release. How many horizontal meters before the target should the package be released in order to hit the target?For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. 55. The trajectory of a bullet is given by x=v0(cos)ty=v0(sin)t12gt2 where v0=500m/s,g=9.8=9.8m/s2 and =30degrees . When will the bullet hit the ground? How far from the gun will the bullet hit the ground?For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. 56. [T] Use technology to sketch the curve represented by x=2tan(t),y=3sec(t),t .For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. 57. [T] Use technology to sketch x=2tan(t),y=3sec(t),t .For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. 58. Sketch the curve known as an epitrochoid, which gives the path of a point on a circle of radius b as it rolls on the outside of a circle of radius a. The equations are x=(a+b)costccos[( a+b)tb]y=(a+b)sintcsin[( a+b)tb] Let a=1,b=2,c=1 .For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. 59. [T] Use technology to sketch the spiral curve given by x=tcost,y=tsin(t) from 2t2 .For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. 60. [T] Use technology to graph the curve given by the parametric equations x=2cot(t),y=1cos(2t),/2t/2 . This curve is known as the witch of Agnesi.For the following exercises, use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. 61. [T] Sketch the curve given by parametric equations x=cosh(t)y=sinh(t) where 2t2 .For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line. 62. x=3+t,y=1t For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line. 63. x=8+2t,y=1 For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line. 64. x=43t,y=2+6t For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line. 65. x=5t+7,y=3t1 For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. 66. x=3sint,y=3cost,t=4 For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. 67. x=cost,y=8sint,t=2 For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. 68. x=2t,y=t3,t=1 For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. 69. x=t+1t,y=t1t,t=1 For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. 70. x=t,y=2t,t=4 For the following exercises, find all points on the curve that have the given slope. 71. x=4cost,y=4sint,slope=0.5 For the following exercises, find all points on the curve that have the given slope. 72. x=2cost,y=8sint,slope=1 For the following exercises, find all points on the curve that have the given slope. 73. x=t+1t,y=t1t,slope=1 For the following exercises, find all points on the curve that have the given slope. 74. x=2+t,y=24t,slope=0 For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. 75. x=et,y=1lnt2,t=1 For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. 76. x=tlnt,y=sin2t,t=4 For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. 77. x=et,y=(t1)2, at (1,1)For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. 78. For x=sin(2t),y=2sint where 0t2 . Find all values of t at which a horizontal tangent line exists.For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. 79. For x=sin(2t),y=2sint where 0t2 . Find all values of t at which a vertical tangent line exists.For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. 80. Find all points on the curve x=4cos(t),y=4sin(t) that have the slope of 12 .For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. 81. Find dydx for x=sin(t),y=cos(t) .For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. 82. Find the equation of the tangent line to x=sin(t),y=cos(t) at t=4 .For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. 83. For the curve x=4t,y=3t2 , find the slope and concavity 0f the curve at t=3 .For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. 84. For the parametric curve whose equation is x=4cos,y=4sin , find the slope and concavity of the curve at =4 .For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. 85. Find the slope and concavity for the curve whose equation is x=2+sec,y=1+2tan at =6 .For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. 86. Find all points on the curve x=t+4,y=t33t at which there are vertical and horizontal tangents.For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. 87. Find all points on the curve x=sec,y=tan at which horizontal and vertical tangents exist.For the following exercises. find d2y/dx2 . 88. x=t41,y=tt2 For the following exercises. find d2y/dx2 . 89. x=sin(t),y=cos(t)For the following exercises. find d2y/dx2 . 90. x=et,y=te2t For the following exercises, find points on the curve at which tangent line is horizontal or vertical. 91. x=t(t23),y=3(t23)For the following exercises, find points on the curve at which tangent line is horizontal or vertical. 92. x=3t1+t3,y=3t21+t3 For the following exercises, find dy/dx at the value of the parameter. 93. x=cost,y=sint,t=34 For the following exercises, find dy/dx at the value of the parameter. 94. x=t,y=2t+4,t=9 For the following exercises, find dy/dx at the value of the parameter. 95. x=4cos(2s),y=3sin(2s),s=14 For the following exercises, find d2y/dx2 at the given point without eliminating the parameter. 96. x=12t2,y=13t3,t=2 For the following exercises, find d2y/dx2 at the given point without eliminating the parameter. 97. x=t,y=2t+4,t=1 For the following exercises, find d2y/dx2 at the given point without eliminating the parameter. 98. Find t intervals on which the curve x=3t2,y=t3t is concave up as well as concave down.For the following exercises, find d2y/dx2 at the given point without eliminating the parameter. 99. Determine the concavity of the curve x=2t+lnt,y=2tlnt .For the following exercises, find d2y/dx2 at the given point without eliminating the parameter. 100. Sketch and find the area under one arch of the cycloid x=r(sin),y=r(1cos) .For the following exercises, find d2y/dx2 at the given point without eliminating the parameter. 101. Find the area bounded by the curve x=cost,y=et,0t2 and the lines y=1 and x=0 .For the following exercises, find d2y/dx2 at the given point without eliminating the parameter. 102. Find the area enclosed by the ellipse x=acos,y=bsin,02 .For the following exercises, find d2y/dx2 at the given point without eliminating the parameter. 103. Find the area of the region bounded by x=2sin2,y=2sin2tan , for 02 .For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter. 104. x=2cot,y=2sin2,0 For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter. 105. [T] x=2acostacos(2t),y=2asintasin(2t),0t2 For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter. 106. [T] x=asin(2t),y=bsin(t),0t2 (the “hourglass”)For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter. 107. [T] x=2acostasin(2t),y=bsint,0t2 (the “teardrop”)For the following exercises, find the arc length of the curve on the indicated interval of the parameter. 108. x=4t+3,y=3t2,0t2 For the following exercises, find the arc length of the curve on the indicated interval of the parameter. 109. x=13t3,y=12t2,0t1 For the following exercises, find the arc length of the curve on the indicated interval of the parameter. 110. x=cos(2t),y=sin(2t),0t2 For the following exercises, find the arc length of the curve on the indicated interval of the parameter. 111. x=1+t2,y=(1+t)3,0t1 For the following exercises, find the arc length of the curve on the indicated interval of the parameter. 112. x=etcost,y=etsint,0t2 (express answer as a decimal rounded to three places)For the following exercises, find the arc length of the curve on the indicated interval of the parameter. 113. x=acos3,y=asin3 on the interval [0,2) (the hypocycloid)For the following exercises, find the arc length of the curve on the indicated interval of the parameter. 114. Find the length of one arch of the cycloid x=4(tsint),y=4(1cost) .For the following exercises, find the arc length of the curve on the indicated interval of the parameter. 115. Find the distance traveled by a particle with position (x,y) as t varies in the given time interval: x=sin2t,y=cos2t,0t3 .For the following exercises, find the arc length of the curve on the indicated interval of the parameter. 116. Find the length of one arch of the cycloid x=sin,y=1cos .For the following exercises, find the arc length of the curve on the indicated interval of the parameter. 117. Show that the total length of the ellipse x=4sin,y=3cos is L=160/21 e 2 sin 2d where e=ca and c=a2b2 .For the following exercises, find the arc length of the curve on the indicated interval of the parameter. 118. Find the length of the curve x=ett,y=4et/2,8t3 .For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis. 119. x=t3,y=t2,0t1 For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis. 120. x=acos3,y=asin3,02 For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis. 121. [T] Use a CAS to find the area of the surface generated by rotating x=t+t3,y=t1t2,1t2 about the x-axis. (Answer to three decimal places.)For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis. 122. Find the surface area obtained by rotating x=3t2,y=2t3,0t5 about the y-axis.For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis. 123. Find the area of the surface generated by revolving x=t2,y=2t,0t4 about the x-axis.For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis. 124. Find the surface area generated by revolving x=t2,y=2t2,0t1 about the y-axis.In the following exercises. plot the point whose polar coordinates are given by first constructing the angle and then marking off the distance r along the ray. 125. (3,6)In the following exercises. plot the point whose polar coordinates are given by first constructing the angle and then marking off the distance r along the ray. 126. (2,53)In the following exercises. plot the point whose polar coordinates are given by first constructing the angle and then marking off the distance r along the ray. 127. (0,76)In the following exercises. plot the point whose polar coordinates are given by first constructing the angle and then marking off the distance r along the ray. 128. (4,34)In the following exercises. plot the point whose polar coordinates are given by first constructing the angle and then marking off the distance r along the ray. 129. (1,4)In the following exercises. plot the point whose polar coordinates are given by first constructing the angle and then marking off the distance r along the ray. 130. (2,56)In the following exercises. plot the point whose polar coordinates are given by first constructing the angle and then marking off the distance r along the ray. 131. (1,2)For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point. 132. Coordinates of point A.For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point. 133. Coordinates of point B.For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point. 134. Coordinates of point C.For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point. 135. Coordinates of point D.For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in (0,2] . Round to three decimal places. 136. (2,2)For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in (0,2] . Round to three decimal places. 137. (3,4)(3,4)For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in (0,2] . Round to three decimal places. 138. (8,15)For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in (0,2] . Round to three decimal places. 139. (6,8)For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in (0,2] . Round to three decimal places. 140. (4,3)For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in (0,2] . Round to three decimal places. 141. (3,3)For the following Exercises, find rectangular coordinates for the given point in polar coordinates. 142. (2,54)For the following Exercises, find rectangular coordinates for the given point in polar coordinates. 143. (2,6)For the following Exercises, find rectangular coordinates for the given point in polar coordinates. 144. (5,3)For the following Exercises, find rectangular coordinates for the given point in polar coordinates. 145. (1,76)For the following Exercises, find rectangular coordinates for the given point in polar coordinates. 146. (3,34)For the following Exercises, find rectangular coordinates for the given point in polar coordinates. 147. (0,2)For the following Exercises, find rectangular coordinates for the given point in polar coordinates. 148. (4.5,6.5)For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the x-axis, the y-axis, or the origin. 149. r=3sin(2)For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the x-axis, the y-axis, or the origin. 150. r2=9cos For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the x-axis, the y-axis, or the origin. 151. r=cos(5)For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the x-axis, the y-axis, or the origin. 152. r=2sec For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the x-axis, the y-axis, or the origin. 153. r=1+cos For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation. 154. r=3 For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation. 155. =4 For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation. 156. r=sec For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation. 157. r=csc For the following exercises, convert the rectangular equation to polar form and sketch its graph. 158. x2+y2=16 For the following exercises, convert the rectangular equation to polar form and sketch its graph. 159. x2y2=16 For the following exercises, convert the rectangular equation to polar form and sketch its graph. 160. x=8 For the following exercises, convert the rectangular equation to polar form and sketch its graph. 161. 3xy=2 For the following exercises, convert the rectangular equation to polar form and sketch its graph. 162. y2=4x For the following exercises, convert the polar equation to rectangular form and sketch its graph. 163. r=4sin For the following exercises, convert the polar equation to rectangular form and sketch its graph. 164. r=6cos For the following exercises, convert the polar equation to rectangular form and sketch its graph. 165. r=For the following exercises, convert the polar equation to rectangular form and sketch its graph. 166. r=cotcsc For the following exercises, sketch a graph of the polar equation and identify any symmetry. 167. r=1+sin For the following exercises, sketch a graph of the polar equation and identify any symmetry. 168. r=32cos For the following exercises, sketch a graph of the polar equation and identify any symmetry. 169. r=22cos For the following exercises, sketch a graph of the polar equation and identify any symmetry. 170. r=54sin For the following exercises, sketch a graph of the polar equation and identify any symmetry. 171. r=3cos(2)For the following exercises, sketch a graph of the polar equation and identify any symmetry. 172. r=3sin(2)For the following exercises, sketch a graph of the polar equation and identify any symmetry. 173. r=2cos(3)For the following exercises, sketch a graph of the polar equation and identify any symmetry. 174. r=3cos(2)For the following exercises, sketch a graph of the polar equation and identify any symmetry. 175. r2=4cos(2)For the following exercises, sketch a graph of the polar equation and identify any symmetry. 176. r2=4sin For the following exercises, sketch a graph of the polar equation and identify any symmetry. 177. r=2 [T] The graph of r=2cos(2)sec() . is called a strophoid. Use a graphing utility to sketch the graph, and, from the graph, determine the asymptote.[T] Use a graphing utility and sketch the graph of r=62sin3cos.[T] Use a graphing utility to graph r=11cos.[T] Use technology to graph r=esin()2cos(4) .[T] Use technology to plot r=sin(37) (use the interval 014 ).Without using technology, sketch the polar curve =23 .[T] Use a graphing utility to plot r=sin for [T] Use technology to plot r=e0.1 for 1010 .[T] There is a curve known as the “Black Hole”. Use technology to plot r=e0.01 for 100100 .[T] Use the results of the preceding two problems to explore the graphs of r=e0.001 and r=e0.0001 for ||100 .For the following exercises, determine a definite integral that represents the area. 188. Region enclosed by r=4 For the following exercises, determine a definite integral that represents the area. 189. Region enclosed by r=3sin For the following exercises, determine a definite integral that represents the area. 190. Region in the first quadrant within the cardioid r=1+sin For the following exercises, determine a definite integral that represents the area. 191. Region enclosed by one petal of r=8sin(2)For the following exercises, determine a definite integral that represents the area. 192. Region enclosed by one petal of r=cos(3)For the following exercises, determine a definite integral that represents the area. 193. Region below the polar axis and enclosed by r=1sin For the following exercises, determine a definite integral that represents the area. 194. Region in the first quadrant enclosed by r=2cos For the following exercises, determine a definite integral that represents the area. 195. Region enclosed by the inner loop of r=23sin For the following exercises, determine a definite integral that represents the area. 196. Region enclosed by the inner loop of r=34cos For the following exercises, determine a definite integral that represents the area. 197. Region enclosed by r=12cos and Outside the inner loop For the following exercises, determine a definite integral that represents the area. 198. Region common to r=3sin and r=2sin For the following exercises, determine a definite integral that represents the area. 199. Region common to r=2 and r=4cos For the following exercises, determine a definite integral that represents the area. 200. Region common to r=3cos and r=3sin For the following exercises, find the area of the described region. 201. Enclosed by r=6sin For the following exercises, find the area of the described region. 202. Above the polar axis enclosed by r=2+sin For the following exercises, find the area of the described region. 203. Below the polar axis and enclosed by r=2cos For the following exercises, find the area of the described region. 204. Enclosed by one petal of r=4cos(3)For the following exercises, find the area of the described region. 205. Enclosed by one petal of r=3cos(2)For the following exercises, find the area of the described region. 206. Enclosed by r=1+sin For the following exercises, find the area of the described region. 207. Enclosed by the inner loop of r=3+6cos For the following exercises, find the area of the described region. 208. Enclosed by r=2+4cos and outside the inner loop For the following exercises, find the area of the described region. 209. Common interior of r=4sin(2) and r=2 For the following exercises, find the area of the described region. 210. Common interior of r=32sin and r=3+2sin For the following exercises, find the area of the described region. 211. Common interior of r=6sin and r=3 For the following exercises, find the area of the described region. 212. Inside r=1+cos and outside r=cos For the following exercises, find the area of the described region. 213. Common interior of r=2+2cos and r=2sin For the following exercises, find a definite integral that represents the arc length. 214. r=4cos on the interval 02 For the following exercises, find a definite integral that represents the arc length. 215. r=1+sin on the interval 02 For the following exercises, find a definite integral that represents the arc length. 216. r=2sec on the interval 03 For the following exercises, find a definite integral that represents the arc length. 217. r=e on the interval 01 For the following exercises, find the length of the curve over the given interval. 218. r=6 on the interval 02 For the following exercises, find the length of the curve over the given interval. 219. r=e3 on the interval 02 For the following exercises, find the length of the curve over the given interval. 220. r=6cos on the interval 0+2 For the following exercises, find the length of the curve over the given interval. 221. r=8+8cos on the interval 0 For the following exercises, find the length of the curve over the given interval. 222. r=1sin on the interval 02 For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve. 223. [T]r=3 on the interval 02 For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve. 224. [T]r=2 on the interval 2 For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve. 225. [T]r=sin2(2) 0n the interval 0 For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve. 226. [T]r=22 on the interval 0 For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve. 227. [T]r=sin(3cos) on the interval 0 For the following exercises, use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral. 228. r=3sin on the interval 0 For the following exercises, use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral. 229. r=sin+cos on the interval 0 For the following exercises, use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral. 230. r=6sin+8cos on the interval 0 For the following exercises, use the familiar formula from geometry to find the length of the curve and then confirm using the definite integral. 231. r=3sin on the interval 0 For the following exercises, use the familiar formula from geometry to find the length of the curve and then confirm using the definite integral. 232. r=sin+cos on the interval 0 For the following exercises, use the familiar formula from geometry to find the length of the curve and then confirm using the definite integral. 233. r=6sin+8cos on the interval 0 For the following exercises, use the familiar formula from geometry to find the length of the curve and then confirm using the definite integral. 234. Verify that if y=rsin=f()sin then dyd=f()sin+f()cos For the following exercises, find the slope of a tangent line to a polar curve r=f() . Let x=rcos=f()cos and y=rsin=f()sin , so the polar equation r=f() is now written in parametric form. 235. Use the de?nition of the derivative dydx=dy/ddx/d and the product rule to derive the derivative of a polar equation.For the following exercises, find the slope of a tangent line to a polar curve r=f() . Let x=rcos=f()cos and y=rsin=f()sin , so the polar equation r=f() is now written in parametric form. 236. r=1sin;(12,6)For the following exercises, find the slope of a tangent line to a polar curve r=f() . Let x=rcos=f()cos and y=rsin=f()sin , so the polar equation r=f() is now written in parametric form. 237. r=4cos;(2,3)For the following exercises, find the slope of a tangent line to a polar curve r=f() . Let x=rcos=f()cos and y=rsin=f()sin , so the polar equation r=f() is now written in parametric form. 238. r=8sin;(4,56)For the following exercises, find the slope of a tangent line to a polar curve r=f() . Let x=rcos=f()cos and y=rsin=f()sin , so the polar equation r=f() is now written in parametric form. 239. r=4+sin;(3,32)For the following exercises, find the slope of a tangent line to a polar curve r=f() . Let x=rcos=f()cos and y=rsin=f()sin , so the polar equation r=f() is now written in parametric form. 240. r=6+3cos;(3,)For the following exercises, find the slope of a tangent line to a polar curve r=f() . Let x=rcos=f()cos and y=rsin=f()sin , so the polar equation r=f() is now written in parametric form. 241. r=4cos(2); tips of the leaves For the following exercises, find the slope of a tangent line to a polar curve r=f() . Let x=rcos=f()cos and y=rsin=f()sin , so the polar equation r=f() is now written in parametric form. 242. r=2sin(3) ; tips of the leaves For the following exercises, find the slope of a tangent line to a polar curve r=f() . Let x=rcos=f()cos and y=rsin=f()sin , so the polar equation r=f() is now written in parametric form. 243. r=2;(2,4)Find the points on the interval at which the cardioid r=1cos has a vertical or horizontal tangent line.For the cardioid r=1+sin , find the slope of the tangent line when =3 .For the following exercises, find the slope of the tangent line to the given polar curve at the point given by the value of . 246. r=3cos,=3 For the following exercises, find the slope of the tangent line to the given polar curve at the point given by the value of . 247. r=,=2 For the following exercises, find the slope of the tangent line to the given polar curve at the point given by the value of . 248. r=ln,=e For the following exercises, find the slope of the tangent line to the given polar curve at the point given by the value of . 249. [T] Use technology: r=2+4cos at =6 For the following Exercises, find the points at which the following polar curves have a horizontal or vertical tangent line. 250. r=4cos For the following Exercises, find the points at which the following polar curves have a horizontal or vertical tangent line. 251. r2=4cos(2)For the following Exercises, find the points at which the following polar curves have a horizontal or vertical tangent line. 252. r=2sin(2)For the following Exercises, find the points at which the following polar curves have a horizontal or vertical tangent line. 253. The cardioid r=1+sin For the following Exercises, find the points at which the following polar curves have a horizontal or vertical tangent line. 254. Show that the curve r=sintan (called a cissoid of Diocles) has the line x=1 as a vertical asymptote.For the following exercises, determine the equation of the parabola using the information given. 255. Focus (4,0) and directrix x=4 For the following exercises, determine the equation of the parabola using the information given. 256. Focus (0,3) and directrix y=3 For the following exercises, determine the equation of the parabola using the information given. 257. Focus (0,0.5) and directrix y=0.5 For the following exercises, determine the equation of the parabola using the information given. 258. Focus (2,3) and directrix x=2 For the following exercises, determine the equation of the parabola using the information given. 259. Focus (0,2) and directrix y=4 For the following exercises, determine the equation of the parabola using the information given. 260. Focus (1,4) and directrix x=5 For the following exercises, determine the equation of the parabola using the information given. 261. Focus (3,5) and directrix y=1 For the following exercises, determine the equation of the parabola using the information given. 262. Focus (52,4) and directrix x=72 For the following Exercises, determine the equation of the ellipse using the information given. 263. Endpoints of major axis at (4,0),(4,0) and foci located at (2,0),(2,0)For the following Exercises, determine the equation of the ellipse using the information given. 264. Endpoints of major axis at (0,5),(0,5) and foci located at (0,3),(0,3)For the following Exercises, determine the equation of the ellipse using the information given. 265. Endpoints of major axis. at (0,2),(0,2) and foci located at (3,0),(3,0)For the following Exercises, determine the equation of the ellipse using the information given. 266. Endpoints of major axis at (3,3),(7,3) and foci located at (2,3),(6,3)For the following Exercises, determine the equation of the ellipse using the information given. 267. Endpoints of major axis at (3,5),(3,3) and foci located at (3,3),(3,1)For the following Exercises, determine the equation of the ellipse using the information given. 268. Endpoints of major axis at (0,0),(0,4) and foci located at (5,2),(5,2)For the following Exercises, determine the equation of the ellipse using the information given. 269. Foci located at (2,0),(2,0) and eccentricity of 12 For the following Exercises, determine the equation of the ellipse using the information given. 270. Foci located at (0,3),(0,3) and eccentricity of 34 For the following exercises, determine the equation of the hyperbole using the information given. 271. Vertices located at (5,0),(5,0) and foci located at (6,0),(6,0)For the following exercises, determine the equation of the hyperbole using the information given. 272. Vertices located at (0,2),(0,2) and foci located at (0,3),(0,3)For the following exercises, determine the equation of the hyperbole using the information given. 273. Endpoints of the conjugate axis located at (0,3),(0,3) and foci located (4,0),(4,0)For the following exercises, determine the equation of the hyperbole using the information given. 274. Vertices located at (0,1),(6,1) and focus located at (8,1)For the following exercises, determine the equation of the hyperbole using the information given. 275. Vertices located at (2,0),(2,4) and focus located at (2,8)For the following exercises, determine the equation of the hyperbole using the information given. 276. Endpoints of the conjugate axis located at (3,2),(3,4) and focus located at (3,7)For the following exercises, determine the equation of the hyperbole using the information given. 277. Foci located at (6,0),(6,0) and eccentricity of 3 For the following exercises, determine the equation of the hyperbole using the information given. 278. (0,10),(0,10) and eccentricity of 2.5 For the following exercises, consider the following polar equations of conics. Determine the eccentricity and identify the conic. 279. r=11+cos For the following exercises, consider the following polar equations of conics. Determine the eccentricity and identify the conic. 280. r=82sin For the following exercises, consider the following polar equations of conics. Determine the eccentricity and identify the conic. 281. r=52+sin For the following exercises, consider the following polar equations of conics. Determine the eccentricity and identify the conic. 282. r=51+2sin For the following exercises, consider the following polar equations of conics. Determine the eccentricity and identify the conic. 283. r=326sin For the following exercises, consider the following polar equations of conics. Determine the eccentricity and identify the conic. 284. r=34+3sin For the following exercises, find a polar equation of the conic with focus at the origin and eccentricity and directrix as given. 285. Directrix: x=4;e=15 For the following exercises, find a polar equation of the conic with focus at the origin and eccentricity and directrix as given. 286. Directrix: x=4;e=5

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