v elearn.usp.ac.fj + M Sign in to yo... M Sent Mail - r... Student Onli... https:/elear... M Results - $11... M Inbox (2,267... Summarize... Al Humanize... H Bypass Turni.. Dashboard IUSL WE YOUI UWII WOLK CAPIESCU III YOUL UWII WOLUS. 8. The lecturer reserves the right to question any student submitting a solution about any portion of the solution. Inability to respond adequately will automatically result in loss of points. Question 1 A). Find the cartesian equation of the curve, sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. [Verify using Mathematica] i). x = sin -0, y = cos-0, -TSOSn. 2 ii). x = - cos0, y = 2sin0, OSOST. [5 + 5 = 10 marks] B). Consider the family of curves x= sint +sinnt, y=cost +cosnt, where n is a positive integer. Graph several members of the family of curves. What features do the curves have in common? What happens as increases? [Verify using Mathematica] [5 marks] C). Consider Figure 1 below. Find parametric equations for the set of all points P as shown in the figure such that |OP = AB . Use the geometric description of the curve to draw a sketch of the curve. [Verify using Mathematical] B x = 2a O a Xv > elearn.usp.ac.fj C + M Sign in to... M Sent Mail... Student O... X https://ele... M Results -... M Inbox (2,2... Summarize.. Al Humani... H Bypass Tu... Screensh... *Homewor... D). Using parametric equations, find the circumference of a circle with radius r. [Verify using Mathematica] [5 marks] E). Find the area enclosed by the loop of r = 2cos0-sec 0. [Verify using Mathematica] [5 marks] Question 2 A). Let z = In(e* +e ). Find: (i). Oz Oz [Verify using Mathematica] Ox ay 2 2 z (1i ) . ax2 ay2 . [Verify using Mathematica] [3 + 7 = 10 marks] B). Let f (tx, ty ) = t" f(x, y). i). Find x2 of + 2xy a f ax2 Oxdy ii) . Show that fx (tx, ty) = t"-f. (x, y). Q Q QQ [5 + 5 = 10 marks]v > elearn.usp.ac.fj C + M Sign in to... M Sent Mail... Student O... https://ele... M Results -... M Inbox (2,2... Summarize.. Al Humani... H Bypass Tu... Screensh... Homewor.. [3 + 7 = 10 marks] B). Let f (tx, ty ) = t" f (x, y). i ) . Find x2 of + 2xy 2 2 f ax2 axdy ii). Show that fx (tx, ty) = t"-ifx (x, y) [5 + 5 = 10 marks] C). In an engineering class, the students were able to inscribe a rectangular box in a sphere of radius r. Using: i). The constraint to eliminate a variable (Second Partials Test (SPT)); and ii). Lagrange multipliers; find the maximum volume of a rectangular box. [Verify using Mathematica] [5 + 5 = 10 marks] D). Find the area of the region inside the cardioid r=1+ cos and outside the circle r =3cos 0. Sketch a graph of the region bounded by the graphs of the equations. Use: (i). Double integral in polar coordinate system. [Verify using Mathematica] (ii). Single integral in polar coordinate system. [Verify using Mathematica] [5 + 5 = 10 marks]