tapia jat4858 HW06 clark 52990 This print out should have 22 questions Multiple choice questions may continue on the next column or page find all choices before answering 001 keywords parametric curve parabola 002 10 0 points Find a Cartesian equation for the curve given in parametric form by 10 0 points If the constant C is chosen so that the curve given parametrically by C2 2 Ct t 0 t 4 8 is the arc of the parabola 8y x2 from 0 0 to 4 2 find the coordinates of the point P on this arc corresponding to t 2 1 1 P 1 8 1 2 P 2 correct 2 1 3 P 1 2 1 2 4 P 2 1 5 P 2 8 1 1 6 P 8 Explanation The point P has coordinates C2 C2 2 2C Ct t 2 t 2 8 t 2 so we need to find C But we are told that the graph of C2 2 Ct t 8 passes through 4 2 when t 4 Thus 4C 4 i e C 1 Consequently P 1 x t 3 cos 4t 1 x2 y 2 9 4 36 2 2 x y 1 2 4 9 36 2 2 x y 1 3 4 9 36 1 4 9x2 4y 2 36 5 4x2 9y 2 36 correct 6 4x2 9y 2 36 Explanation We have to eliminate the parameter t from the equations for x and y Now cos2 sin2 1 Thus x2 y 2 1 9 4 But then after simplification the curve has Cartesian form 4x2 9y 2 36 003 10 0 points Describe the motion of a particle with position P x y when x 5 sin t 2C C2 2 2 1 2 y t 2 sin 4t y 3 cos t as t varies in the interval 0 t 2 tapia jat4858 HW06 clark 52990 1 Moves along the line x y 1 5 3 starting at 0 3 and ending at 5 0 2 Moves once counterclockwise along the ellipse 5x 2 3y 2 1 starting and ending at 0 3 3 Moves once clockwise along the ellipse 2 from x 0 to x 5 while y t decreases from y 3 to y 0 in particular the particle moves from a point on the positive y axis to a point on the positive x axis so it is moving clockwise In the same way we see that as t increases from 2 to the particle moves to a point on the negative y axis then to a point on the negative x axis as t increases from to 3 2 until finally it returns to its starting point on the positive y axis as t increases from 3 2 to 2 Consequently the particle moves clockwise once around the ellipse 5x 2 3y 2 1 x2 y 2 1 25 9 starting and ending at 0 3 4 Moves along the line x y 1 5 3 starting at 5 0 and ending at 0 3 5 Moves once counterclockwise along the ellipse x2 y 2 1 25 9 starting and ending at 0 3 starting and ending at 0 3 keywords motion on curve ellipse 004 10 0 points Which one of the following could be the graph of the curve given parametrically by 6 Moves once clockwise along the ellipse x t y t x2 y 2 1 25 9 when the graphs of x t and y t are shown in starting and ending at 0 3 correct Explanation Since cos2 t sin2 t 1 1 1 for all t the particle travels along the curve given in Cartesian form by x2 y 2 1 25 9 this is an ellipse centered at the origin At t 0 the particle is at 5 sin 0 3 cos 0 i e at the point 0 3 on the ellipse Now as t increases from t 0 to t 2 x t increases t 0 0 1 1 x t y t tapia jat4858 HW06 clark 52990 y 1 1 3 y 5 1 1 1 x x 0 0 0 1 0 1 1 1 y y 2 1 6 1 x 1 correct 1 x 0 0 1 0 0 1 1 1 Explanation Since y 3 1 1 x 1 y 1 0 x 0 1 y 0 0 x 0 0 1 1 the graph passes through the origin and the point 1 0 This already eliminates three of the graphs There are two ways of deciding which of the remaining three it is i since y 4 1 x 1 1 2 3 4 y 1 2 1 the graph must pass also through 3 4 1 x 0 0 1 1 ii the graph of y t is increasing near t 0 at the same rate as it is decreasing near t 1 while the graph of x t is decreasing faster near t 1 than it is near t 0 So the graph of x t y t is decreasing faster near t 1 than it is increasing near t 0 Consequently the graph of x t y t is tapia jat4858 HW06 clark 52990 and y 1 4 1 6t 5 8t 1 are x t y t linear functions of t This already eliminates four of the possible answers On the other hand the first of these starts at P and ends at Q while the second starts at Q and ends at P Consequently only x 0 0 1 5 6t 3 8t 1 keywords parametric curve graph 005 represents the line segment in parametric form by linear functions x t y t so that 10 0 points Find parametric equations representing the line segment joining P 5 3 to Q 1 5 as x t y t P x 0 y 0 0 t 1 006 where P x 0 y 0 Q x 1 y 1 10 0 points Find the path x t y t of a particle that moves once clockwise around the curve and x t y t are linear functions of t t t 1 5 6 sin2 5 8 cos2 2 2 2 5 6t 3 8t correct Q x 1 y 1 x2 y 5 2 49 starting at 7 5 1 7 cos t 5 7 sin t 0 t 3 1 6t 5 8t 2 7 cos t 5 7 sin t 2 t 2 t 4 1 6 sin 3 8 cos 2 2 3 7 cos t 5 7 sin t 0 t 2 correct 5 1 6t2 5 8t2 4 7 cos t 5 7 sin t 6 5 6t2 3 8t2 5 7 cos t 5 7 sin t Explanation All six possible answers represent the line segment joining P 5 3 to Q 1 5 in parametric form x t y t 6 7 cos t 5 7 sin t 0 t 2 0 t 2 0 t Explanation The graph of 0 t 1 x2 y 5 2 49 But only in the cases 5 …