18.02a Problem Set 7(due in class on Thurs., Nov. 6)Part I (32 points)TB = Simmons; SN = 18.02A Supplementary Notes (all have solutions)The problems marked ’other’ are not to be handed in.Topic 22 (M, Nov. 3) Parametric equations continued.Read: TB: 17.4Hand in: 1J/1ac, 3, 4ab, 5, 6, 9abc, find the curvature of the helix in 1J/6.Others: 1J/2, 7Topic 23 (Not done in class) Kepler’s second law.Read: SN: KHand in: NoneOthers: NoneTopic 24 (T, Nov. 4) Functions of several variables, partial derivatives.Read: TB: 19.1Hand in: 2A/1abeOthers: 2A/1cdComing nextTopic 25 (W, Nov. 5) Tangent plane, level curves, contour surfaces.Read: TB: 19.2 SN: TAContinuation: (R, Nov. 6) Discussion, review and catch up.Problem section: (F, Nov. 7)No class: (M, Nov. 10) Veterans DayNo class: (T, Nov. 11) Veterans DayContinuation: (W, Nov. 12) ReviewExam: (R, Nov. 13) Exam 3 (covers 17-24)Part II (33 points)Problem 1 (Class 22: 5 pts: 3,1,1)(a) Find the unit tangent vector, unit normal vector, radius of curvature and center ofcurvature to the parabola (x, y) = (at2, 2at), where a is a constant.(b) Find the radius of curvature at a general point (x, y) on the graph of y = 2x + 3.(c) Find the point of maximum curvature on the parabola y = x2.118.02a Problem Set 7 2Problem 2 (Classes 22 (3 pts))(a) Define the cycloid and derive parametric equations for it.(b) Compute the arclength of one arch of the cycloid.Problem 3 (Class 20 (4 pts: 2,2))Find the center of the unique circle through the three points (1, 0, 0), (0, 2, 0) and (0, 0, 1).Problem 4 (Class 24, 3 pts:1,2)Place a unit cube in the corner of the first octant with edges along the axes. For thisproblem consider the front face diagonal containing (1, 0, 1) and the right face diagonalcontaining (0, 1, 0). These two lines are skew; the problem is to find the length and positionof the shortest line segment joining them.(a) Draw a picture and write parametric equations for the two lines containing these twodiagonals. For clarity, use different variables, t and u, as the parameters for the two lines.(b) Let w(t, u) be the square of the distance between a point on A on the front diagonaland a point B on the side diagonal. Find the (unique) point where ∇w = 0. (This is calleda critical point.)Problem 5 (Class 22: 8 pts: 4,2,2) A hockey puck of radius 1 slides along the ice at aspeed 10√2 in the direction of the vector (1, 1). As it slides, it spins in a counterclockwisedirection at 2 revolutions per unit time. At time t = 0, the puck’s center is at the origin(0, 0).(a) Find the parametric equations for the trajectory of the point P at the edge of the puckinitially at (1, 0).(b) Find the velocity v of the point P .(c) What is the minimum speed of the point P , and what is the direction of the velocityat the corresponding time?Problem 6 (Class 22: 8 pts: 2,2,2,2) Consider the helical trajectory with position vectorr = sin(4t)i + cos(4t)j + 3tk.(a) Calculate the velocity vector v and the unit tangent vector T.(b) Calculate the speed ds/dt and the arclength traced out between the points at t = 0and t = 2π.(c) Calculate the curvature κ.(d) Show that the curve makes a constant angle with the k-direction.Problem 7 (Class 24: 2 pts) Show that z = tan−1(y/x) satisfies zxx+ zyy= 0.End of pset
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