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 4 Hand in 1J 1ac 3 4ab 5 6 9abc find the curvature of the helix in 1J 6 Others 1J 2 7 Topic 23 Not done in class Kepler s second law Read SN K Hand in None Others None Topic 24 T Nov 4 Functions of several variables partial derivatives Read TB 19 1 Hand in 2A 1abe Others 2A 1cd Coming next Topic 25 W Nov 5 Tangent plane level curves contour surfaces Read TB 19 2 SN TA Continuation R Nov 6 Discussion review and catch up Problem section F Nov 7 No class M Nov 10 Veterans Day No class T Nov 11 Veterans Day Continuation W Nov 12 Review Exam 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 of curvature 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 1 18 02a Problem Set 7 2 Problem 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 this problem consider the front face diagonal containing 1 0 1 and the right face diagonal containing 0 1 0 These two lines are skew the problem is to find the length and position of the shortest line segment joining them a Draw a picture and write parametric equations for the two lines containing these two diagonals 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 diagonal and a point B on the side diagonal Find the unique point where w 0 This is called a critical point Problem 5 Class 22 8 pts 4 2 2 A hockey puck of radius 1 slides along the ice at a speed 10 2 in the direction of the vector 1 1 As it slides it spins in a counterclockwise direction 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 puck initially 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 velocity at the corresponding time Problem 6 Class 22 8 pts 2 2 2 2 Consider the helical trajectory with position vector r 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 0 and t 2 c Calculate the curvature 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 7
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