MATH 32A SECOND MIDTERM EXAMINATION February 27th 2012 Please show your work You will receive little or no credit for a correct answer to a problem which is not accompanied by sufficient explanations If you have a question about any particular problem please raise your hand and one of the proctors will come and talk to you At the completion of the exam please hand the exam booklet to your TA If you have any questions about the grading of the exam please see the instructor within 15 calendar days of the examination I certify that the work appearing on this exam is completely my own Signature Name Section 1 2 3 4 5 Total R 1 2 32A MIDTERM 1 RADKO Problem 1 10 points Let r t t2 2 3 t t 3 1 1 pt Find T 1 Solution 2t 2t2 1 1 4t2 4t4 2 2 1 T 1 3 T t 2 2 pt Find N 1 Solution Since N 1 is the unit vector in the directin perpendicular to that of T 1 we find 1 2 2 N 1 3 3 3 3 2 pts Use the formula r r k t r 3 to compute k 1 Solution 2t 2t2 1 2 4t 0 2 2 1 2 4 0 4 2 4 6 27 6 2 k 1 27 9 4 2 pt Write down the equation of the line in the direction of N 1 going through the point 1 32 1 Solution 2 1 2 2 r 1 1 t 3 3 3 3 r r r 1 r 1 r 1 r 1 r 1 r 1 r 3 1 32A MIDTERM 1 Radko 3 5 2 pts Find the equation of the osculating circle at t 0 Solution The radius of the circle is R k1 12 Since B 0 T 0 N 0 0 0 1 1 0 0 0 1 0 the circle lies in the xz plane The equation of the circle is 2 1 1 z2 x 2 4 4 32A MIDTERM 1 RADKO Problem 2 10 pts A ball is thrown from the ground at the angle of 6 at the initial speed of v0 10m s The acceleration of gravity is g 9 8m s2 Find how long it takes before the ball hits the ground again Express your answer in seconds Solution Choose a coordinate system for which the ball is initially at the origin the x axis is parallel to the ground and the y axis is perpendicular to the ground pointing up Let r t denote the position of the ball at time t The acceleration of the ball is then r t Since we are in a free fall the acceleration is equal to the acceleration of gravity which is a vector pointing downward and of magnitude g So we know that r t g Integrating we get r t v gt j 0 1 r t r0 v0 t gt2 j 2 The initial conditions imply that r0 0 and that v0 is the initial velocity Thus we know that the magnitude of v0 is 10 and that it makes the angle of 6 with the x axis Thus v0 10 cos 6 10 sin 6 5 3 5 It follows that 1 r t 5 3 t 5t gt2 2 1 2 The ball hits the ground again when 5t 2 gt 0 i e 5 21 gt so that t 10 g 1 02s 32A MIDTERM 1 Radko Problem 3 10 pts Use linear approximations to estimate the value of 3 8 2 e0 1 1 3 pts Write down the function you will use for the linear approximation f x y 3 y ex 2 2 pts Write down the values of the variables you are going to use x 8 y 0 3 5 pts Finish the computation for the approximate value 1 x fx 3 y e 3 3 2 fx 8 0 3 x 1 fy e 3 2 3 y 1 fy 8 0 6 f 8 0 2 1 2 3 8 2 e0 1 2 0 2 0 1 3 6 0 8 0 1 0 9 2 2 6 6 1 2 15 2 12 5 6 32A MIDTERM 1 RADKO Problem 4 Limits of functions of several variables 1 5 pts Does the limit x lim x y 0 1 x y 1 exist Justify your answer If the limit exists prove this and compute the limit If the limit does not exist explain why not Solution Let y 1 Then x x 1 1 as x 0 x y 1 x Let x 0 Then x 0 0 x y 1 Thus the limit does not exist 2 5 pts Does the limit as y 1 xy 2y 2 3z 2 exist Justify your answer If the limit exists prove this and compute the limit If the limit does not exist explain why not Solution Consider the line z 0 x y Then 1 x2 f x x 0 2 2 x 2x 3 Consider the line x 0 z y Then lim x y z 0 0 0 x2 f 0 y y 0 Since the limits along the two lines are not equal the limit does not exist 32A MIDTERM 1 Radko 7 Problem 5 10 points Multiple choice questions For each of the questions below circle the right answer The right answers are in bold face 1 Circle the correct statement about curvature a The curvature of the curve given by y x 1 3 at 1 0 is 0 b The curvature of a curve can be positive zero or negative c If the speed of a particle the magnitude of the velocity is constant the curvature of the trajectory is 0 d The only curve with constant curvature is a circle e None of the above 2 The quadric surface with the equation x2 4y 2 z 0 is a an ellipsoid b an elliptic paraboloid c a hyperbolic paraboloid d a hyperboloid of one sheets e a cone 3 Circle one statement which is true a For any differentiable vector function r t we have d r t r t dt b The curve with vector equation r t t3 1 i t3 j k is a straight line 8 32A MIDTERM 1 RADKO c If two curves lie in the xy plane and are tangent to each other in their point of intersection then they have the same unit binormal vector B d If r t 1 for all t then v t is constant e None of the above 32A MIDTERM 1 Radko 9 4 Circle one statement about motion in space which is true a If a particle moves with constant speed the velocity vector is perpendicular to the accelaration vector b If a body moves in the plane given by x y z 1 in the field of gravity of the Sun positioned at 0 0 0 then the velocity vector is parallel to the position vector c Vectors r t v t and …