Math32a 2 R Kozhan Final Dec 14 2012 Name UID Circle your TA and discussion session 2A Tues Jordan Greenblatt 2C Tues William Rosenbaum 2E Tues Silas Richelson 2B Thur Jordan Greenblatt 2D Thur William Rosenbaum 2F Tues Silas Richelson Instructions If you get stuck move on to the next question You don t have a lot of time Show all work if you want to get full credit You have an extra scratch paper on the last page If you need more raise your hand and ask No books notes electronics incl calculators and cell phones are allowed Good luck and have fun Question Max Your score 1 12 2 10 3 18 4 10 5 14 6 10 7 10 8 12 9 10 10 16 11 10 Total 132 1 Problem 1 a 4 points Consider vector valued functions a t b t c t Express d a t b t c t dt in terms of a t b t c t and a 0 t b 0 t c 0 t only b 4 points Sketch y 2 2y 4x in R2 and find equation of its axis c 4 points Find the linearization function of f x y 2x 4y 3 at the point 1 1 2 Problem 2 a 2 points Write the formula for proj v w b 2 points State Clairaut s theorem with all of its conditions c 2 points Complete the definition f x y is called continuous at a b if d 4 points True or false if f x y log x log y log z and u is a unit vector then 3 D u f 1 1 1 3 Justify answer prove or show a counterexample No credit if no explanation is given 3 Problem 3 Questions do not depend on each other Let L be the line x 1 t y 1 2t z 3 2t a 6 points Find equation of the plane that contains L and the point 3 1 3 b 6 points Decompose the vector w h 3 1 1i into the sum w w where w is parallel to L and w is orthogonal to L i e find w and w c 6 points Find the distance from 3 1 3 to the line L 4 Problem 4 A particle travels in R3 according to the law r t Suppose that at time t 0 its position and velocity are 1 1 0 and h1 1 1i respectively Its acceleration at time t is a t ht 1 ti a 5 points Find the velocity and speed of the particle at t 1 b 5 points Find the position of the particle at t 1 5 Problem 5 If you do not solve a c you may still solve part d Let r t cos t t sin t i sin t t cos t j 3 2 2 t k for t 0 a 2 points Find the unit tangent vector T t at time t b 4 points Find the unit normal vector N t and unit binormal vector B t at time t c 2 points Find the equations of osculating plane and of the normal plane at time t 0 d 6 points Reparametrize r t with respect to the arc length starting from t 0 in other words find the natural parametrization 6 Problem 6 a 5 points Find all critical points of f x y e4y 4 x 2 y 2 b 5 points Use the second derivative test to classify them i e local min local max saddle point 7 Problem 7 a 5 points Find the limit or show that the limit doesn t exist x3 x y 0 0 x2 y 2 lim b 5 points Find the limit or show that the limit doesn t exist xy 2 z 2 x y z 0 0 2 x4 4y 4 9 z 2 4 lim 8 Problem 8 Consider the surface x y z y z a 2 points What are the possible values of z if x 0 and y 1 b 3 points Viewing z z x y as an implicitly defined function of x y around the point 0 1 1 compute 0 1 1 z x at c 3 points Viewing z z x y as an implicitly defined function of x y around the point 0 1 1 compute 0 1 1 z y at d 2 points Find the equation of the tangent plane to this surface at 0 1 1 e 2 points Find the equation of the normal line to this surface at 0 1 1 9 Problem 9 10 points Consider a rocket orbiting a planet on an elliptical orbit given by the intersection of x2 y 2 8 and 2x 2y 4z 4 Find all the points where the rocket is closest to the point 0 0 1 and all the points where it s farthest from 0 0 1 guessing the right points from a picture is possible but will not be considered a solution 10 Problem 10 a 2 points Let D be a set in R2 Finish the definition D is called bounded if b 2 points Let D be a set in R3 A point a b c is called a boundary point of D if c 4 points For each of the following sets in R2 state whether it is closed and whether it is bounded answers only no justification needed i x y y x2 ii x y y 0 1 x 1 iii x y 1 x2 y 2 4 iv x y y 2012 x 2013 d 8 points Let D be the triangle with vertices 0 0 0 2 2 0 Find the global maximum and global minimum values attained by the function f x y x2 x y 2 y on D and all the points where these values are attained 11 Problem 11 Let u u x y where x st and y st a 4 points Compute u t b 6 points Compute 2u t2 your answer will contain ux and uy your answer will contain various derivatives of u with respect to x and y 12 Extra scratch paper 13