Math32a 1 R Kozhan Final Exam Dec 06 2011 Name UID Circle your TA and discussion session 1A Tues Mike O Brien 1C Tues Jeff Lin 1E Tues Jordy Greenblatt 1B Thur Mike O Brien 1D Thur Jeff Lin 1F Thur Jordy Greenblatt 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 I reserve the right to take off points if I cannot see how you arrived at your answer even if your final answer is correct No books notes electronics incl calculators and cell phones are allowed Good luck Question Max Your score 1 12 2 12 3 12 4 12 5 8 6 12 7 9 8 8 9 12 10 8 11 8 Total 113 1 Problem 1 For the questions on this page no justification is needed a 2 points Find the midpoint between 2 8 11 and 2 6 1 b 2 points Write the equation of an ellipse in R2 that passes through points 2 0 and 0 1 there are many of them but you have to find at least one c 2 points Compute the scalar triple product of vectors h0 2 0i h1 0 0i h0 0 3i d 2 points Suppose the curve in R2 is given parametrically as x f t y g t Suppose that g is a positive function and f is increasing Write the formula for the area between this graph and the x axis when t is between and t e 2 points Classify and sketch in R3 the surface z 2 2z x2 0 f 2 points Suppose u is unit and f x y z is a function of three variables Write the definition of D u f a b c 2 Problem 2 Let a and b be two vectors in R3 which are perpendicular to each other Suppose that the length of a is 2 and the length of b is 4 a 4 points Find 2 a b a b b 4 points Find the length of 2 a b 2 a c 4 points Find the angle between a and a b 3 a Hint good picture should make this question very easy 3 Problem 3 a 4 points Find an equation of the plane that passes through the point 5 4 9 that is parallel to the xz plane b 4 points Find an equation of the line through the origin that is perpendicular to the plane x y 3z 7 c 4 points Prove that lines x 1 y 2 z 2 and x 2 y 1 z 3 intersect 4 Problem 4 There re at least two different methods of solving c in particular if you can t solve b you can still solve c Consider surface z x2 y 2 a 1 point What is the name of this surface b 6 points Find a parametrization hx y zi hf t g t h t i of the curve of intersection of this surface with the plane z 8x 6y 0 Hint get rid of z parametrize x and y in the usual way then find z in terms of your parameter t c 5 points Find a tangent vector to this curve at the point 9 3 90 5 Problem 5 Consider the curve r t h1 t3 2 2t3 4 2t3 i in R3 a 5 points Compute the arc length function s t of r measured from the point 1 2 4 in the direction of increasing t Reminder recall that s t is the length of the part of the curve between the initial point 1 2 4 and the point r t b 3 points Reparametrize the curve with respect to arc length measured from the point 1 2 4 in the direction of increasing t 6 Problem 6 The two questions are unrelated a 6 points Determine the set of points at which the following function is continuous Justify your answer explain what theorem you re using if any Sketch this set in R2 p 1 x y x y b 6 points Show that the limit doesn t exist xy z 3 x y x y z 0 0 3 x4 y 4 z 3 4 lim Hint lines in R3 through 0 0 3 7 Problem 7 a 6 points Find the linearization function L x y z of f x y z e6 xyz at the point 1 2 3 b 3 points Compute fzzz x y z 8 Problem 8 8 points Let f x y z be some function of three variables and g r s some function of two variables Assume that x y f x y z g y z Suppose g 21 1 2 gr 12 1 4 gs 12 1 4 Evaluate f 1 2 2 9 Problem 9 Consider f x y 2e x ln y The following questions can be solved in any order a 4 points Find the minimum directional derivative of f at the point 0 1 and the direction at which it occurs b 2 points Is f differentiable at 0 1 Explain c 2 points Find the directional derivative of f at 0 1 in the direction of v h2 2i d 4 points Consider surface z 2e x ln y Find the normal line to this surface at the point 0 1 0 10 Problem 10 8 points Find and classify all of the critical points of f x y 3x 12y x3 y 3 11 Problem 11 8 points Find the maximum and the minimum values of the function f x y xy if x y is restricted to lie on the ellipse 2x2 y 2 2 Find all the points where the maximum and the minimum are achieved 12 Extra scratch paper 13