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Unformatted text preview: 18.02 PRACTICE MIDTERM #1A BJORN POONEN Please turn cell phones off completely and put them away. No books, notes, or electronic devices are permitted during this exam. The exam runs from 1:05pm to 1:55pm (50 minutes). Generally, you must show your work to receive credit. Name: Student ID number: Recitation instructor’s last name: Recitation time (e.g., 10am): (Do not write below this line.) 1 out of 15 2 out of 15 3 out of 25 4 out of 20 5 out of 25 Total out of 100 1) Find the equation of the tangent plane to the graph of f ( x, y ) := x 2 y 3 y 2 at the point where x = 2 and y = 1. 2) Given that x 1 , x 2 , x 3 , y 1 , y 2 , y 3 are real numbers satisfying the conditions x 2 1 + x 2 2 + x 2 3 = 4 y 2 1 + y 2 2 + y 2 3 = 9 , what is the range of possible values of x 1 y 1 + x 2 y 2 + x 3 y 3 ? 3) Let f ( x, y ) = x 3 + 3 xy y 3 . (a) (15 pts.) Find all points (if any) at which f ( x, y ) has a local minimum. (b) (10 pts.) Find all points (if any) at which f ( x, y ) is at its global minimum. (Explain how you know that your answer is correct.) 4) Given that a, b, c, d, e, f, g, h, i are real numbers such that 1 2 0 3 0 4 0 5 6 a b c d e f g h i = 1 0 0 0 1 0 0 0 1 , find b . 5) What is the distance between the two lines in R 3 given in parametric form by the equations r 1 ( t ) = t i + (3 t + 8) j and r 2 ( t ) = ( t 5) i + 2 t k ? This is the end! If you finish during the last 5 minutes of the exam, please remain in your seat until the end, and then pass your exams towards the aisles, and continue to wait silently in your seat until the proctors have collected all exams. SOLUTIONS TO 18.02 PRACTICE MIDTERM #1A BJORN POONEN 1) Find the equation of the tangent plane to the graph of f ( x, y ) := x 2 y 3 y 2 at the point where x = 2 and y = 1. Solution: We have f x = 2 xy and f y = x 2 6 y . At (2 , 1), we have f = 1, f x = 4, f y = 2. So the tangent plane at (2 , 1 , 1) is z 1 = 4( x 2) + ( 2)( y 1) . Alternative forms of the answer include z = 4 x 2 y 5 and 4 x 2 y z = 5 . 2) Given that x 1 , x 2 , x 3 , y 1 , y 2 , y 3 are real numbers satisfying the conditions x 2 1 + x 2 2 + x 2 3 = 4 y 2 1 + y 2 2 + y 2 3 = 9 , what is the range of possible values of x 1 y 1 + x 2 y 2 + x 3 y 3 ? Solution: Equivalently, given that ~x is a vector of length 2, and ~ y is a vector of length 3, what is the range of possible values of ~x · ~ y ? The angle θ between ~x and ~ y can lie anywhere in the range [0 , π ], so cos θ can lie anywhere in [ 1 , 1], so ~x · ~ y =  ~x  ~ y  cos θ = 6 cos θ can lie anywhere in the range [ 6 , 6]. 3) Let f ( x, y ) = x 3 + 3 xy y 3 . (a) (15 pts.) Find all points (if any) at which f ( x, y ) has a local minimum....
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 Fall '11
 Preston
 Vector Calculus, Manifold, Vector field, BJORN POONEN

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