termtest - as possible. Which point or points on the island...

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MAT237Y – Multivariable Calculus Summer 2009 Term Test Thursday, June 25 from 6:10pm to 8:00pm. Instructors: A. Hammerlindl and J. Uren 1. (a) [5 marks] DeFne the directional derivative . (b) [5 marks] Give the deFnition of a convex set. 2. [10 marks] Is the set { ( x, y, z ) R 3 : y = e x , z = sin 2 ( x ) } connected? Explain. 3. (a) [5 marks] Consider the set C R 3 which is the solution of the system of equations arctan( y ) + ( x + 1) 5 = x 4 ( z 1) 3 cos( y ) + x 2 = 1 + ( y 3) z. Show that in a neighbourhood of the origin, y and z can be rep- resented as functions of x . (b) [5 marks] ±ind the line tangent to C at the origin. Give it as a subset of R 3 . 4. [10 marks] You are vacationing on the South PaciFc island of Laga Ranga with a coastline given by the set { ( x, y ) R 2 : 4 x 2 + y 2 = 16 } . Just as you are about to order another pi˜na colada from your resort’s swim-up bar, the ground violently shakes. Mt. Leibniz, located at ( 1 , 0), has erupted and will soon cover the entire island with molten lava. As you have no wish to see your hedonistic life come to an end, you decide to a head to a point on the island as far from the volcano
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Unformatted text preview: as possible. Which point or points on the island qualify? 5. [10 marks] Classify the critical points of the function f : R 2 R , ( x, y ) m ( x 2)( x 2 y 2 ) . 1 6. [10 marks] Suppose that U is the non-convex set U = { ( x, y ) R 2 : y < x 2 + 1 } and f : U R is a C 1 function such that b f ( x ) b 3 for x U . Using the Mean Value Theorem, show that f (3 , 4) f ( 3 , 0) + 30. 7. [10 marks] DeFne g : R 2 R , ( x, y ) m sin( x 2 ) cos( y ) . ind g (0 , 0) for all multi-indices of order six. Note: Once you have a numerical answer, you need not simplify it. That is, 4! / 2! is just as acceptable an answer as 12. 8. Bonus Question: [5 marks] Let A be an n n positive deFnite matrix and B j an n n positive semi-deFnite matrix for j = 1 , . . . , . Show that the sum A + s j =0 B j is a positive deFnite matrix. 2...
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termtest - as possible. Which point or points on the island...

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