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# 237q1 - Remark Questions are not necessarily in the order...

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Remark: Questions are not necessarily in the order of a difficulty level. 1. [4 marks] Find the shortest distance from the point ) 4 , 1 , 1 ( - - P to the line tangent to the curve ) 6 , 3 , 2 ( ) ( t t e e t t - + - = r at the point ) 5 , 2 , 0 ( 0 - P . 2

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2. (a) [5 marks] Let S be the subset of 2 R defined by } 1 : ) , 0 {( \ } 4 : ) , {( 2 2 < + = y y y x y x S . (i) Is S open, closed or neither ? Justify your choice, no marks for guessing. (ii) Describe in the set notation: = int S = S = S (b) [3 marks] Prove that if x is in the closure of n R S then there is a sequence of points in S that converges to x . 3
3. (a) [4 marks] Evaluate the limit or prove that it does not exist: 2 2 3 2 ) 0 , 0 ( ) , ( lim y x y x y x + + . (b) [3 marks] It can be easily proved that if ) , ( y x f is continuous on 2 R , then the functions of one variable ) 0 , ( ) ( t f t g = and ) , 0 ( ) ( t f t h = are continuous on R . Show that the c onverse is false. 4

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4. Consider the function R R f 2 : defined by 2 2 4 1 5 ) , ( y x y x f + + = . (a) [2 marks] Draw the level curve c L of f for 1 = c .
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237q1 - Remark Questions are not necessarily in the order...

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