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235test1sol

# 235test1sol - MAT235-CALCULUS II TERM TEST 1 SOLUTIONS...

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MAT235-CALCULUS II TERM TEST 1, NOVEMBER 10 2010 SOLUTIONS Total Marks: 60 Problem 1 (5 marks) Find parametric equations of the curve that is intersection of the ellip- soid x 2 + y 2 + 2 z 2 = 1 with the plane x + y = 1. Solution: Plugging the relation y = 1 - x into the equation of the ellipsoid we obtain that x 2 + (1 - x ) 2 + 2 z 2 = 1 or 2 x 2 - 2 x + 1 + 2 z 2 = 1 which after completing the square gives that 2( x - 1 2 ) 2 + 2 z 2 = 1 2 or 4( x - 1 2 ) 2 + 4 z 2 = 1 2 or (2 x - 1) 2 + (2 z ) 2 = 1 . Using the identity cos 2 t + sin 2 t = 1 we can set 2 x - 1 = cos t , 2 z = sin t or x = 1+cos t 2 , z = sin t 2 , and therefore y = 1 - x = 1 - cos t 2 . Therefore, the equations x = 1+cos t 2 , y = 1 - cos t 2 , z = sin t 2 are parametric equations of the curve. Problem 2 (5 marks) Sketch the level curves of the function f ( x, y ) = 1 2 x 2 + y 2 +2 y +3 . Let c be a real number. We want f ( x, y ) = c or 1 2 x 2 + y 2 +2 y +3 = c or 2 x 2 + y 2 + 2 y + 3 = 1 c 2 with c 0 or 2 x 2 + ( y + 1) 2 = 1 c 2 - 2 with c 0 (since the square root is always positive we are interested only positive values of c give possibly nonempty sets). If 1 c 2 - 2 < 0 or c > 1 2 (recall that c 0) this is impossible. If 1 c 2 - 2 = 0 or c = 1 2 then 2 x 2 + ( y + 1) 2 = 1 c 2 - 2 = 0 so x = 0, y = - 1. If 1 c 2 - 2 > 0 or c < 1 2 then the level curves are ellipses. A sketch of the level curves is given below in figure 1. Problem 3 Let f ( x, y ) = ( xy 3 x 4 + y 4 + x, if ( x, y ) 6 = (0 , 0) 0 , if ( x, y ) = (0 , 0) . a) (5 marks) Show that f is not continuous at (0 , 0). b) (5 marks) Use the definition of partial derivative to show that ∂f ∂x (0 , 0) exists and compute it.

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235test1sol - MAT235-CALCULUS II TERM TEST 1 SOLUTIONS...

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