prelim1 - i f x y =(0 0(a Compute the partial derivative...

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Mathematics 192 Fall 2006: Prelim Exam 1 September 28, 2006 Groundrules: The exam is closed book. No notes, calculators, etc. 1. Consider the vectors v = i + 2 j + a k and w = i + j + k . (a) Find all values of the parameter a (if any) such that v is perpendicular to w . (b) Find all values of the parameter a (if any) such that the area of the parallelogram determined by v and w is equal to 6. 2. Consider the plane P containing the points A = (1 , 0 , 0), B = (2 , 1 , 1) and C = (1 , 0 , 2). (a) Find a unit vector perpendicular to P . (b) Find the intersection of P with the line perpendicular to P that contains the point D = (1 , 1 , 1). 3. Let r ( t ) be the vector function defined by r ( t ) = i + 1 2 t 2 j + 1 3 t 3 k . (a) Compute the arc length parameter s ( t ) of r ( t ) starting from t = 0. (b) Show that the ratio | r ( t ) | s ( t ) 1 as t → ∞ . 4. Let f ( x, y ) = 1 ( x 2 + y 2 ) xy . (a) What is the domain of f ? What is the boundary of the domain? (b) Is the domain of f open, closed, both or neither? Is the domain of f bounded or unbounded? Explain your answers. 5. Let f ( x, y ) = x 3 + xy x 2 + y 2 if ( x, y
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Unformatted text preview: i f ( x, y ) = (0 , 0) . (a) Compute the partial derivative ∂f ∂y at points % = (0 , 0). (b) Is f continuous at (0 , 0)? Explain your answer. (c) Do the partial derivatives of f exist at (0 , 0)? Explain your answer. 6. (Homework Exercises, Sections 14.4, 14.5) (a) The equation F ( x, y, z ) = sin( x + y )+sin( y + z )+sin( x + z ) = 0 implicitly de±nes z as a function of x and y . Find the value of ∂z ∂x at the point ( π, π, π ). (b) Let g ( x, y, z ) = xe y + z 2 . Find the direction in which g increases most rapidly at the point P = (1 , ln 2 , 1 / 2). 7. The plane L is tangent to the sphere x 2 + y 2 + z 2 = 1 at the point P = ³ 1 3 , √ 8 3 , ´ . The plane L is also tangent to the sphere ( x − a ) 2 + y 2 + ( z − c ) 2 = 4 at the point Q = ³ − 7 3 , 2 √ 8 3 , ´ . Find a and c ....
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