Exam 1 Fall 2007 - Math 318 Exam 1 Show all your work for...

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Math 318, Exam # 1, September 28, 2007 Show all your work for full credit! 50 points maximum. 1. (4) Let g : R R 4 be defined by g ( t ) = ( t 3 + sin t, 2 t 2 - t + 3 , t 2 + 2 t, t 2 ). Calculate g (0) and g (0) and show that g (0) g (0). 2.(2) What is a normal vector for the plane defined byz= 3x+ 2y-6? 3. (5) Let -→ x = ( x 1 , ..., x n ) R n and define g ( -→ x ) = | -→ x | 3 . For i between 1 and n calculate ∂g ∂x i ( -→ x ) . 4. (12) (a) Let E = { ( x, y ) : x > 0 and x 2 + y 2 1 } . Determine the boundary of E and also determine the interior points of E . (b) In the four examples of sets in R 2 determine whether the set is open, closed, neither open nor closed, or both open and closed. E 1 = { ( x, y ) : x 2 + y 2 1 } E 2 = { ( x, y ) : x 2 < 1 } E 3 = { ( x, y ) : x > 0 and y = 0 } E 4 = { ( x, y ) : x = 1 or y = 4 } 5. (8) Mark each statement as true or false. For (a)-(c) let D R n be an open set, let f : D R , and -→ x 0 D . (a) If f is differentiable at -→ x 0 , then f is continuous at -→ x 0 . (b) If f and all of its partial derivatives are continuous in D and if -→ v R n , then the derivative of f in the direction of -→ v , ∂f -→ v ( -→ x 0 ) , exists. (c) If all partial derivatives of f exist at -→ x 0 , then
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Unformatted text preview: f is continuous at-→ x . (d) Every limit point of a set E ⊆ R n is an interior point of E . 6. (10) Let f ( x, y, z ) = x 2 y 3 + (1 + zx ) 4 . (a) Calculate ∇ f ( x, y, z ). (b) Explain how you know that f is differentiable at (1 , ,-2). (c) For-→ v = (1 , 2 ,-3) calculate ∂f ∂-→ v (1 , ,-2). (d) Write down the expression for the first degree Taylor approximation to f at (1 , ,-2). 7. (4) Let f ( x, y ) = xy ( x 2-y 2 ) ( x 2 + y 2 ) 2 . Show that lim ( x,y ) → (0 , 0) f ( x, y ) does not exist. This is what the problem should have been. On the sheet I originally had the typo where f ( x, y ) = xy ( x 2-y 2 ) x 2 + y 2 . In this case the limit exists and equals 0. 8. (5) Let f ( x, y ) = 3 + 2 x ( y 2-x 2 ) x 2 + y 2 if ( x, y ) 6 = (0 , 0) and f (0 , 0) = 3. Calculate ∂f ∂x (0 , 0). Typeset by A M S-T E X 1...
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  • Derivative, #, Closure, General topology, Boundary

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