272t2sol

# 272t2sol - D f = - + + =- = 7. Verify that ( , ) sin( ) u x...

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Mat 272 Test 2 Dr. Firoz 1 1. Find and sketch the domain of 1 ( , ) 2 x y f x y x + - = - Solution: 1 0, 2 x y x + - ≥ 2. Draw the rough sketch of the function 2 2 ( , ) 81 f x y x y = - - and draw minimum six level curves. Solution: 3. Evaluate 2 2 2 2 ( , ) (0,0) lim ( )ln( ) x y x y x y + + Solution: 2 2 2 2 2 2 2 ( , ) (0,0) 0 0 0 ln lim ( )ln( ) lim ln 2lim 2lim( 1/ ) 0 1/ x y r r r r x y x y r r r r + + = = = - = by L’Hoepital rule. 4. ( , ) sin , find , 1 x f f f x y y x y = + . Solution: 2 cos , cos (1 ) 1 1 f y y f y x x x y x - = = + + + 5. Find the equation of the tangent plane to 2 2 2 4 18 x y z + + = at P(1, 2, 1) and determine the acute angle that the plane makes with the xy plane. Solution: tangent plane ( 1) ( 2) ( 1) 0 2( 1) 16( 2) 2( 1) 0 8 18 0 x y z f x f y f z x y z z x y - + - + - = - + - + - = + + + = Now 1 2 (1,2,1) 2 16 2 , , cos 1/ 66 n f i j k n k θ = ∇ = + + = =

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Mat 272 Test 2 Dr. Firoz 2 6. Find the directional derivative of ( , , ) , y f x y z x z = + at the point P(2, 1, -1) in the direction from P to Q(-1, 2, 0) Solution: 3 , / | | (2,1, 1) 3/ 11 u PQ i j k u PQ PQ
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Unformatted text preview: D f = - + + =- = 7. Verify that ( , ) sin( ) u x t x at =-, satisfies 2 2 2 2 2 u u a t x -= . 8. The gas law for a fixed mass m of an ideal gas at absolute temperature T , pressure P and volume V is given by PV mRT = where R is the gas constant. Show that P V T mR T T = . 9. Determine the dimensions of a rectangular box, open at the top, having a volume of 32 cubic feet, and requiring the least amount of material for its construction. Answer: (4,4,2) 10. Find the points on the sphere 2 2 2 36 x y z + + = that are closest to and farthest from the point (1, 2, 2). Use Lagrange multiplier method. Answer: Closest (2,4,4) and furthest (-2,-4,-4) . You may look at your text book for calculation and work....
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## This note was uploaded on 05/09/2008 for the course MAT 272 taught by Professor Firoz during the Spring '08 term at ASU.

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272t2sol - D f = - + + =- = 7. Verify that ( , ) sin( ) u x...

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