math2004_ft06_test3c_sol

# math2004_ft06_test3c_sol - MATH 2004 TEST 3 Version C...

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Unformatted text preview: MATH 2004 TEST 3 Version C November 2006 I N S T R U C T I O N S This test has one page and is worth 30 marks (plus 3 bonus marks ). It cannot be taken out of the exam room. Test duration: 50 minutes . Write your full name and student number onto the exam booklet provided by your tutor. Show all your work in this booklet and explain carefully your solutions. No calculators are allowed. 1. Suppose the function T ( x, y ) = 4 x 3 + 2 xy + y 2 describes the temperature T on a plate at the position ( x, y ). [5 marks] (a) Find the direction of the steepest increase of temperature at the position (1 , 1). Solution: The direction of steepest increase of T ( x, y ) at (1 , 1) is given by the gradient vector ∇ T (1 , 1): 1 mark for knowledge that gradient plays are role 1 mark for knowledge of correct direction ∇ T ( x, y ) = ( T x ( x, y ) , T y ( x, y )) = (12 x 2 + 2 y, 2 x + 2 y ) ⇒ ∇ T (1 , 1) = (14 , 4) . 2 marks one mark for each partial derivative 1 mark for evaluation at point [3 marks] (b) Find the directional derivative of T at (1 , 1) in the direction u = ( 3 5 ,- 4 5 ). Solution: Since u is already a unit vector, the directional derivative of T at(1 , 1) in the direction u is: D u T (1 , 1) = ∇ T (1 , 1) • u = ( 14 , 4 ) • (- 3 5 , 4 5 ) =- 42 5 + 16 5 =- 26 5 ....
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math2004_ft06_test3c_sol - MATH 2004 TEST 3 Version C...

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