math192_hw5sp06

# math192_hw5sp06 - HW5 Solutions 14.5 Directional...

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HW5 Solutions 14.5 Directional Derivatives and Gradient Vectors 4. gradient: g = 2 i j ; g ( 2 , 1 ) = 1 2 1 2 = x 2 2 y 2 2 or 1 = x 2 y 2 is the level curve. 8. f = parenleftBig 3+2 2 parenrightBig i + 3 2 j 1 2 k 12. ( D u h ) P 0 = h · u = 3 2 13 13. u = v | v | = 3 i +6 j 2 k 3 2 +6 2 +( 2) 2 = 3 7 i + 6 7 2 7 k ; f x ( x,y,z ) = y + z f x (1 , 1 , 2) = 1; f y ( x,y,z ) = x + z f y (1 , 1 , 2) = 3; f z ( x,y,z ) = y + x f z (1 , 1 , 2) = 0 ⇒ ∇ f = i + 3 j ( D u f ) P 0 = f · u = 3 7 + 18 7 = 3 20. g increases most rapidly in the direction u = 2 3 i + 2 3 j + 1 3 k and decreases most rapidly in the direction of u = 2 3 i 2 3 j 1 3 k ; ( D u g ) P 0 = 3 and ( D u g ) P 0 = 3 24. f = 2 x i j f ( 2 , 1 ) = 2 2 i j Tangent line: 2 2 ( x 2 ) ( y 1) = 0 y = 2 2 x 3 32. (a) We are given that the derivative is the greatest in the direction of v = i + j k and has a value of 2 3. Since the gradient points in the direction of maximum change, v is in the same direction as f (i.e.

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