math226fall10mt1s

# math226fall10mt1s - 1(a The dimensions x y z of a...

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1 . (a) The dimensions x,y,z of a rectangular box are measured as 70 cm, 50 cm and 40 cm respectively with a possible error of 0.1 in each dimension. Use diﬀerentials to estimate the maximum error in the calculated volume V = xyz of the box. Answer . The diﬀerential dV = V x dx + V y dy + V z dz = yz ( dx ) + xz ( dy ) + xy ( dz ) and the calculated error Δ V yz x ) + xz y ) + xy z ) = 50 · 40(0 . 1) + 70 · 40(0 . 1) + 70 · 50(0 . 1) = 830 (cm 3 ) . (b) Find the linearization of f ( x,y ) = ln( x 2 - 3 y ) at (1 , 0) and use it to approximate f (1 . 1 , - 0 . 1). Write the equation of the tangent plane to z = ln( x 2 - 3 y ) at (1 , 0 , 0). Answer . Linearization L ( x,y ) = f (1 , 0) + f x (1 , 0)( x - 1) + f y (1 , 0) y. We ﬁnd f x = 2 x x 2 - 3 y ,f y = - 3 x 2 - 3 y , f (1 , 0) = 0 ,f x (1 , 0) = 2 ,f y (1 , 0) = - 3 , and L ( x,y ) = 2( x - 1) - 3 y. The approximation f (1 . 1 , - 0 . 1) L (1 . 1 , - 0 . 1) = 2(0 . 1) - 3( - 0 . 1) = 0 . 5 . The equation of the tangent plane is

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## This note was uploaded on 12/16/2011 for the course MATH 226 at USC.

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math226fall10mt1s - 1(a The dimensions x y z of a...

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