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math-51-f09-hw8-revised

# math-51-f09-hw8-revised - 1 Problem set 8(This version of...

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1 Problem set 8 (This version of the problem set has corrected the typo on Question 3.) The notation in this problem set follows Colley. 1. Let f ( x ) = ( x + 1 , x 2 + 9) and let h ( r, s, t ) = r 2 t 2 + rs . Use the Chain Rule to compute the D ( f h )(2 , 0 , 3) . 2. Let g ( q, r, s ) = q 3 +2 rs , and let f ( u ) = ( u 2 , u 2 + u +1 , ln( u 2 +1) ) . Compute D ( f g ) in two ways. First, write out the composition and differentiate, and second, use the Chain Rule. 3. Let f : R 3 R 2 be defined by f ( x, y, z ) = ( e x 2 + y 2 + z 2 2 z, x 2 y y 2 z ) and let g : R 2 R satisfy Dg ( e 2 , 1) = [12 17] . If h = g f , compute Dh (1 , 1 , 0) . 4. A rectangular box has width 4 . 0 cm, length 6 . 0 cm, and height 3 . 0 cm. At what rate is the volume of the box changing if the width is decreasing at the rate 1 . 5 cm/s, the length is increasing at the rate 2 . 5 cm/s and the height is increasing at the rate 0 . 2 cm/s. At what rate is the surface area changing? 5. Let f ( x, y ) = 1 xy , v = bracketleftbigg 3 2 bracketrightbigg , and a = bracketleftbigg 1 2 bracketrightbigg . Let u

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math-51-f09-hw8-revised - 1 Problem set 8(This version of...

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