15.3Ex61-62

15.3Ex61-62 - 652 C H A P T E R 15 D I F F E R E N T I AT I...

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652 CHAPTER 15 DIFFERENTIATION IN SEVERAL VARIABLES (ET CHAPTER 14) SOLUTION The function ρ ( 33 , T ) is concave up (concave down) if T ( 33 , T ) is an increasing (decreasing) function of T . We use Table 1 to estimate whether the function T ( 33 , T ) is increasing or decreasing. We compute the following values: T ( 33 , 2 ) ( 33 , 4 ) ( 33 , 2 ) 2 = 26 . 23 26 . 38 2 =− 0 . 075 T ( 33 , 4 ) ( 33 , 6 ) ( 33 , 4 ) 2 = 26 26 . 23 2 0 . 115 T ( 33 , 6 ) ( 33 , 8 ) ( 33 , 6 ) 2 = 25 . 73 26 2 0 . 135 T ( 33 , 8 ) ( 33 , 10 ) ( 33 , 8 ) 2 = 25 . 42 25 . 73 2 0 . 155 T ( 33 , 10 ) ( 33 , 12 ) ( 33 , 10 ) 2 = 25 . 07 25 . 42 2 0 . 175 These values indicate that T ( 33 , T ) is a decreasing function of T , which means that the second derivative is negative, i.e., 2 T 2 ( 33 , T )< 0 and the graph of ( 33 , T ) is concave down. 61. Compute f xyz for f ( x , y , z ) = sin ( yx ) + tan à z + z 1 x x 1 Hint: Use a well-chosen order of differentiation on each term. At the points where the derivatives are continuous, the partial derivative f may be performed in any order. To simplify the computation we Frst differentiate with respect to y .Thisgives f y ( x , y , z ) = y sin ( ) + 0 = cos ( ) y ( ) = x cos ( ) We now differentiate f y with respect to z : f yz ( x , y , z ) = z ( x cos ( ) ) = 0 Hence, f yzx ( x , y , z ) = 0 We conclude that at the points where the partial derivatives are continuous,
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This note was uploaded on 02/13/2010 for the course MATH MATH 32A taught by Professor Park during the Fall '09 term at UCLA.

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