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Unformatted text preview: 1 . a) Differentiating implicitly yz = ln( x + z ) (assuming this equation defines z as a function x and y ), find the rate of change of z in the direction of xaxis, when x = 0 ,y = 0 ,z = 1. Answer . Differentiating both sides of yz = ln( x + z ) (thinking of z as a function of x,y ) we get y z x = 1 x + z (1 + z x ) . At x = 0 ,y = 0 ,z = 1 , 1 0 + 1 (1 + z x ) = 0 or z x = 1 . b) Show that if u ( t,x ) = f ( x + ct ) + g ( x ct ) , then u tt = c 2 u xx . Answer . By chain rule, u t ( t,x ) = cf ( x + ct ) cg ( x ct ) ,u x ( t,x ) = f ( x + ct ) + g ( x ct ) , u tt ( t,x ) = c 2 f 00 ( x + ct ) + c 2 g 00 ( x + ct ) ,u xx ( t,x ) = f 00 ( x + ct ) + g 00 ( x + ct ) . 2 . The temperature at a point ( x,y ) is T ( x,y ) and T x (5 , 1) = 2 ,T y (5 , 1) = 3. A bug crawls so that its position after t seconds is given by x = 1 + t 2 ,y = cos( t ). How fast is the temperature rising on the bugs path in 2 seconds. Answer . d dt T  t =2 = T x (5 , 1) dx dt  t =2 + T y (5 , 1) dy dt  t =2 = 2(2 t )  t =2 +3( sin t )  t =2 = 8 . 3 . A right circular cylinder is measured to have a radius of 10 . 02 inches and a height of 6 . 01 inches. Estimate the error in the volume calculation (the formula V = r 2 h was used) ....
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 '07
 KAMIENNY
 Math, Calculus, Rate Of Change

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