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math226fall07pmt2s

math226fall07pmt2s - 1 a Dierentiating implicitly yz = ln(x...

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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 x -axis, 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 0 ( x + ct ) - cg 0 ( x - ct ) , u x ( t, x ) = f 0 ( x + ct ) + g 0 ( 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 ( πt ). How fast is the temperature rising on the bug’s 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 πt ) π | t =2 = 8 . 3 . A right circular cylinder is measured to have a radius of 10 ± 0 . 02 inches and a height of 6 ± 0 . 01 inches. Estimate the error in the volume calculation (the formula V = πr 2 h was used) . Answer . Differential of f ( r, h ) = πr 2 h is dV = f r ( r, h ) dr + f h ( r, h ) dh = 2 πrhdr + πr 2 dh. The error Δ V f r (10 , 6)0 . 02 + f h (10 , 6)0 . 01 = 2 π · 10 · 6 · 0 . 02 + π 10 2 0 . 01 = 3 . 4 π 4 . Write the equation of the tangent plane to the level surface S : f ( x, y, z ) = ln ( xy
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