Dx l f x dx 3 where f x is as in problem c show that

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) dx = L 0 f ( x ) dx,
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3 where f ( x ) is as in Problem C. Show that if u solves Problem C , then v ( x, t ) = u ( x, t ) - u ( x ) solves v t ( x, t ) = kv xx ( x, t ) , 0 < x < L, t > 0 , v x (0 , t ) = 0 , v x ( L, t ) = 0 , t > 0 , v ( x, 0) = f ( x ) - u ( x ) , 0 < x < L, and use this to explain (prove?) that lim t →∞ u ( x, t ) = u ( x ). (d) Calculus question: In equation (1), on the left hand side the derivative is denoted by d dt , on the right hand side by ∂t . Why the change in notation?
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