In pipeline design for oil transport, pressure losses must be carefully estimated. They are directly proportional...

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In pipeline design for oil transport, pressure losses must be carefully estimated. They are directly proportional to a positive friction factor fD, whose inverse square root x =1/√f is given by a non-linear equation. In fact, it is known that x depends only on the Reynolds number R. For a turbulent flow at a given R, the equation to find x is x +1.931n(1.32x/R) = 0, in which In(x) denotes the natural logarithm. In this exercise, we will set R = 5000 and use fixed-point iterations to approximate x, according to x(k+1)=g(x(k)), with g(x):=-1.931n(1.32x/5000). Let us also define the iteration error as ek x(k)x(k-1) Based on that, you are asked to: a) Compute g'(x) and use the result to determine whether g(x), x>0, is an increasing, decreasing or a non-monotonic function. b) Calculate the maximum and minimum values of g(x) in the interval x € [e, e³]. In pipeline design for oil transport, pressure losses must be carefully estimated. They are directly proportional to a positive friction factor fD, whose inverse square root x =1/√f is given by a non-linear equation. In fact, it is known that x depends only on the Reynolds number R. For a turbulent flow at a given R, the equation to find x is x +1.931n(1.32x/R) = 0, in which In(x) denotes the natural logarithm. In this exercise, we will set R = 5000 and use fixed-point iterations to approximate x, according to x(k+1)=g(x(k)), with g(x):=-1.931n(1.32x/5000). Let us also define the iteration error as ek x(k)x(k-1) Based on that, you are asked to: a) Compute g'(x) and use the result to determine whether g(x), x>0, is an increasing, decreasing or a non-monotonic function. b) Calculate the maximum and minimum values of g(x) in the interval x € [e, e³].

Answer rating: 100% (QA)

a To compute gx we first apply the chain rule gx 1931 1132 1x 1132x5000 132500... View the full answer