 # A popular method for curve fitting is the “Least Squared&rdqu...

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A popular method for curve fitting is the “Least Squared” method. a) Determine the value of the constant k which best approximates a specified function f(x) on the interval 𝑎 ≤ 𝑥 ≤ 𝑏. Use as a cost function to be minimized: 𝐿(𝑘) = 1 2 ∫ [𝑘 − 𝑓(𝑥)] 2𝑑𝑥 𝑏 𝑎 Verify that a minimum is achieved by checking the 2nd order optimality condition. b) Determine the optimal value of 𝑘 and 𝐿(𝑘) in fractional form for 𝑓(𝑥) = 𝑥 2 , 𝑎 = 0, 𝑎𝑛𝑑 𝑏 = 1. c) For the best linear fit, determine the optimal value of 𝑘, 𝑚 𝑎𝑛𝑑 𝐿(𝑘) in fractional form for 𝑓(𝑥) = 𝑥 2 , 𝑎 = 0, 𝑎𝑛𝑑 𝑏 = 1 where, 𝐿(𝑘, 𝑚) = 1 2 ∫ [𝑚𝑥 + 𝑘 − 𝑓(𝑥)] 2𝑑𝑥 𝑏 𝑎 Hint: The answers to these problems can be confirmed using MATLAB’s or Excel’s curve fitting. In part a. we need to solve 𝑑𝐿(𝑘) 𝑑𝑘 = 0

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