Algebraic Modeling subject x−3−2−101234y427494341236108324 What is the exponential regression of the data? y=?

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lamanocornudaW · Accepted Answer

Step 1 Given a dataset where an exponential function y=abx where a and b are constants. are to be obtained. Taking logarithm on both sides of the above equation, log⁡ y=log⁡ a+xlog⁡ b ⇒υ=A+Bx where υ=log⁡ y A=log⁡ a B=log⁡ b Hence, the exponential function is reduced to a linear function between υ and x and can be solved by least square method. By principle of least squares, the normal equations for estimating A and B is given by, ∑υ=nA+B∑x ∑υx=A∑x+B∑x2 Step 2 The following is table is formed to calculate the value of A and B yxυ=log⁡ yx2υ427−3−0.829392.487949−2−0.352240.704443−10.12491−0.1249400.6021001211.079211.07923621.556343.112610832.033496.100232442.51051610.0420Total46.72494423.4014 Hence, putting the values obtained from the table to the normal equations we get, 1) 6.7249=8A+4B 2) 23.4014=4A+44B On solving the simultaneous equations, by method of comparison, it is obtained that, from (1) 3) A=6.7249−aB8 from (2) 4) A=23.4014−55B4 Comparing (3) and (4) 6.7249−4B8=23.4014−44B4 Hence solving the above equation, it is obtained that, B=0.4771 ⇒b=Antilog⁡ B =2.9999 Then putting the value of B in (3), it is obtained that, A=0.6021 ⇒a=Antilog⁡ A =4.0004 Therefore, the required exponential regression of the data is y=4.0005(2.9999)x

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