Math Problem Statement

it a least square curve 𝑦 = 𝑎𝑥𝑏 for the following data: X 2.2 2.7 3.5 4.1 Y 65 60 53 50

Solution

To find the least squares curve of the form y=axby = ax^b that best fits the given data points, we'll use the logarithmic transformation to convert this nonlinear equation into a linear form.

  1. The equation is y=axby = ax^b.

  2. Take the natural logarithm of both sides: ln(y)=ln(a)+bln(x)\ln(y) = \ln(a) + b \ln(x) This is now a linear equation of the form ln(y)=A+Bln(x)\ln(y) = A + B \ln(x), where A=ln(a)A = \ln(a) and B=bB = b.

  3. Now, we will treat this as a linear least squares problem to find AA and BB. Once we find AA and BB, we can compute a=eAa = e^A and b=Bb = B.

Steps:

  1. Compute ln(x)\ln(x) and ln(y)\ln(y) for each data point.
  2. Set up the least squares regression to find AA and BB.
  3. Use the values of AA and BB to find aa and bb.

Let’s proceed with the calculations.

The given data is:

  • X=[2.2,2.7,3.5,4.1]X = [2.2, 2.7, 3.5, 4.1]
  • Y=[65,60,53,50]Y = [65, 60, 53, 50]

I'll perform the calculations to find the least squares fit.The least squares fit for the curve y=axby = ax^b based on the given data is:

y=91.43x0.43y = 91.43x^{-0.43}

Here:

  • a=91.43a = 91.43
  • b=0.43b = -0.43

This means that the curve fitting the data follows the equation y=91.43x0.43y = 91.43x^{-0.43}.

Would you like further details on the steps, or any clarifications?

Here are 5 related questions to explore this concept further:

  1. How does the least squares method work for linear versus nonlinear models?
  2. What is the significance of the exponent bb in this curve?
  3. How can the goodness of fit (like R2R^2) be interpreted for this model?
  4. What happens if more data points are added—how does it affect the curve fitting?
  5. Can this method be applied to other forms of equations like y=ax2+bx+cy = ax^2 + bx + c?

Tip: When fitting nonlinear models, transforming the equation to a linear form (like using logarithms) can greatly simplify the least squares method.

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Math Problem Analysis

Mathematical Concepts

Least Squares Method
Exponential Functions
Logarithmic Transformation

Formulas

y = ax^b
ln(y) = ln(a) + b * ln(x)
Linear Least Squares Equation

Theorems

Least Squares Theorem
Logarithmic Function Properties

Suitable Grade Level

Grades 10-12