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 that best fits the given data points, we'll use the logarithmic transformation to convert this nonlinear equation into a linear form.
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The equation is .
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Take the natural logarithm of both sides: This is now a linear equation of the form , where and .
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Now, we will treat this as a linear least squares problem to find and . Once we find and , we can compute and .
Steps:
- Compute and for each data point.
- Set up the least squares regression to find and .
- Use the values of and to find and .
Let’s proceed with the calculations.
The given data is:
I'll perform the calculations to find the least squares fit.The least squares fit for the curve based on the given data is:
Here:
This means that the curve fitting the data follows the equation .
Would you like further details on the steps, or any clarifications?
Here are 5 related questions to explore this concept further:
- How does the least squares method work for linear versus nonlinear models?
- What is the significance of the exponent in this curve?
- How can the goodness of fit (like ) be interpreted for this model?
- What happens if more data points are added—how does it affect the curve fitting?
- Can this method be applied to other forms of equations like ?
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
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