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

To find the least squares curve of the form \( y = ax^b \) 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 \( y = ax^b \).
2. Take the natural logarithm of both sides:
   \[
   \ln(y) = \ln(a) + b \ln(x)
   \]
   This is now a linear equation of the form \( \ln(y) = A + B \ln(x) \), where \( A = \ln(a) \) and \( B = b \).

3. Now, we will treat this as a linear least squares problem to find \( A \) and \( B \). Once we find \( A \) and \( B \), we can compute \( a = e^A \) and \( b = B \).

### Steps:

1. Compute \( \ln(x) \) and \( \ln(y) \) for each data point.
2. Set up the least squares regression to find \( A \) and \( B \).
3. Use the values of \( A \) and \( B \) to find \( a \) and \( b \).

Let’s proceed with the calculations.

The given data is:
- \( X = [2.2, 2.7, 3.5, 4.1] \)
- \( Y = [65, 60, 53, 50] \)

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

\[
y = 91.43x^{-0.43}
\]

Here:
- \( a = 91.43 \)
- \( b = -0.43 \)

This means that the curve fitting the data follows the equation \( y = 91.43x^{-0.43} \).

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 \( b \) in this curve?
3. How can the goodness of fit (like \( R^2 \)) 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 \( y = ax^2 + bx + c \)?

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

it a least square curve 𝑦 = 𝑎𝑥^𝑏 for the following data X 2.2 2.7 3.5 4.1, Y 65 60 53 50

Learn how to use the least squares method to find the curve of the form y = ax^b for the data points X = [2.2, 2.7, 3.5, 4.1] and Y = [65, 60, 53, 50]. This problem explores key mathematical concepts like logarithmic transformations and the least squares method for nonlinear models. Suitable for students in Grades 10-12.

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