function f ( x ) f(x) which is the best fit for the data given in Table 1 recorded by a student is f ( x )

The goal is to find the value of $c$ that minimizes the Sum of Squared Errors (SSE) between the given function $f(x) = -(x - 1)^2 (x - 3)(x - 5)(x - 7) + c$ and the data points in the table.

Given Data:

• $(x_1, y_1) = (1, 4)$
• $(x_2, y_2) = (2, 8)$
• $(x_3, y_3) = (3, 4)$
• $(x_4, y_4) = (4, -24)$
• $(x_5, y_5) = (5, 3)$

Approach:

1. Expression for SSE: $SSE(c) = \sum_{i=1}^{5} \left( y_i - f(x_i) \right)^2$ Where $f(x_i) = -(x_i - 1)^2 (x_i - 3)(x_i - 5)(x_i - 7) + c$, for each $i$.

2. Objective: Minimize $SSE(c)$, which is a function of $c$. To minimize the SSE, we compute its derivative with respect to $c$, set the derivative equal to zero, and solve for $c$.

3. Derivative of SSE: Since the derivative of the squared error with respect to $c$ is: $\frac{d}{dc} \left( (y_i - f(x_i))^2 \right) = -2(y_i - f(x_i)) \cdot \frac{d}{dc} f(x_i)$ The derivative of $f(x_i)$ with respect to $c$ is just $1$, since the term involving $c$ is linear: $\frac{d}{dc} f(x_i) = 1$ Thus, the derivative of $SSE$ with respect to $c$ is: $\frac{d}{dc} SSE(c) = -2 \sum_{i=1}^{5} (y_i - f(x_i))$ To minimize the SSE, set this derivative equal to zero: $\sum_{i=1}^{5} (y_i - f(x_i)) = 0$

4. Final equation to solve for $c$: $\sum_{i=1}^{5} y_i = \sum_{i=1}^{5} f(x_i)$ Substituting $f(x_i) = -(x_i - 1)^2 (x_i - 3)(x_i - 5)(x_i - 7) + c$ into the above equation, we get: $\sum_{i=1}^{5} y_i = \sum_{i=1}^{5} \left[ -(x_i - 1)^2 (x_i - 3)(x_i - 5)(x_i - 7) + c \right]$ Now, solve for $c$.

Calculation:

Let's first calculate the sum of $y_i$'s: $\sum_{i=1}^{5} y_i = 4 + 8 + 4 - 24 + 3 = -5$

Next, calculate the sum of $-(x_i - 1)^2 (x_i - 3)(x_i - 5)(x_i - 7)$ for each data point:

• For $x = 1$: $-(1 - 1)^2 (1 - 3)(1 - 5)(1 - 7) = 0$

• For $x = 2$: $-(2 - 1)^2 (2 - 3)(2 - 5)(2 - 7) = -1^2 (-1)(-3)(-5) = -15$

• For $x = 3$: $-(3 - 1)^2 (3 - 3)(3 - 5)(3 - 7) = 0$

• For $x = 4$: $-(4 - 1)^2 (4 - 3)(4 - 5)(4 - 7) = -(3)^2 (1)(-1)(-3) = -27$

• For $x = 5$: $-(5 - 1)^2 (5 - 3)(5 - 5)(5 - 7) = 0$

Now, sum these results: $\sum_{i=1}^{5} -(x_i - 1)^2 (x_i - 3)(x_i - 5)(x_i - 7) = 0 + (-15) + 0 + (-27) + 0 = -42$

Now substitute into the equation: $-5 = -42 + 5c$

Solving for $c$: $5c = 37$ $c = \frac{37}{5} = 7.4$

Final Answer:

The value of $c$ that minimizes the SSE is $c = 7.4$.

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Would you like further clarification or details on any step? Here are some related questions that might expand your understanding:

1. How does the derivative of SSE relate to finding the minimum of a function?
2. What are the implications of the sum of squared errors in regression analysis?
3. How does the choice of function form impact the minimization of SSE?
4. What happens if the data points deviate significantly from the expected model form?
5. How can we apply this technique to higher-degree polynomials or other types of fitting functions?

Tip: To minimize error in curve fitting, always check if your model is well-suited to the data distribution—sometimes a simple linear model might perform poorly for non-linear data.