Let's analyze the image and extract the problems step-by-step for solving:

Extracted Problems:

1. Problem 1:

   • Create the following table of values where y1, y2, y3, and y4 are the first, second, third, and fourth differences of the function.
   • The given table seems to have the following values:

   x	f(x)
   1	10
   2	21
   3	36
   4	55
   5	78

2. Problem 2:

   • Find a cubic polynomial which fits the following table:

   x	y
   1	3
   2	6
   3	19
   4	58

3. Problem 3:

   • Determine if a polynomial fits the data and find the values if it does:

   x	y
   0	6
   1	11
   2	18
   3	27
   4	38

Let's solve these problems step-by-step:

Solution 1: Difference Table

To create the difference table, we calculate the first, second, third, etc., differences between consecutive values of f(x).

Solution 2: Cubic Polynomial Fitting

To find a cubic polynomial of the form:

$y = ax^3 + bx^2 + cx + d$

We set up the system of equations using the provided values and solve for $a, b, c,$ and $d$.

Solution 3: Quadratic Polynomial Fitting

For the given table, we try to fit it using a quadratic polynomial of the form:

$y = ax^2 + bx + c$

By setting up a system of linear equations using the given points, we can determine if a quadratic polynomial fits the data and solve for the coefficients $a, b,$ and $c$.

Let's proceed with detailed calculations for each of these solutions.

Solution 1: Calculating Differences (Detailed)

Let's compute the values for the first problem to fill out the differences:

1. First Differences (y1):

   • $y1_2 = f(2) - f(1) = 21 - 10 = 11$
   • $y1_3 = f(3) - f(2) = 36 - 21 = 15$
   • $y1_4 = f(4) - f(3) = 55 - 36 = 19$
   • $y1_5 = f(5) - f(4) = 78 - 55 = 23$

2. Second Differences (y2):

   • $y2_3 = y1_3 - y1_2 = 15 - 11 = 4$
   • $y2_4 = y1_4 - y1_3 = 19 - 15 = 4$
   • $y2_5 = y1_5 - y1_4 = 23 - 19 = 4$

3. Third Differences (y3):

   • $y3_4 = y2_4 - y2_3 = 4 - 4 = 0$
   • $y3_5 = y2_5 - y2_4 = 4 - 4 = 0$

4. Fourth Differences (y4):

   • $y4_5 = y3_5 - y3_4 = 0 - 0 = 0$

Would you like me to provide detailed steps for solving the other two problems or focus on a specific part?

Relative Questions

1. How do we find the equation of a polynomial that fits a set of data points?
2. What is the significance of difference tables in polynomial interpolation?
3. How can we solve a system of linear equations to find polynomial coefficients?
4. What are some real-world applications of polynomial interpolation?
5. How does Newton's Divided Difference method differ from Lagrange Interpolation?

Tip:

When working with polynomials and fitting curves to data, always double-check the degree of the polynomial to ensure it appropriately matches the number of data points and the observed behavior of the data.