To determine which finite differences are constant for the polynomial function $y = 3x^2 + 4x - 2x^4 + 5$, we need to follow these steps:

Step 1: Understanding Finite Differences

• First finite differences are the differences between consecutive $y$-values.
• Second finite differences are the differences between consecutive first finite differences.
• Third finite differences are the differences between consecutive second finite differences, and so on.

For a polynomial function:

• A linear function has constant first finite differences.
• A quadratic function has constant second finite differences.
• A cubic function has constant third finite differences.
• A quartic function (degree 4, as in this case) has constant fourth finite differences.

Step 2: Identifying the Degree of the Polynomial

The given polynomial is: $y = 3x^2 + 4x - 2x^4 + 5$ This is a quartic polynomial because the highest degree of $x$ is 4 (from the $-2x^4$ term).

Step 3: Determine the Constant Finite Differences

For a quartic polynomial, the fourth finite differences will be constant.

Step 4: Find the Value of the Constant Finite Differences

To find the actual value of the constant fourth finite difference, we can compute successive finite differences for several values of $x$.

Let's compute this using a small set of points numerically.### Results: For the polynomial $y = 3x^2 + 4x - 2x^4 + 5$, we calculated the finite differences as follows:

• $y$-values for $x$ from -2 to 3: $[-23, 2, 5, 10, -7, -118]$
• First finite differences: $[25, 3, 5, -17, -111]$
• Second finite differences: $[-22, 2, -22, -94]$
• Third finite differences: $[24, -24, -72]$
• Fourth finite differences: $[-48, -48]$

a) Which finite differences are constant?

The fourth finite differences are constant.

b) Value of the constant finite differences:

The constant value of the fourth finite differences is $-48$.

Would you like more details or have any other questions?

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Here are five related questions to expand on this topic:

1. How do finite differences relate to derivatives in calculus?
2. What is the significance of the degree of a polynomial in determining finite differences?
3. Can finite differences be used to approximate the derivative of a function?
4. How do you compute finite differences for non-polynomial functions?
5. How does the symmetry of a polynomial affect its finite differences?

Tip: Finite differences are particularly useful in numerical methods for approximating solutions to differential equations.