To solve this problem, we need to find the interpolating polynomial $P_3(x) = ax^3 + bx^2 + cx + d$ using the Lagrange interpolation method based on the data provided:

Given Data: $x: \{0, 1, 2, 3\}, \quad f(x): \{1, 2, 1, 10\}$

Steps to Solve:

1. Lagrange Basis Polynomials: Define the Lagrange basis polynomials $L_i(x)$ for $i = 0, 1, 2, 3$: $L_i(x) = \prod_{\substack{j=0 \\ j \neq i}}^3 \frac{x - x_j}{x_i - x_j}$

2. Interpolating Polynomial: The interpolating polynomial is: $P_3(x) = \sum_{i=0}^3 f(x_i) L_i(x)$

3. Calculate the Basis Polynomials: Substitute the values of $x_0 = 0, x_1 = 1, x_2 = 2, x_3 = 3$:

   • $L_0(x) = \frac{(x - 1)(x - 2)(x - 3)}{(0 - 1)(0 - 2)(0 - 3)} = -\frac{1}{6}(x - 1)(x - 2)(x - 3)$
   • $L_1(x) = \frac{(x - 0)(x - 2)(x - 3)}{(1 - 0)(1 - 2)(1 - 3)} = \frac{1}{2}(x)(x - 2)(x - 3)$
   • $L_2(x) = \frac{(x - 0)(x - 1)(x - 3)}{(2 - 0)(2 - 1)(2 - 3)} = -\frac{1}{2}(x)(x - 1)(x - 3)$
   • $L_3(x) = \frac{(x - 0)(x - 1)(x - 2)}{(3 - 0)(3 - 1)(3 - 2)} = \frac{1}{6}(x)(x - 1)(x - 2)$

4. Form the Polynomial: Combine the results to find $P_3(x)$: $P_3(x) = f(0)L_0(x) + f(1)L_1(x) + f(2)L_2(x) + f(3)L_3(x)$

5. Substitute $x = 0.5$: Plug in $x = 0.5$ into the polynomial and calculate $f(0.5)$.

Let me calculate this and provide the result. sympy import symbols, expand, simplify

Define the variable

x = symbols('x')

Given points

points = [(0, 1), (1, 2), (2, 1), (3, 10)]

Lagrange basis polynomials

L0 = ((x - 1)(x - 2)(x - 3)) / ((0 - 1)(0 - 2)(0 - 3)) L1 = ((x - 0)(x - 2)(x - 3)) / ((1 - 0)(1 - 2)(1 - 3)) L2 = ((x - 0)(x - 1)(x - 3)) / ((2 - 0)(2 - 1)(2 - 3)) L3 = ((x - 0)(x - 1)(x - 2)) / ((3 - 0)(3 - 1)(3 - 2))

Function values

f0, f1, f2, f3 = [p[1] for p in points]

Interpolating polynomial

P3 = simplify(f0 * L0 + f1 * L1 + f2 * L2 + f3 * L3)

Evaluate the polynomial at x = 0.5

value_at_05 = P3.subs(x, 0.5)

P3, value_at_05The interpolating polynomial using the Lagrange approach is:

$P_3(x) = 2x^3 - 7x^2 + 6x + 1$

Substituting $x = 0.5$, the estimated value is:

$f(0.5) = 2.5$

If you'd like, I can further elaborate on the steps or explain any specific part. Let me know!

Related Questions:

1. How do the Lagrange basis polynomials ensure interpolation at given points?
2. Can you derive the general formula for Lagrange interpolation?
3. What is the significance of the degree of the interpolating polynomial in this case?
4. How does Lagrange interpolation compare to Newton interpolation?
5. What are some practical applications of polynomial interpolation?

Tip:

For numerical stability, always simplify intermediate expressions when calculating interpolation polynomials!