Final answer:
Constructing interpolation polynomials of degree at most one and two for f(x) = sin(πx) to approximate f(1.4) involves calculating the values of the function at given points and solving for the polynomials that fit these points. We then compare the polynomial approximations for f(1.4) with the true value to find the absolute errors.
Step-by-step explanation:
In this mathematical problem, we are asked to construct interpolation polynomials of degree at most one (linear) and at most two (quadratic) to approximate the value of the function f(x) = sin(πx) at x = 1.4, using the given points x₀ = 1, x₁ = 1.25, and x₂ = 1.6. Additionally, we need to calculate the absolute error of these approximations compared to the true value of f(1.4).
To solve the mathematical problem completely, we start with the linear interpolation using points x₀ and x₁. The equation of the line passing through the points (x₀, f(x₀)) and (x₁, f(x₁)) is given by: P1(x) = f(x₀) + ¶◁f/◁x · (x - x₀), where ◁f/◁x is the difference in y-values divided by the difference in x-values. We compute the value P1(1.4) and then the absolute error by finding the difference between the true value of f(1.4) = sin(π· 1.4) and P1(1.4).
For the quadratic interpolation using points x₀, x₁, and x₂, we need to calculate the coefficients of the quadratic polynomial P2(x) such that P2(x₀) = f(x₀), P2(x₁) = f(x₁), and P2(x₂) = f(x₂). By solving the system of equations, we obtain the expression for P2(x) to approximate f(1.4). Lastly, we calculate the absolute error in the same manner by comparing P2(1.4) with f(1.4).
The actual computation of these polynomials and errors involves substituting the values of x and calculating sin(πx) at these points. We would also apply mathematical calculations to find the final results.