Question

(1.1/1.2: Interpolating polynomials) Say we want to find a polynomialf(x) ofdegree 3,f(x) =a0+a1x+a2x2+a3x3,satisfying some interpolation conditions. In each case below, write a system of linearequations whose solutions are (a0,a1,a2,a3). You don’t need to solve the system.(a) We wantf(x) to pass through the points (−1,−1),(1,2),(2,1) and (3,5).(b) We wantf(x) to pass through (1,0) with derivative +2 and (2,3) withderivative−1

Answer 1

(a) If

f(x) = a₀ + a₁ x + a₂ x ² + a₃ x ³

then from the given conditions we get the system of equations,

f (-1) = a₀ - a₁ + a₂ - a₃ = -1

f (1) = a₀ + a₁ + a₂ + a₃ = 2

f (2) = a₀ + 2a₁ + 4a₂ + 8a₃ = 1

f (3) = a₀ + 3a₁ + 9a₂ + 27x ³ = 5

(b) Similarly, if

f(x) = a₀ + a₁ x + a₂ x ² + a₃ x ³

then

f'(x) = a₁ + 2a₂ x + 3a₃ x ²

so that the given conditions yield the system,

f (1) = a₀ + a₁ + a₂ + a₃ = 0

f' (1) = a₁ + 2a₂ + 3a₃ = 2

f (2) = a₀ + 2a₁ + 4a₂ + 27a₃ = 3

f' (2) = a₁ + 4a₂ + 12a₃ = -1