Equation for i=2: h1*z1 + 2*(h1+h2)*z2 + h2*z3 = 6*(slope2 - slope1) → 1*0 + 2*(1+2)*z2 + 2*z3 = 6*(-0.5 - 1) → 6*z2 + 2*z3 = -9

(x=1 to 2, h=1): a = 2 b = (3-2)/1 - 1/6*(2 0 + (-1.92857)) = 1 - (1/6) (-1.92857) = 1 + 0.32143 = 1.32143 c = 0/2 = 0 d = (-1.92857 - 0)/(6*1) = -0.32143

[ (x_i - x_i-1) z_i-1 + 2(x_i+1 - x_i-1) z_i + (x_i+1 - x_i) z_i+1 = 6 \left( \fracy_i+1 - y_ix_i+1 - x_i - \fracy_i - y_i-1x_i - x_i-1 \right) ]

Manual solution: From first: z2 = (-9 - 2*z3)/6 Sub into second: 2*[(-9 - 2*z3)/6] + 10*z3 = 9 → (-18 - 4*z3)/6 + 10*z3 = 9 → -3 - (2/3)z3 + 10*z3 = 9 → (28/3)z3 = 12 → z3 = 9/7 ≈ 1.285714 Then z2 = (-9 - 2*(9/7))/6 = (-9 - 18/7)/6 = (-81/7)/6 = -81/42 = -27/14 ≈ -1.92857

Equation for i=3: h2*z2 + 2*(h2+h3)*z3 + h3*z4 = 6*(slope3 - slope2) → 2*z2 + 2*(2+3)*z3 + 3*0 = 6*(1 - (-0.5)) → 2*z2 + 10*z3 = 9

(x=2 to 4, h=2): a = 3 b = (2-3)/2 - 2/6*(2*(-1.92857) + 1.285714) = -0.5 - (1/3) (-3.85714 + 1.285714) = -0.5 - (1/3) (-2.57143) = -0.5 + 0.85714 = 0.35714 c = -1.92857/2 = -0.964285 d = (1.285714 - (-1.92857))/(6*2) = (3.214284)/12 = 0.267857

[ S(x) = a + b(x-x_i) + c(x-x_i)^2 + d(x-x_i)^3 ]