Question

2. (15 points) Let $f$ be a function defined on an interval $[a, b]$ and let the nodes $a = x_0 < x_1 < x_2 = b$ be given. A quadratic spline $s(x)$ interpolating function $f(x)$ consists of the quadratic polynomials $s_0(x) = a_0 + b_0(x - x_0) + c_0(x - x_0)^2$ on $[x_0, x_1]$ and $s_1(x) = a_1 + b_1(x - x_1) + c_1(x - x_1)^2$ on $[x_1, x_2]$ such that the following conditions are satisfied: (a) $s(x_i) = f(x_i)$, for $i = 0, 1, 2$ (b) $s_0(x_1) = s_1(x_1)$ (continuity) (c) $s_0'(x_1) = s_1'(x_1)$ (smoothness) (d) one other condition (e.g. boundary condition) that will lead to a unique solution (e.g. $s'(x_0) = const$). Given the data $f(0) = 0$, $f(1) = 1$ and $f(2) = 2$ determine the quadratic spline that interpolates this data and satisfies $s'(0) = 2$.

          2. (15 points) Let $f$ be a function defined on an interval $[a, b]$ and let the nodes $a = x_0 < x_1 < x_2 = b$ be given. A quadratic spline $s(x)$ interpolating function $f(x)$ consists of the quadratic polynomials

$s_0(x) = a_0 + b_0(x - x_0) + c_0(x - x_0)^2$ on $[x_0, x_1]$
and
$s_1(x) = a_1 + b_1(x - x_1) + c_1(x - x_1)^2$ on $[x_1, x_2]$
such that the following conditions are satisfied:
(a) $s(x_i) = f(x_i)$, for $i = 0, 1, 2$
(b) $s_0(x_1) = s_1(x_1)$ (continuity)
(c) $s_0'(x_1) = s_1'(x_1)$ (smoothness)
(d) one other condition (e.g. boundary condition) that will lead to a unique solution
(e.g. $s'(x_0) = const$).
Given the data $f(0) = 0$, $f(1) = 1$ and $f(2) = 2$ determine the quadratic spline that interpolates this data and satisfies $s'(0) = 2$.

2. (15 points) Let f be a function defined on an interval [a, b] and let the nodes a = x0 < x1 < x2 = b be given. A quadratic spline s(x) interpolating function f(x) consists of the quadratic polynomials

s0(x) = a0 + b0(x - x0) + c0(x - x0)^2 on [x0, x1]
and
s1(x) = a1 + b1(x - x1) + c1(x - x1)^2 on [x1, x2]
such that the following conditions are satisfied:
(a) s(xi) = f(xi), for i = 0, 1, 2
(b) s0(x1) = s1(x1) (continuity)
(c) s0'(x1) = s1'(x1) (smoothness)
(d) one other condition (e.g. boundary condition) that will lead to a unique solution
(e.g. s'(x0) = const).
Given the data f(0) = 0, f(1) = 1 and f(2) = 2 determine the quadratic spline that interpolates this data and satisfies s'(0) = 2.

Added by Erin H.

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