91Ó°ÊÓ

Decide whether the equations form a cubic spline. (a) \(S(x)= \begin{cases}x^{3}+x-1 & \text { on }[0,1] \\\ -(x-1)^{3}+3(x-1)^{2}+3(x-1)+1 & \text { on }[1,2]\end{cases}\) (b) \(S(x)= \begin{cases}2 x^{3}+x^{2}+4 x+5 & \text { on }[0,1] \\\ (x-1)^{3}+7(x-1)^{2}+12(x-1)+12 & \text { on }[1,2]\end{cases}\)

List the Chebyshev interpolation nodes \(x_{1}, \ldots, x_{n}\) in the given interval. (a) \([-1,1], n=6\) (b) \([-2,2], n=4\) (c) \([4,12], n=6\) (d) \([-0.3,0.7], n=5\)

Use Lagrange interpolation to find a polynomial that passes through the points. (a) \((0,1),(2,3),(3,0)\) (b) \((-1,0),(2,1),(3,1),(5,2)\) (c) \((0,-2),(2,1),(4,4)\)

(a) Given the data points \((1,0),(2, \ln 2),(4, \ln 4)\), find the degree 2 interpolating polynomial. (b) Use the result of (a) to approximate \(\ln 3\). (c) Use Theorem \(3.3\) to give an error bound for the approximation in part (b). (d) Compare the actual error to your error bound.

How many degree \(d\) polynomials pass through the four points \((-1,3),(1,1),(2,3),(3,7)\) ? Write one down if possible. (a) \(d=2\) (b) \(d=3\) (c) \(d=6\).

Assume that the polynomial \(P_{9}(x)\) interpolates the function \(f(x)=e^{-2 x}\) at the 10 evenly spaced points \(x=0,1 / 9,2 / 9,3 / 9, \ldots, 8 / 9,1\). (a) Find an upper bound for the error \(\left|f(1 / 2)-P_{9}(1 / 2)\right|\). (b) How many decimal places can you guarantee to be correct if \(P_{9}(1 / 2)\) is used to approximate \(e^{-1} ?\)

Assume that Chebyshev interpolation is used to find a fifth degree interpolating polynomial \(Q_{5}(x)\) on the interval \([-1,1]\) for the function \(f(x)=e^{x}\). Use the interpolation error formula to find a worst-case estimate for the error \(\left|e^{x}-Q_{5}(x)\right|\) that is valid for \(x\) throughout the interval \([-1,1] .\) How many digits after the decimal point will be correct when \(Q_{5}(x)\) is used to approximate \(e^{x}\) ?

(a) Find a polynomial \(P(x)\) of degree 3 or less whose graph passes through the points \((0,0),(1,1),(2,2),(3,7)\). (b) Find two other polynomials (of any degree) that pass through these four points. (c) Decide whether there exists a polynomial \(P(x)\) of degree 3 or less whose graph passes through the points \((0,0),(1,1),(2,2),(3,7)\), and \((4,2)\).

Find an upper bound for the error on \([0,2]\) when the degree 3 Chebyshev interpolating polynomial is used to approximate \(f(x)=\sin x\).

(a) Find a polynomial \(P(x)\) of degree 3 or less whose graph passes through the four data points \((-2,8),(0,4),(1,2),(3,-2)\). (b) Describe any other polynomials of degree 4 or less which pass through the four points in part (a).