91Ó°ÊÓ

Find the parabolically terminated cubic spline through the data points \((0,1),(1,1),(2,1)\), \((3,1),(4,1)\). Is this spline also not-a-knot? natural?

Find \(P(0)\), where \(P(x)\) is the degree 10 polynomial that is zero at \(x=1, \ldots, 10\) and satisfies \(P(12)=44\).

Let \(T_{n}(x)\) denote the degree \(n\) Chebyshev polynomial. Find a formula for \(T_{n}(0)\).

Let \(P(x)\) be the degree 5 polynomial that takes the value 10 at \(x=1,2,3,4,5\) and the value 15 at \(x=6\). Find \(P(7)\).

Can a degree 3 polynomial intersect a degree 4 polynomial in exactly five points? Explain.

(a) Find \(b_{1}\) and \(c_{3}\) in the cubic spline $$ S(x)= \begin{cases}-1+b_{1} x-\frac{5}{9} x^{2}+\frac{5}{9} x^{3} & \text { on }[0,1] \\ \frac{14}{9}(x-1)+\frac{10}{9}(x-1)^{2}-\frac{2}{3}(x-1)^{3} & \text { on }[1,2] \\ 2+\frac{16}{9}(x-2)+c_{3}(x-2)^{2}-\frac{1}{9}(x-2)^{3} & \text { on }[2,3]\end{cases} $$ (b) Is this spline natural? (c) This spline satisfies "clamped" endpoint conditions. What are the values of the two clamps?

The estimated mean atmospheric concentration of carbon dioxide in earth's atmosphere is given in the table that follows, in parts per million by volume. Find the degree 3 interpolating polynomial of the data and use it to estimate the \(\mathrm{CO}_{2}\) concentration in (a) 1950 and (b) 2050 . (The actual concentration in 1950 was \(310 \mathrm{ppm}\).) $$ \begin{array}{||c|c||} \hline \text { year } & \mathrm{CO}_{2}(\mathrm{ppm}) \\ \hline 1800 & 280 \\ 1850 & 283 \\ 1900 & 291 \\ 2000 & 370 \\ \hline \end{array} $$

The case \(n=2\) for parabolically terminated cubic splines is not covered by Theorem \(3.8\). Discuss existence and uniqueness for the cubic spline in this case.

Discuss the existence and uniqueness of a not-a-knot cubic spline when \(n=2\) and \(n=3\).