91Ó°ÊÓ

Use the Lagrange interpolating polynomial of degree three or less and four- digit chopping arithmetic to approximate \(\cos 0.750\) using the following values. Find an error bound for the approximation. $$ \cos 0.698=0.7661 \quad \cos 0.733=0.7432 \quad \cos 0.768=0.7193 \quad \cos 0.803=0.6946 $$ The actual value of \(\cos 0.750\) is \(0.7317\) (to four decimal places). Explain the discrepancy between the actual error and the error bound.

Construct the Lagrange interpolating polynomials for the following functions, and find a bound for the absolute error on the interval \(\left[x_{0}, x_{n}\right]\). a. \(f(x)=e^{2 x} \cos 3 x, \quad x_{0}=0, x_{1}=0.3, x_{2}=0.6, n=2\) b. \(\quad f(x)=\sin (\ln x), \quad x_{0}=2.0, x_{1}=2.4, x_{2}=2.6, n=2\) c. \(f(x)=\ln x, \quad x_{0}=1, x_{1}=1.1, x_{2}=1.3, x_{3}=1.4, n=3\) d. \(\quad f(x)=\cos x+\sin x, \quad x_{0}=0, x_{1}=0.25, x_{2}=0.5, x_{3}=1.0, n=3\)

Suppose you need to construct eight-decimal-place tables for the common, or base-10, logarithm function from \(x=1\) to \(x=10\) in such a way that linear interpolation is accurate to within \(10^{-6}\). Determine a bound for the step size for this table. What choice of step size would you make to ensure that \(r=10\) is included in the tahle?

Suppose that \(f(x)\) is a polynomial of degree 3 . Show that \(f(x)\) is its own clamped cubic spline, but that it cannot be its own natural cubic spline.

It is suspected that the high amounts of tannin in mature oak leaves inhibit the growth of the winter moth (Operophtera bromata L., Geometridae) larvae that extensively damage these trees in certain years. The following table lists the average weight of two samples of larvae at times in the first 28 days after birth. The first sample was reared on young oak leaves, whereas the second sample was reared on mature leaves from the same tree. a. Use Lagrange interpolation to approximate the average weight curve for each sample. b. Find an approximate maximum average weight for each sample by determining the maximum of the interpolating polynomial. $$ \begin{array}{l|c|c|c|c|c|c|c} \text { Day } & 0 & 6 & 10 & 13 & 17 & 20 & 28 \\ \hline \text { Sample 1 average weight (mg) } & 6.67 & 17.33 & 42.67 & 37.33 & 30.10 & 29.31 & 28.74 \\ \text { Sample 2 average weight }(\mathrm{mg}) & 6.67 & 16.11 & 18.89 & 15.00 & 10.56 & 9.44 & 8.89 \end{array} $$

It is suspected that the high amounts of tannin in mature oak leaves inhibit the growth of the winter moth (Operophtera bromata \(L\), Geometridae) larvae that extensively damage these trees in certain years. The following table lists the average weight of two samples of larvae at times in the first 28 days after birth. The first sample was reared on young oak leaves, whereas the second sample was reared on mature leaves from the same tree. a. Use a natural cubic spline to approximate the average weight curve for each sample. b. Find an approximate maximum average weight for each sample by determining the maximum of the spline. \begin{tabular}{l|c|c|c|c|c|c|c} Day & 0 & 6 & 10 & 13 & 17 & 20 & 28 \\ \hline Sample 1 average weight \((\mathrm{mg})\) & \(6.67\) & \(17.33\) & \(42.67\) & \(37.33\) & \(30.10\) & \(29.31\) & \(28.74\) \\ \hline Sample 2 average weight \((\mathrm{mg})\) & \(6.67\) & \(16.11\) & \(18.89\) & \(15.00\) & \(10.56\) & \(9.44\) & \(8.89\) \end{tabular}

The upper portion of this noble beast is to be approximated using clamped cubic spline interpolants. The curve is drawn on a grid from which the table is constructed. Use Algorithm \(3.5\) to construct the three clamped cubic splines. \begin{aligned} &\begin{array}{cccccccccccc} & \multicolumn{3}{c}{\text { Curve } 1} & \multicolumn{5}{c}{\text { Curve 2 }} & \multicolumn{5}{c}{\text { Curve 3 }} \\ \hline i & x_{i} & f\left(x_{i}\right) & f^{\prime}\left(x_{i}\right) & i & x_{i} & f\left(x_{i}\right) & f^{\prime}\left(x_{i}\right) & i & x_{i} & f\left(x_{i}\right) & f^{\prime}\left(x_{i}\right) \\ \hline 0 & 1 & 3.0 & 1.0 & 0 & 17 & 4.5 & 3.0 & 0 & 27.7 & 4.1 & 0.33 \\ 1 & 2 & 3.7 & & 1 & 20 & 7.0 & & 1 & 28 & 4.3 & \\ 2 & 5 & 3.9 & & 2 & 23 & 6.1 & & 2 & 29 & 4.1 & \\ 3 & 6 & 4.2 & & 3 & 24 & 5.6 & & 3 & 30 & 3.0 & -1.5 \\ 4 & 7 & 5.7 & & 4 & 25 & 5.8 & & & & & \\ 5 & 8 & 6.6 & & 5 & 27 & 5.2 & & & & & \\ 6 & 10 & 7.1 & & 6 & 27.7 & 4.1 & -4.0 & & & & \\ 7 & 13 & 6.7 & & & & & & & & & \\ 8 & 17 & 4.5 & -0.67 & & & & & & & & \\ \hline \end{array}\\\ &\text { Curve } 3 \end{aligned}