91Ó°ÊÓ

Use the Composite Trapezoidal rule with the indicated values of \(n\) to approximate the following integrals. a. \(\int_{1}^{2} x \ln x d x, \quad n=4\) b. \(\int_{-2}^{2} x^{3} e^{x} d x, \quad n=4\) c. \(\int_{0}^{2} \frac{2}{x^{2}+4} d x, \quad n=6\) d. \(\int_{0}^{\pi} x^{2} \cos x d x, \quad n=6\) e. \(\int_{0}^{2} e^{2 x} \sin 3 x d x, \quad n=8\) f. \(\int_{1}^{3} \frac{x}{x^{2}+4} d x, \quad n=8\) g. \(\int_{3}^{5} \frac{1}{\sqrt{x^{2}-4}} d x, \quad n=8\) h. \(\int_{0}^{3 \pi / 8} \tan x d x, \quad n=8\)

Approximate the following integrals using Gaussian quadrature with \(n=2\), and compare your results to the exact values of the integrals. a. \(\int_{1}^{1.5} x^{2} \ln x d x\) b. \(\int_{0}^{1} x^{2} e^{-x} d x\) c. \(\int_{0}^{0.35} \frac{2}{x^{2}-4} d x\) d. \(\int_{0}^{\pi / 4} x^{2} \sin x d x\) e. \(\int_{0}^{\pi / 4} e^{3 x} \sin 2 x d x\) f. \(\int_{1}^{1.6} \frac{2 x}{x^{2}-4} d x\) g. \(\int_{3}^{3.5} \frac{x}{\sqrt{x^{2}-4}} d x\) h. \(\int_{0}^{\pi / 4}(\cos x)^{2} d x\)

Use the forward-difference formulas and backward-difference formulas to determine each missing entry in the following tables. $$ \begin{aligned} &\text { a. }\\\ &\begin{array}{c|c|c} x & f(x) & f^{\prime}(x) \\ \hline 0.5 & 0.4794 & \\ 0.6 & 0.5646 & \\ 0.7 & 0.6442 & \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { b. }\\\ &\begin{array}{c|c|c} x & f(x) & f^{\prime}(x) \\ \hline 0.0 & 0.00000 & \\ 0.2 & 0.74140 & \\ 0.4 & 1.3718 & \end{array} \end{aligned} $$

Approximate the following integrals using the Trapezoidal rule. a. \(\int_{0.5}^{1} x^{4} d x\) b. \(\int_{0}^{0.5} \frac{2}{x-4} d x\) c. \(\int_{1}^{1.5} x^{2} \ln x d x\) d. \(\int_{0}^{1} x^{2} e^{-x} d x\) e. \(\int_{1}^{1.6} \frac{2 x}{x^{2}-4} d x\) f. \(\quad \int_{0}^{0.35} \frac{2}{x^{2}-4} d x\) g. \(\int_{0}^{\pi / 4} x \sin x d x\) h. \(\int_{0}^{\pi / 4} e^{3 x} \sin 2 x d x\)

Use Adaptive quadrature to approximate the following integrals to within \(10^{-5}\). a. \(\int_{1}^{3} e^{2 x} \sin 3 x d x\) b. \(\int_{1}^{3} e^{3 x} \sin 2 x d x\) c. \(\int_{0}^{5}\left(2 x \cos (2 x)-(x-2)^{2}\right) d x\) d. \(\quad \int_{0}^{5}\left(4 x \cos (2 x)-(x-2)^{2}\right) d x\)

Use Simpson's Composite rule with \(n=4,6,8, \ldots\), until successive approximations to the following integrals agree to within \(10^{-6}\). Determine the number of nodes required. Use the Adaptive Quadrature Algorithm to approximate the integral to within \(10^{-6}\), and count the number of nodes. Did Adaptive quadrature produce any improvement? a. \(\int_{0}^{x} x \cos x^{2} d x\) b. \(\int_{0}^{\pi} x \sin x^{2} d x\) c. \(\int_{0}^{\pi} x^{2} \cos x d x\) d. \(\int_{0}^{\pi} x^{2} \sin x d x\)

Use the most accurate three-point formula to determine each missing entry in the following tables. $$ \begin{aligned} &\mathrm{a} .\\\ &\begin{array}{c|c|c} x & f(x) & f^{\prime}(x) \\ \hline-0.3 & -0.27652 & \\ -0.2 & -0.25074 & \\ -0.1 & -0.16134 & \\ 0 & 0 & \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { b. }\\\ &\begin{array}{l|c|c} x & f(x) & f^{\prime}(x) \\ \hline 7.4 & -68.3193 & \\ 7.6 & -71.6982 & \\ 7.8 & -75.1576 & \\ 8.0 & -78.6974 & \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { c. }\\\ &\begin{array}{c|c|c} x & f(x) & f^{\prime}(x) \\ \hline 1.1 & 1.52918 & \\ 1.2 & 1.64024 & \\ 1.3 & 1.70470 & \\ 1.4 & 1.71277 & \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { d. }\\\ &\begin{array}{c|c|c} x & f(x) & f^{\prime}(x) \\ \hline-2.7 & 0.054797 & \\ -2.5 & 0.11342 & \\ -2.3 & 0.65536 & \\ -2.1 & 0.98472 & \end{array} \end{aligned} $$

Sketch the graphs of \(\sin (1 / x)\) and \(\cos (1 / x)\) on \([0.1,2]\). Use Adaptive quadrature to approximate the following integrals to within \(10^{-3}\). a. \(\int_{0.1}^{2} \sin \frac{1}{x} d x\) b. \(\int_{0.1}^{2} \cos \frac{1}{x} d x\)

a. Show that $$ \lim _{h \rightarrow 0}\left(\frac{2+h}{2-h}\right)^{1 / h}=e $$ b. Compute approximations to \(e\) using the formula \(N(h)=\left(\frac{2+h}{2-h}\right)^{1 / h}\), for \(h=0.04,0.02\), and \(0.01\). c. Assume that \(e=N(h)+K_{1} h+K_{2} h^{2}+K_{3} h^{3}+\cdots .\) Use extrapolation, with at least 16 digits of precision, to compute an \(O\left(h^{3}\right)\) approximation to \(e\) with \(h=0.04\). Do you think the assumption is correct? d. Show that \(N(-h)=N(h)\). e. Use part (d) to show that \(K_{1}=K_{3}=K_{5}=\cdots=0\) in the formula $$ e=N(h)+K_{1} h+K_{2} h^{2}+K_{3} h^{3} K_{4} h^{4}+K_{5} h^{5}+\cdots $$ so that the formula reduces to $$ e=N(h)+K_{2} h^{2}+K_{4} h^{4}+K_{6} h^{6}+\cdots $$ f. Use the results of part (e) and extrapolation to compute an \(O\left(h^{6}\right)\) approximation to \(e\) with \(h=0.04\).

Determine the values of \(n\) and \(h\) required to approximate $$ \int_{0}^{2} \frac{1}{x+4} d x $$ to within \(10^{-5}\) and compute the approximation. Use a. Composite Trapezoidal rule. b. Composite Simpson's rule. c. Composite Midpoint rule.