91Ó°ÊÓ

Apply Romberg Integration to find \(R_{33}\) for the integrals \(\begin{array}{llll}\text { (a) } \int_{0}^{1} x^{2} d x & \text { (b) } \int_{0}^{x / 2} \cos x d x & \text { (c) } \int_{0}^{1} e^{x} d x\end{array}\)

Apply Adaptive Quadrature by hand, using the Trapezoid Rule with tolerance \(\mathrm{TOL}=0.05\) to approximate the integrals. Find the approximation error. (a) \(\int_{0}^{1} x^{2} d x\) (b) \(\int_{0}^{x / 2} \cos x d x\) (c) \(\int_{0}^{1} e^{x} d x\)

Use the two-point forward-difference formula to approximate \(f^{\prime}(1)\), and find the approximation error, where \(f(x)=\ln x\), for (a) \(h=0.1\) (b) \(h=0.01\) (c) \(h=0.001\).

Use the two-point forward-difference formula to approximate \(f^{\prime}(\pi / 3)\), where \(f(x)=\sin x\), and find the approximation error. Also, find the bounds implied by the error term and show that the approximation error lies between them (a) \(h=0.1\) (b) \(h=0.01\) (c) \(h=0.001 .\)

Apply the composite Simpson's Rule with \(m=1,2\), and 4 panels to the integrals, and report the errors. (a) \(\int_{0}^{1} x e^{x} d x\) (b) \(\int_{0}^{1} \frac{d x}{1+x^{2}} d x\) (c) \(\int_{0}^{\pi} x \cos x d x\)

Use the three-point centered-difference formula for the second derivative to approximate \(f^{\prime \prime}(1)\), where \(f(x)=x^{-1}\), for (a) \(h=0.1\) (b) \(h=0.01\) (c) \(h=0.001\). Find the approximation error.

Use the three-point centered-difference formula for the second derivative to approximate \(f^{\prime \prime}(0)\), where \(f(x)=\cos x\), for (a) \(h=0.1\) (b) \(h=0.01\) (c) \(h=0.001\). Find the approximation error.

Develop a formula for a two-point backward-difference formula for approximating \(f^{\prime}(x)\), including error term.

Find the Legendre polynomials up to degree 3 and compare with Example \(5.13\).

Find a second-order formula for approximating \(f^{\prime}(x)\) by applying extrapolation to the two-point forward-difference formula.