91Ó°ÊÓ

For each of the following equations, determine an interval \([a, b]\) on which fixed-point iteration will converge. Estimate the number of iterations necessary to obtain approximations accurate to within \(10^{-5}\), and perform the calculations. a. \(\quad x=\frac{2-e^{x}+x^{2}}{3}\) b. \(\quad x=\frac{5}{x^{2}}+2\) c. \(\quad x=\left(e^{x} / 3\right)^{1 / 2}\) d. \(\quad x=5^{-x}\) e. \(\quad x=6^{-x}\) f. \(\quad x=0.5(\sin x+\cos x)\)

For each of the following equations, use the given interval or determine an interval \([a, b]\) on which fixed-point iteration will converge. Estimate the number of iterations necessary to obtain approximations accurate to within \(10^{-5}\), and perform the calculations. a. \(\quad 2+\sin x-x=0 \quad\) use \([2,3]\) b. \(\quad x^{3}-2 x-5=0 \quad\) use \([2,3]\) c. \(\quad 3 x^{2}-e^{x}=0\) d. \(\quad x-\cos x=0\)

Find all the zeros of \(f(x)=x^{2}+10 \cos x\) by using the fixed-point iteration method for an appropriate iteration function \(g .\) Find the zeros accurate to within \(10^{-4}\).

Use Newton's method to approximate, to within \(10^{-4}\), the value of \(x\) that produces the point on the graph of \(y=x^{2}\) that is closest to \((1,0) .\left[\right.\) Hint \(:\) Minimize \([d(x)]^{2}\), where \(d(x)\) represents the distance from \(\left(x, x^{2}\right)\) to \(\left.(1,0) .\right]\)

Use a fixed-point iteration method to determine a solution accurate to within \(10^{-4}\) for \(x=\tan x\), for \(x\) in \([4,5]\).

A sequence \(\left\\{p_{n}\right\\}\) is said to be superlinearly convergent to \(p\) if $$ \lim _{n \rightarrow \infty} \frac{\left|p_{n+1}-p\right|}{\left|p_{n}-p\right|}=0 $$ a. Show that if \(p_{n} \rightarrow p\) of order \(\alpha\) for \(\alpha>1\), then \(\left\\{p_{n}\right\\}\) is superlinearly convergent to \(p\). b. Show that \(p_{n}=\frac{1}{n^{n}}\) is superlinearly convergent to 0 but does not converge to 0 of order \(\alpha\) for any \(\alpha>1\)

Use Newton's method to approximate, to within \(10^{-4}\), the value of \(x\) that produces the point on the graph of \(y=1 / x\) that is closest to \((2,1)\).

Let \(A\) be a given positive constant and \(g(x)=2 x-A x^{2}\). a. Show that if fixed-point iteration converges to a nonzero limit, then the limit is \(p=1 / A\), so the inverse of a number can be found using only multiplications and subtractions. b. Find an interval about \(1 / A\) for which fixed-point iteration converges, provided \(p_{0}\) is in that interval.

A trough of length \(L\) has a cross section in the shape of a semicircle with radius \(r\). (See the accompanying figure.) When filled with water to within a distance \(h\) of the top, the volume \(V\) of water is $$ V=L\left[0.5 \pi r^{2}-r^{2} \arcsin (h / r)-h\left(r^{2}-h^{2}\right)^{1 / 2}\right] $$ Suppose \(L=10 \mathrm{ft}, r=1 \mathrm{ft}\), and \(V=12.4 \mathrm{ft}^{3}\). Find the depth of water in the trough to within \(0.01 \mathrm{ft}\).

A particle starts at rest on a smooth inclined plane whose angle \(\theta\) is changing at a constant rate $$ \frac{d \theta}{d t}=\omega<0 $$ At the end of \(t\) seconds, the position of the object is given by $$ x(t)=-\frac{g}{2 \omega^{2}}\left(\frac{e^{w t}-e^{-w t}}{2}-\sin \omega t\right) $$ Suppose the particle has moved \(1.7 \mathrm{ft}\) in \(1 \mathrm{~s}\). Find, to within \(10^{-5}\), the rate \(\omega\) at which \(\theta\) changes. Assume that \(g=32.17 \mathrm{ft} / \mathrm{s}^{2}\).