91Ó°ÊÓ

Use the Bisection method to find solutions, accurate to within \(10^{-5}\) for the following problems. a. \(\quad 3 x-e^{x}=0\) for \(1 \leq x \leq 2\) b. \(\quad 2 x+3 \cos x-e^{x}=0 \quad\) for \(0 \leq x \leq 1\) c. \(\quad x^{2}-4 x+4-\ln x=0 \quad\) for \(1 \leq x \leq 2 \quad\) and \(\quad 2 \leq x \leq 4\) d. \(x+1-2 \sin \pi x=0 \quad\) for \(0 \leq x \leq 0.5 \quad\) and \(\quad 0.5 \leq x \leq 1\)

Use a fixed-point iteration method to determine a solution accurate to within \(10^{-2}\) for \(x^{3}-x-1=0\) on \([1,2]\). Use \(p_{0}=1\).

Show that the following sequences converge linearly to \(p=0\). How large must \(n\) be before \(\left|p_{n}-p\right| \leq\) \(5 \times 10^{-2} ?\) a. \(\quad p_{n}=\frac{1}{n}, \quad n \geq 1\) b. \(\quad p_{n}=\frac{1}{n^{2}}, \quad n \geq 1\)

a. Show that for any positive integer \(k\), the sequence defined by \(p_{n}=1 / n^{k}\) converges linearly to \(p=0\) b. For each pair of integers \(k\) and \(m\), determine a number \(N\) for which \(1 / N^{k}<10^{-m}\).

a. Sketch the graphs of \(y=x\) and \(y=\tan x\). b. Use the Bisection method to find an approximation to within \(10^{-5}\) to the first positive value of \(x\) with \(x=\tan x\).

Use each of the following methods to find a solution in \([0.1,1]\) accurate to within \(10^{-4}\) for $$ 600 x^{4}-550 x^{3}+200 x^{2}-20 x-1=0 $$ a. Bisection method c. Secant method e. Müller's method b. Newton's method d. method of False Position

a. Construct a sequence that converges to 0 of order 3 . b. Suppose \(\alpha>1\). Construct a sequence that converges to 0 zero of order \(\alpha\).

Suppose \(p\) is a zero of multiplicity \(m\) of \(f\), where \(f^{(m)}\) is continuous on an open interval containing \(p\). Show that the following fixed- point method has \(g^{\prime}(p)=0\) : $$ g(x)=x-\frac{m f(x)}{f^{\prime}(x)} $$

Let \(f(x)=(x+2)(x+1)^{2} x(x-1)^{3}(x-2)\). To which zero of \(f\) does the Bisection method converge when applied on the following intervals? a. \(\quad[-1.5,2.5]\) b. \(\quad[-0.5,2.4]\) c. \([-0.5,3]\) d. \([-3,-0.5]\)

Let \(f(x)=(x+2)(x+1) x(x-1)^{3}(x-2)\). To which zero of \(f\) does the Bisection method converge when applied on the following intervals? a. \(\quad[-3,2.5]\) b. \(\quad[-2.5,3]\) c. \(\quad[-1.75,1.5]\) d. \(\quad[-1.5,1.75]\)