91Ó°ÊÓ

Find the approximations to within \(10^{-4}\) to all the real zeros of the following polynomials using Newton's method. a. \(\quad f(x)=x^{3}-2 x^{2}-5\) b. \(\quad f(x)=x^{3}+3 x^{2}-1\) c. \(\quad f(x)=x^{3}-x-1\) d. \(\quad f(x)=x^{4}+2 x^{2}-x-3\) e. \(\quad f(x)=x^{3}+4.001 x^{2}+4.002 x+1.101\) f. \(f(x)=x^{5}-x^{4}+2 x^{3}-3 x^{2}+x-4\)

Use algebraic manipulation to show that each of the following functions has a fixed point at \(p\) precisely when \(f(p)=0\), where \(f(x)=x^{4}+2 x^{2}-x-3\). a. \(\quad g_{1}(x)=\left(3+x-2 x^{2}\right)^{1 / 4}\) b. \(\quad g_{2}(x)=\left(\frac{x+3-x^{4}}{2}\right)^{1 / 2}\) c. \(\quad g_{3}(x)=\left(\frac{x+3}{x^{2}+2}\right)^{1 / 2}\) d. \(\quad g_{4}(x)=\frac{3 x^{4}+2 x^{2}+3}{4 x^{3}+4 x-1}\)

Use Newton's method to find solutions accurate to within \(10^{-5}\) to the following problems. a. \(\quad x^{2}-2 x e^{-x}+e^{-2 x}=0, \quad\) for \(0 \leq x \leq 1\) b. \(\quad \cos (x+\sqrt{2})+x(x / 2+\sqrt{2})=0, \quad\) for \(-2 \leq x \leq-1\) c. \(\quad x^{3}-3 x^{2}\left(2^{-x}\right)+3 x\left(4^{-x}\right)-8^{-x}=0, \quad\) for \(0 \leq x \leq 1\) d. \(\quad e^{6 x}+3(\ln 2)^{2} e^{2 x}-(\ln 8) e^{4 x}-(\ln 2)^{3}=0, \quad\) for \(-1 \leq x \leq 0\)

Consider the function \(f(x)=e^{6 x}+3(\ln 2)^{2} e^{2 x}-(\ln 8) e^{4 x}-(\ln 2)^{3}\). Use Newton's method with \(p_{0}=0\) to approximate a zero of \(f\). Generate terms until \(\left|p_{n+1}-p_{n}\right|<0.0002\). Construct the sequence \(\left\\{\hat{p}_{n}\right\\}\). Is the convergence improved?

Find approximations to within \(10^{-5}\) to all the zeros of each of the following polynomials by first finding the real zeros using Newton's method and then reducing to polynomials of lower degree to determine any complex zeros. a. \(f(x)=x^{4}+5 x^{3}-9 x^{2}-85 x-136\) b. \(\quad f(x)=x^{4}-2 x^{3}-12 x^{2}+16 x-40\) c. \(\quad f(x)=x^{4}+x^{3}+3 x^{2}+2 x+2\) d. \(\quad f(x)=x^{5}+11 x^{4}-21 x^{3}-10 x^{2}-21 x-5\) e. \(f(x)=16 x^{4}+88 x^{3}+159 x^{2}+76 x-240\) f. \(\quad f(x)=x^{4}-4 x^{2}-3 x+5\) g. \(\quad f(x)=x^{4}-2 x^{3}-4 x^{2}+4 x+4\) h. \(f(x)=x^{3}-7 x^{2}+14 x-6\)

Use the Bisection method to find solutions accurate to within \(10^{-2}\) for \(x^{3}-7 x^{2}+14 x-6=0\) on each interval. a. \([0,1]\) b. \(\quad[1,3.2]\) c. \([3.2,4]\)

Use the Bisection method to find solutions accurate to within \(10^{-2}\) for \(x^{4}-2 x^{3}-4 x^{2}+4 x+4=0\) on each interval. a. \([-2,-1]\) b. \(\quad[0,2]\) c. \([2,3]\) d. \(\quad[-1,0]\)

The following four methods are proposed to compute \(7^{1 / 5}\). Rank them in order, based on their apparent speed of convergence, assuming \(p_{0}=1\). a. \(\quad p_{n}=p_{n-1}\left(1+\frac{7-p_{n-1}^{5}}{p_{n-1}^{2}}\right)^{3}\) b. \(\quad p_{n}=p_{n-1}-\frac{p_{n-1}^{5}-7}{p_{n-1}^{2}}\) c. \(\quad p_{n}=p_{n-1}-\frac{p_{n-1}^{5}-7}{5 p_{n-1}^{4}}\) d. \(\quad p_{n}=p_{n-1}-\frac{p_{n-1}^{5}-7}{12}\)

Use the Bisection method to find solutions accurate to within \(10^{-5}\) for the following problems. a. \(\quad x-2^{-x}=0 \quad\) for \(0 \leq x \leq 1\) b. \(\quad e^{x}-x^{2}+3 x-2=0 \quad\) for \(0 \leq x \leq 1\) c. \(\quad 2 x \cos (2 x)-(x+1)^{2}=0 \quad\) for \(-3 \leq x \leq-2 \quad\) and \(\quad-1 \leq x \leq 0\) d. \(x \cos x-2 x^{2}+3 x-1=0 \quad\) for \(0.2 \leq x \leq 0.3 \quad\) and \(\quad 1.2 \leq x \leq 1.3\)

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 \(p_{0}=1\).