# Question   APPLY FIXED-POINT ITERATION SIX TIMES WITH STARTING POINT $$X_{0}=10$$ TO FIND AN APPROXIMATED SOLUTION FOR $$x=20 \cdot \cos \left(\frac{x}{40}\right)$$ THIS WILL RESULT IN a sPIRALING CONVERGENCE. DOES THIS MEAN THAT THE ANSWER IS TOO SMALL OR TOO HIGH?

Transcribed Image Text: APPLY FIXED-POINT ITERATION SIX TIMES WITH STARTING POINT $$X_{0}=10$$ TO FIND AN APPROXIMATED SOLUTION FOR $$x=20 \cdot \cos \left(\frac{x}{40}\right)$$ THIS WILL RESULT IN a sPIRALING CONVERGENCE. DOES THIS MEAN THAT THE ANSWER IS TOO SMALL OR TOO HIGH?
Transcribed Image Text: APPLY FIXED-POINT ITERATION SIX TIMES WITH STARTING POINT $$X_{0}=10$$ TO FIND AN APPROXIMATED SOLUTION FOR $$x=20 \cdot \cos \left(\frac{x}{40}\right)$$ THIS WILL RESULT IN a sPIRALING CONVERGENCE. DOES THIS MEAN THAT THE ANSWER IS TOO SMALL OR TOO HIGH?
&#12304;General guidance&#12305;The answer provided below has been developed in a clear step by step manner.Step1/2Solution:Given: $$\mathrm{{f{{\left({x}\right)}}}={20}{\cos{{\left(\frac{{x}}{{40}}\right)}}}{\quad\text{and}\quad}{x}_{{0}}={10}}$$$$\mathrm{\Rightarrow\phi{\left({x}\right)}={20}{\cos{{\left(\frac{{x}}{{40}}\right)}}}}$$Using the Fixed Point Iteration Method,$$\mathrm{{x}_{{1}}=\phi{\left({x}_{{0}}\right)}=\phi{\left({10}\right)}={19.378}}$$$$\mathrm{{x}_{{2}}=\phi{\left({x}_{{1}}\right)}=\phi{\left({19.378}\right)}={17.699}}$$$$\mathrm{{x}_{{3}}=\phi{\left({x}_{{2}}\right)}=\phi{\left({17.699}\right)}={18.074}}$$\( \mathrm{{x}_{{4}}=\phi{\left({x}_{{3}}\ ... See the full answer