Lachlan's question via email about the Bisection Method

[Prove It]

Perform four iterations of the Bisection Method to find an approximate solution to the equation

$\displaystyle 11\cos{ \left( x \right) } = 1 - 2\,\mathrm{e}^{-x/10} $

when it is known there is a solution in the interval $\displaystyle x \in \left[ 4.65, 4.82 \right] $

The Bisection Method solves equations of the form $\displaystyle f\left( x \right) = 0 $ so we must write the equation as $\displaystyle 11\cos{ \left( x \right) } - 1 + 2\,\mathrm{e}^{-x/10} = 0 $. We can then see that $\displaystyle f\left( x \right) = 11\cos{ \left( x \right) } - 1 + 2\,\mathrm{e}^{-x/10} $.

I have used my CAS to solve this problem.

View attachment 9654

View attachment 9655

View attachment 9656

After four iterations, we accept $\displaystyle c_4 = 4.68188 $ as the root.

I also told the CAS to solve the equation, and this solution is correct to two decimal places, which is very reasonable considering how slowly the Bisection Method converges.

 

[jvicens]

Great work on using the Bisection Method to find an approximate solution to this equation! It is important to note that the Bisection Method is an iterative process, so the more iterations you perform, the closer you will get to the exact solution. In this case, it seems like four iterations was enough to get a reasonable approximation. However, if you wanted a more accurate solution, you could continue performing more iterations until you reach a desired level of precision. Keep up the good work!