Lachlan's question via email about the Bisection Method

In summary, the conversation discusses the use of the Bisection Method to find an approximate solution to an equation involving cosine and exponential functions. Four iterations were performed and a root of 4.68188 was accepted as the solution. It is noted that the Bisection Method is an iterative process and more iterations can lead to a more accurate solution. The CAS was also used to solve the equation and the solution was correct to two decimal places.
  • #1
Prove It
Gold Member
MHB
1,465
24
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.
 

Attachments

  • lbm1.jpg
    lbm1.jpg
    23 KB · Views: 85
  • lbm2.jpg
    lbm2.jpg
    23.4 KB · Views: 76
  • lbm3.jpg
    lbm3.jpg
    25.2 KB · Views: 77
Mathematics news on Phys.org
  • #2


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!
 

FAQ: Lachlan's question via email about the Bisection Method

What is the Bisection Method?

The Bisection Method is a numerical method used to find the root of a function. It involves repeatedly dividing an interval in half and checking which half contains the root until the root is approximated to a desired accuracy.

How does the Bisection Method work?

The Bisection Method works by first choosing an interval [a, b] where the function changes sign. Then, the midpoint of the interval is calculated and the function is evaluated at this point. Depending on the sign of the function at the midpoint, the interval is either halved on the left or right side. This process is repeated until the root is approximated within a desired accuracy.

What are the advantages of using the Bisection Method?

The Bisection Method is relatively simple to implement and does not require any knowledge of the derivative of the function. It is also guaranteed to converge to a root if the function is continuous and changes sign on the chosen interval.

What are the limitations of the Bisection Method?

The Bisection Method can be slow to converge compared to other numerical methods, especially for functions with multiple roots. It also requires the initial interval to be chosen carefully in order to converge to the desired root.

In what situations is the Bisection Method commonly used?

The Bisection Method is commonly used to find roots of polynomials or other functions where finding the exact solution is difficult or impossible. It is also useful for finding roots of functions that are not easily differentiable.

Similar threads

Replies
1
Views
10K
Replies
1
Views
10K
Replies
1
Views
10K
Replies
1
Views
10K
Replies
1
Views
9K
Replies
4
Views
10K
Replies
1
Views
9K
Replies
1
Views
10K
Replies
2
Views
10K
Back
Top