MHB Hayldiburasomas' question via email about Secant Method

AI Thread Summary
The discussion focuses on using the Secant Method to approximate a solution for the equation sin(1.8x) = (1/2)x^2 - 10, with initial estimates x0 = 4.43 and x1 = 4.63. The equation is reformulated as f(x) = (1/2)x^2 - 10 - sin(1.8x) to apply the method. The Secant Method formula is utilized to perform three iterations, leading to an approximate solution of x4 = 4.66053. The results from the calculations align with those obtained from a calculator. The discussion highlights the effectiveness of the Secant Method in finding numerical solutions.
Prove It
Gold Member
MHB
Messages
1,434
Reaction score
20
Use three iterations of the Secant Method to find an approximate solution of the equation

$\displaystyle \sin{\left( 1.8\,x \right) } =\frac{1}{2}\,x^2 - 10 $

if your initial estimates are $\displaystyle x_0 = 4.43 $ and $\displaystyle x_1 = 4.63 $.

The Secant Method is a numerical scheme to solve equations of the form $\displaystyle f\left( x \right) = 0 $, so we must rewrite the equation as $\displaystyle 0 = \frac{1}{2}\,x^2 - 10 - \sin{ \left( 1.8\,x \right) } $.

Thus $\displaystyle f\left( x \right) = \frac{1}{2}\,x^2 - 10 - \sin{ \left( 1.8\,x \right) } $.

The Secant Method is $\displaystyle x_{n+1} = x_n - f\left( x_n \right) \left[ \frac{x_n - x_{n-1}}{f\left( x_n \right) - f\left( x_{n-1}\right) } \right] $.

I have used my CAS to solve this problem.

View attachment 9651

View attachment 9652

So after three iterations your solution is approximately $\displaystyle x_4 = 4.66053 $.

I also included the calculator's answer, which matches.
 

Attachments

  • sm1.jpg
    sm1.jpg
    27.3 KB · Views: 140
  • sm2.jpg
    sm2.jpg
    27.2 KB · Views: 118
Mathematics news on Phys.org
Thanks for the help and support as usual Hayden!
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Back
Top