Exercise · ZW-SECANT-METHOD-006 · ★★☆ Intermediate
With \(f(x)=x^2-4\), \(x_0=1\) and \(x_1=3\), what is \(x_2\) under the secant method?
Choose one option.
Show answer
- D. 1.75
Explanation
We have \(f(3)=5\) and \(f(1)=-3\), so \(x_2=3-5\cdot\frac{3-1}{5-(-3)}=3-\frac{10}{8}=1.75\). Adding the term instead of subtracting it gives 2.75; the midpoint gives 2.
Review the concept: Secant method
The secant method is an iterative procedure for approximating roots of a function \(f(x)=0\) using two previous approximations, without needing to compute the derivative.
\[x_{n+1}=x_n-f(x_n)\frac{x_n-x_{n-1}}{f(x_n)-f(x_{n-1})}\]
More practice problems: secant method
- ZW-SECANT-METHOD-001 Which method replaces Newton-Raphson's derivative with a difference quotient?
- ZW-SECANT-METHOD-002 How many starting values does the secant method need?
- ZW-SECANT-METHOD-003 What geometric object defines the secant step?
- ZW-SECANT-METHOD-004 What happens if f(x_n)=f(x_n-1) in the formula?
- ZW-SECANT-METHOD-005 Approximately what order of convergence does the secant method have?
- ZW-SECANT-METHOD-007 If f(x)=x^3-x-2 with x_0=1 and x_1=2, what is f(x_1)-f(x_0)?
- ZW-SECANT-METHOD-008 For f(x)=x^2-2 with x_0=1 and x_1=2, what is x_2?
- ZW-SECANT-METHOD-009 Which initial condition should be checked before iterating with the secant method?