Exercise · ZW-FIXED-POINT-ITERATION-008 · ★★☆ Intermediate
For \( g(x) = \sqrt{x + 6} \), what is the positive fixed point?
Choose one option.
Show answer
- A. \( 3 \)
Explanation
Solving \( x = \sqrt{x+6} \) gives \( x^2 = x + 6 \), or \( (x-3)(x+2) = 0 \). The positive root is \( 3 \); the other, \( -2 \), fails because \( \sqrt{x+6} \geq 0 \).
Review the concept: Fixed-point iteration
Iterative method that approximates a solution of the equation \( x = g(x) \) by generating the sequence \( x_{n+1} = g(x_n) \) from an initial value \( x_0 \).
\[x_{n+1} = g(x_n), \quad |g'(p)| < 1 \Rightarrow \text{convergencia local}\]
More practice problems: fixed-point iteration
- ZW-FIXED-POINT-ITERATION-001 What is the general formula of fixed-point iteration?
- ZW-FIXED-POINT-ITERATION-002 What condition defines a fixed point p of g ?
- ZW-FIXED-POINT-ITERATION-003 If |g'(p)| < 1 , what type of convergence is expected near p ?
- ZW-FIXED-POINT-ITERATION-004 What does the fixed-point theorem state about the existence of a fixed point?
- ZW-FIXED-POINT-ITERATION-005 What is x_1 if g(x) = cos (x) and x_0 = 0 ?
- ZW-FIXED-POINT-ITERATION-006 If g(x) = (x^2 + 2)/(3) and x_0 = 1 , what is x_2 ?
- ZW-FIXED-POINT-ITERATION-007 What is the linear convergence constant when g'(p) = -0.5 ?
- ZW-FIXED-POINT-ITERATION-009 If g(x) = (x)/(2) + 1 , what is the limit of x_n starting from any x_0 ?
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