Question

1. Consider the function $g(x) = \frac{1}{3}(x^2 + 1)$. a. By hand, find the two fixed points of $g(x)$. b. Sketch the graphs of $g(x)$ and $y = x$ on the same set of axes, confirming the fixed points. c. For one of the fixed points, find an interval $[a, b]$ such that $g([a, b]) \subset [a, b]$. d. Use the midpoint of the interval you found in c. as a starting point for a fixed point iteration. Do ten iterations (use a spreadsheet) for each starting point. Do the iterations seem to be converging? For the other fixed point, choose a starting point that's close (within 0.2), and do ten iterations. Do the iterations seem to be converging? e. Explain the results you found in d. Are the conditions of the contraction mapping theorem satisfied? 2. Consider the three functions below $g_1(x) = \sqrt{x^4 - 2}$ $g_2(x) = \sqrt[4]{x^2 + 2}$ $g_3(x) = \sqrt{\frac{2}{x^2} + 1}$ a. Verify that $\alpha = \sqrt{2}$ is a fixed point for each of the g functions. b. Explain why two of these fixed point iterations will converge and one will diverge, if $x_0$ is close to $\alpha$. For the two that converge, which converges faster? Why? c. Start with initial interval $1.2 < x < 1.6$. For the two that converge, determine how many iterations would be necessary to estimate $\alpha$ with error less than $< 10^{-8}$ with $x_0 = 1.5$. Hint: Use a graph to figure out $\lambda$.

          1. Consider the function $g(x) = \frac{1}{3}(x^2 + 1)$.
a. By hand, find the two fixed points of $g(x)$.
b. Sketch the graphs of $g(x)$ and $y = x$ on the same set of axes, confirming the fixed points.
c. For one of the fixed points, find an interval $[a, b]$ such that $g([a, b]) \subset [a, b]$.
d. Use the midpoint of the interval you found in c. as a starting point for a fixed point iteration. Do ten iterations (use a spreadsheet) for each starting point. Do the iterations seem to be converging? For the other fixed point, choose a starting point that's close (within 0.2), and do ten iterations. Do the iterations seem to be converging?
e. Explain the results you found in d. Are the conditions of the contraction mapping theorem satisfied?
2. Consider the three functions below
$g_1(x) = \sqrt{x^4 - 2}$
$g_2(x) = \sqrt[4]{x^2 + 2}$
$g_3(x) = \sqrt{\frac{2}{x^2} + 1}$
a. Verify that $\alpha = \sqrt{2}$ is a fixed point for each of the g functions.
b. Explain why two of these fixed point iterations will converge and one will diverge, if $x_0$ is close to $\alpha$.
For the two that converge, which converges faster? Why?
c. Start with initial interval $1.2 < x < 1.6$. For the two that converge, determine how many iterations would be necessary to estimate $\alpha$ with error less than $< 10^{-8}$ with $x_0 = 1.5$. Hint: Use a graph to figure out $\lambda$.

1. Consider the function g(x) = (1)/(3)(x^2 + 1).
a. By hand, find the two fixed points of g(x).
b. Sketch the graphs of g(x) and y = x on the same set of axes, confirming the fixed points.
c. For one of the fixed points, find an interval [a, b] such that g([a, b]) ⊂ [a, b].
d. Use the midpoint of the interval you found in c. as a starting point for a fixed point iteration. Do ten iterations (use a spreadsheet) for each starting point. Do the iterations seem to be converging? For the other fixed point, choose a starting point that's close (within 0.2), and do ten iterations. Do the iterations seem to be converging?
e. Explain the results you found in d. Are the conditions of the contraction mapping theorem satisfied?
2. Consider the three functions below
g1(x) = √(x^4 - 2)
g2(x) = √(x^2 + 2)
g3(x) = √((2)/(x^2) + 1)
a. Verify that α = √(2) is a fixed point for each of the g functions.
b. Explain why two of these fixed point iterations will converge and one will diverge, if x0 is close to α.
For the two that converge, which converges faster? Why?
c. Start with initial interval 1.2 < x < 1.6. For the two that converge, determine how many iterations would be necessary to estimate α with error less than < 10^-8 with x0 = 1.5. Hint: Use a graph to figure out λ.

Added by Stephen B.

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