Suppose f has a real root r and Newton's method is used to approximate r with an initial approximation xo. The basin of attraction of r is the set of initial approximations that produce a sequence that converges to r. Points near r are often in the basin of attraction to r-but not always. Sometimes an initial approximation x may produce a sequence that doesn't converge, and sometimes an initial approximation x may produce a sequence that converges to a distant root. Let f(x)=(x+3)(x-2.5)(x-3.5), which has roots x = -3, 2.5, and 3.5. Use Newton's method with initial approximations on the interval [-4,4] and determine (approximately) the basin of each root.
Suppose f has a real root r and Newton's method is used to approximate r with an initial approximation xo. The basin of attraction of r is the set of initial approximations that produce a sequence that converges to r. Points near r are often in the basin of attraction to r-but not always. Sometimes an initial approximation x may produce a sequence that doesn't converge, and sometimes an initial approximation x may produce a sequence that converges to a distant root. Let f(x)=(x+3)(x-2.5)(x-3.5), which has roots x = -3, 2.5, and 3.5. Use Newton's method with initial approximations on the interval [-4,4] and determine (approximately) the basin of each root.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 9T
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![Suppose f has a real root r and Newton's method is used to approximate r with an initial approximation xo. The basin of attraction of r is the set of initial
approximations that produce a sequence that converges to r. Points near r are often in the basin of attraction to r-but not always. Sometimes an initial
approximation x may produce a sequence that doesn't converge, and sometimes an initial approximation x may produce a sequence that converges to a distant
root. Let f(x)=(x+3)(x-2.5)(x-3.5), which has roots x = -3, 2.5, and 3.5. Use Newton's method with initial approximations on the interval [-4,4] and
determine (approximately) the basin of each root.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5dc918a1-00e0-4845-bd69-bff677bf3d9f%2F1840a5b5-78bc-4508-82f7-dfd4ac2449a2%2Frkifteq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose f has a real root r and Newton's method is used to approximate r with an initial approximation xo. The basin of attraction of r is the set of initial
approximations that produce a sequence that converges to r. Points near r are often in the basin of attraction to r-but not always. Sometimes an initial
approximation x may produce a sequence that doesn't converge, and sometimes an initial approximation x may produce a sequence that converges to a distant
root. Let f(x)=(x+3)(x-2.5)(x-3.5), which has roots x = -3, 2.5, and 3.5. Use Newton's method with initial approximations on the interval [-4,4] and
determine (approximately) the basin of each root.
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