Start with an initial guess x = a₁. Then define a2 to be the x-intercept of the tangent of f(x) at a₁, which can be computed by the following equation f(a₁) - 0 a1a₂ f(x) at a₁ = fo(a₁) = slope of tangent of a2 = a1 f(a₁) f'(a₁). Repeat this process to get a sequence {an} satisfying the relation an+1 = an - f(an) f'(an). (?) The sequence {an} will usually converge to the root r, provided that the initial guess a₁ is close enough to r. Consider the root of e²x - x - 6 = 0. (a) Show that the above equation has at least one root in the interval (0,1). (b) To apply the Newton-Raphson method, define the function f(x) and write down the corresponding relation (?). 7 (c) Choosing the initial guess a₁ = 1, compute a2a3,a4 by the Newton-Raphson method (correct to 4 decimal places).
Start with an initial guess x = a₁. Then define a2 to be the x-intercept of the tangent of f(x) at a₁, which can be computed by the following equation f(a₁) - 0 a1a₂ f(x) at a₁ = fo(a₁) = slope of tangent of a2 = a1 f(a₁) f'(a₁). Repeat this process to get a sequence {an} satisfying the relation an+1 = an - f(an) f'(an). (?) The sequence {an} will usually converge to the root r, provided that the initial guess a₁ is close enough to r. Consider the root of e²x - x - 6 = 0. (a) Show that the above equation has at least one root in the interval (0,1). (b) To apply the Newton-Raphson method, define the function f(x) and write down the corresponding relation (?). 7 (c) Choosing the initial guess a₁ = 1, compute a2a3,a4 by the Newton-Raphson method (correct to 4 decimal places).
Chapter1: Equations, Inequalities, And Mathematical Modeling
Section1.1: Graphs Of Equations
Problem 9ECP
Related questions
Question
Transcribed Image Text:Start with an initial guess x = a1₁. Then define a2 to be the x-intercept of the tangent
of f(x) at a₁, which can be computed by the following equation
f(a₁) - 0
a₁a₂
f(x) at a₁ =
fo(a₁) = slope of tangent of
a2 = a1
f(a₁)
f'(a₁).
Repeat this process to get a sequence {an} satisfying the relation
an+1 = an -
f(an)
f'(an).
(?)
The sequence {an} will usually converge to the root r, provided that the initial guess
a₁ is close enough to r.
Consider the root of
e²x - x - 6 = 0.
(a) Show that the above equation has at least one root in the interval (0,1).
(b) To apply the Newton-Raphson method, define the function f(x) and write down
the corresponding relation (?).
7
(c) Choosing the initial guess a₁ = 1, compute a2a3,a4 by the Newton-Raphson
method (correct to 4 decimal places).
Expert Solution
Step 1: Relation and equation
Step by step
Solved in 4 steps with 4 images
Recommended textbooks for you
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning