current new approximation approximation function f. This step is repeated forn = 0, 1, 2, . . . , until a termination condition is met 2 he discussed). We have derived the general step of Newton's method for approximating roots of a Newton's Method for Approximating Roots of f(x) 0 FASCED URE Chouse an initial approximation x as close to a root as possible. 01, 2,. . f(x,) f' (x,) п provided f'(x) 0. 3. End the calculations when a termination condition is met. QUICK CHECK 1 Verify that setting y = 0 in the equation y - f(x, )= f'(x,)(x - xn) and solving for x gives the formula for Newton's method. Applying Newton's method Approximate the roots of f(x)= x-5x + 1 (Figure 4.83) using seven steps of Newton's method. Use 4 as initial approximations. EXAMPLE 1 3 X 1, and xo -3, xo SOLUTION Noting that f'(x) = 3x-5, Newton's method takes the form f (x,) .3 5x, 1 1 2x3 п n 3x2 -5 3x2- 5 'n+1 п n п f'(x,) H0 where n 0, 1, 2, ... , and x is specified. With an initial approximation of e first approximation is Хо 2(-3)3 1 3x-5 3(-3)2 -5 2x0 1 -2.5 e second approximation is 2x- 1 2(-2.5)3 1 -2.345455. 2 .2
current new approximation approximation function f. This step is repeated forn = 0, 1, 2, . . . , until a termination condition is met 2 he discussed). We have derived the general step of Newton's method for approximating roots of a Newton's Method for Approximating Roots of f(x) 0 FASCED URE Chouse an initial approximation x as close to a root as possible. 01, 2,. . f(x,) f' (x,) п provided f'(x) 0. 3. End the calculations when a termination condition is met. QUICK CHECK 1 Verify that setting y = 0 in the equation y - f(x, )= f'(x,)(x - xn) and solving for x gives the formula for Newton's method. Applying Newton's method Approximate the roots of f(x)= x-5x + 1 (Figure 4.83) using seven steps of Newton's method. Use 4 as initial approximations. EXAMPLE 1 3 X 1, and xo -3, xo SOLUTION Noting that f'(x) = 3x-5, Newton's method takes the form f (x,) .3 5x, 1 1 2x3 п n 3x2 -5 3x2- 5 'n+1 п n п f'(x,) H0 where n 0, 1, 2, ... , and x is specified. With an initial approximation of e first approximation is Хо 2(-3)3 1 3x-5 3(-3)2 -5 2x0 1 -2.5 e second approximation is 2x- 1 2(-2.5)3 1 -2.345455. 2 .2
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.2: Trigonometric Equations
Problem 96E
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Question
For example 1, how did they get 2X^3 - 1 in the numerator?
Transcribed Image Text:current
new
approximation approximation
function f. This step is repeated forn = 0, 1, 2, . . . , until a termination condition is met
2 he discussed).
We have derived the general step of Newton's method for approximating roots of a
Newton's Method for Approximating Roots of f(x) 0
FASCED URE
Chouse an initial approximation x as close to a root as possible.
01, 2,. .
f(x,)
f' (x,)
п
provided f'(x) 0.
3. End the calculations when a termination condition is met.
QUICK CHECK 1 Verify that setting y = 0 in the equation y - f(x, )= f'(x,)(x - xn) and
solving for x gives the formula for Newton's method.
Applying Newton's method Approximate the roots of
f(x)= x-5x + 1 (Figure 4.83) using seven steps of Newton's method. Use
4 as initial approximations.
EXAMPLE 1
3
X
1, and xo
-3, xo
SOLUTION Noting that f'(x) = 3x-5, Newton's method takes the form
f (x,)
.3
5x, 1
1
2x3
п
n
3x2 -5
3x2- 5
'n+1
п
n
п
f'(x,)
H0
where n 0, 1, 2, ... , and x is specified. With an initial approximation of
e first approximation is
Хо
2(-3)3 1
3x-5 3(-3)2 -5
2x0
1
-2.5
e second approximation is
2x- 1 2(-2.5)3 1
-2.345455.
2
.2
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