New ton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x 0 and successively defines a sequence z n + 1 = x n − f ( x n ) f ' ( x n ) . For the given choice of f and x 0 . write out the formula for x n + 1 . If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that x n agrees with x up to four decimal places.
New ton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x 0 and successively defines a sequence z n + 1 = x n − f ( x n ) f ' ( x n ) . For the given choice of f and x 0 . write out the formula for x n + 1 . If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that x n agrees with x up to four decimal places.
New ton’s method seeks to approximate a solution f(x) = 0 that starts with an initial approximation x0and successively defines a sequence
z
n
+
1
=
x
n
−
f
(
x
n
)
f
'
(
x
n
)
. For the given choice of f and x0. write out the formula for
x
n
+
1
. If the sequence appeals to converge, give an exact formula for the solution x. then identify the limit x accurate to four decimal places and the smallest ii such that xnagrees with x up to four decimal places.
5. Consider the equation f(x) = x – 2- = 0.
(a) Show that the above equation has a solution on the interval [0, 1].
(b) Prove that the sequence rk+1 = 2¬** converges to the solution.
Let f(x) = >r". For what values of r does the expression
n=0
converge? For these values of r, write f(x) in the form of an elementary
function.
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).
Using & Understanding Mathematics: A Quantitative Reasoning Approach (7th Edition)
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