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.
Consider the function g(x) = x^2 + 3/16(a) This function has two fixed points, what are they?(b) Consider the fixed point iteration xk+1 = g(xk) for this g. For which of the points you found in(a) can you be sure that the iterations will converge to that fixed point? Justify your answer.(c) For the point(s) you found in (b), roughly how many iterations will be required to reduce theconvergence error by a factor of 10?
A bank teller series customers standing in the queue one by
one. Suppose that the service time X, for customer i has
теаn E(X)
= 2 (minutes) and V(X) = 1. We assume that
service times for different bank customers are independent.
Let Y be the total time the bank teller spends services 50
customers. Find P(90 < Y< 110).
4. Consider the function, f(x) = x³ = x² - 9x +9. Answer the following:
State the exact roots of f(x).
(a)
(b)
Construct three different fixed point functions g(x) such that f(x) = 0. (Make sure that one of the
g(x)'s that you constructed converges to at least a root).
(c)
(d)
Find the approximate root, x, of the above function using fixed point iterations up to 4 significant
figures within the error bound of 1 × 10-³ using xo = 0 and any fixed point function g(x) from part(b) that
converges to the root(s).
Find the convergence rate/ratio for g(x) constructed in previous part and also find which root it is
converging to?
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