Question

let ?(?)=(?+2)(?+1)?(?−1)3(?−2)f(x)=(x+2)(x+1)x(x−1)3(x−2).

To which zero of ?f does the *Bisection method* converges
when applied on the interval [−3,2.5]

Answer #1

1- Let the bisection method be applied to a continuous function,
resulting in the intervals[a0,b0],[a1,b1], and so on. Letcn=an+bn2,
and let r=lim n→∞cn be the corresponding root. Let en=r−c
a. 1-1) Show that|en|≤2−n−1(b0−a0).
b. Show that|cn−cn+1|=2−n−2(b0−a0).
c Show that it is NOT necessarily true that|e0|≥|e1|≥···by
considering the function f(x) =x−0.2on the interval[−1,1].

Let f(x) = x^3 + x - 4
a. Show that f(x) has a root on the interval [1,4]
b. Find the first three iterations of the bisection method on f
on this interval
c. Find a bound for the number of iterations needed of bisection
to approximate the root to within 10^-4

find solution bisection method
x^2-5x+2 limit 3%

Q1: Use bisection method to ﬁnd solution accurate to within
10^−4 on the interval [0, 1] of the function f(x) = x−2^−x
Q3: Find Newton’s formula for f(x) = x^(3) −3x + 1 in [1,3] to
calculate x5, if x0 = 1.5. Also, ﬁnd the rate of convergence of the
method.
Q4: Solve the equation e^(−x) −x = 0 by secant method, using x0
= 0 and x1 = 1, accurate to 10^−4.
Q5: Solve the following system using the...

Consider the function f(x) = 1 2 |x|.
a) Can we use bisection search to find one of its roots? Why or
why not?
b) Can we use Newton’s method to find one of its roots? Why or
why not?

Suppose that r is a double zero of the C2 function f, i.e., f(r)
= f′(r) = 0 but f′′(r) is not 0. Show that Newton’s method applied
to f converges linearly with the asymptotic constant 1/2, i.e.,
show that
lim n->infinity | x(n+1)−r | / | x(n)−r | = 1/2.

Let f(x)=〖2x〗^3-6x^2-18x+2
Find the interval(s) on which f is increasing and the interval
(s) on which f is decreasing.

for f=(x^4)-(6.4*x^3)+(6.45*x^2)+(20.538*x)- 31.752;
find the roots using bisection for five iterations

Suppose we modify the bisection method into the following
variation: for each step, with bracketing interval [a,b],
approximations are chosen at the location (2a + b)/3, but the
interval is cut into two at the diﬀerent location (a + 3b)/4.
(a) Calculate the ﬁrst 2 approximations p1,p2 for this variation
when f(x) = cosx−x with starting interval [0,π/2].
(b) Bound the absolute errors of the approximations pn for a
starting interval of length L.

Let f have a power series representation, S. Suppose that
f(0)=1, f’(0)=3, f’’(0)=2 and f’’’(0)=5.
a. If the above is the only information we have, to what degree
of accuracy can we estimate f(1)?
b. If, in addition to the above information, we know that S
converges on the interval [-2,2] and that |f’’’’(x)|< 11 on that
interval, then to what degree of accuracy can we estimate
f(1)?

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