Question

Consider the function **g (x) = 12x + 4 - cos x.**
Given **g (x) = 0** has a unique solution **x =
b in the interval (−1/2, 0)**, and you can use this without
justification.

(a) Show that Newton's method of starting point
**x _{0} = 0** gives a number sequence with

(The word "curvature" should be included in the argument!)

(b) Calculate

(c) Find a suitable function that you can perform fix piont iteration to find

Answer #1

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Consider the function
g (x) = 12x + 4 - cos x. Given
g (x) = 0 has a unique solution
x = b in the interval (−1/2, 0), and you can use this
without justification.
(a) Show that Newton's method of starting point
x0
= 0 gives a number sequence with
b <··· <xn+1
<xn
<··· <x1
<x0
= 0
(The word "curvature" should be included in the argument!)
(b) Calculate
x1
and x2.
Use theorem 2 in section...

Consider the function g (x) = 12x + 4 - cos x.
Given g (x) = 0 has a unique solution x =
b in the interval (−1/2, 0), and you can use this without
justification.
(a) Show that Newton's method of starting point
x0 = 0 gives a number sequence with
b <··· <xn+1 <xn <···
<x1 <x0 = 0
(The word "curvature" should be included in the argument!)
(b) Calculate x1 and x2. Use
theorem 2 in section...

Newton's method: For a function ?(?)=ln?+?2−3f(x)=lnx+x2−3
a. Find the root of function ?(?)f(x) starting with
?0=1.0x0=1.0.
b. Compute the ratio |??−?|/|??−1−?|2|xn−r|/|xn−1−r|2, for
iterations 2, 3, 4 given ?=1.592142937058094r=1.592142937058094.
Show that this ratio's value approaches
|?″(?)/2?′(?)||f″(x)/2f′(x)| (i.e., the iteration converges
quadratically). In error computation, keep as many digits as you
can.

Part A.
Consider the nonlinear equation
x5-x=15
Attempt to find a root of this equation with Newton's method
(also known as Newton iteration).
Use a starting value of x0=4 and apply Newton's
method once to find x1
Enter your answer in the box below correct to four
decimal places.
Part B.
Using the value for x1 obtained in Part A, apply
Newton's method again to find x2
Note you should not round x1 when computing
x2

If we want to minimize a function f(x) = e^(x^2)
over R, then it is equivalent to finding the root of f '(x).
Starting with x0 = 1, can you perform 4 iterations of Newton's
method to estimate the minimizer
of f(x)? (Correct to four decimal places at each iteration).

3.8/3.9
5. Use Newton's Method to approximate the zero(s) of the
function. Continue the iterations until two successive
approximations differ by less than 0.001. Then find the zero(s) to
three decimal places using a graphing utility and compare the
results.
f(x) = 3 − x + sin(x)
Newton's Method: x=
Graphing Utility: x=
6. Find the tangent line approximation T to the graph
of f at the given point. Then complete the table. (Round
your answer to four decimal places.)...

Calculate two iterations of Newton's Method to approximate a
zero of the function using the given initial guess. (Round your
answers to four decimal places.)
f(x) = cos x, x1 = 0.8
n
xn
f(xn)
f '(xn)
f(xn)
f '(xn)
xn −
f(xn)
f '(xn)
1
2

2. (a) For the equation e^x = 3 - 2 x , find a function, f(x),
whose x-intercept is the solution of the equation (i.e. a function
suitable to use in Newton’s Method), and use it to set up xn+1 for
Newton’s Method.
(b) Use Newton's method to find x3 , x4 and x5 using the initial
guess x1 = 0 . How many digits of accuracy are you certain of from
these results?
(c) Use x1+ ln 2 and show...

Consider the function, f(x) = - x4 - 2x3 -
8x2 - 5x
Use parabolic interpolation (x0 = -2, x1 =
-1, x2= 1, iterations = 4). Select new points
sequentially as in the secant method.

Consider a function x2 − 3 = 0 . Then with the
starting point x0 = 1 , if we perform three iterations
of Newton-Rhapson Method, we have the following:
(Note that your answer format should be x.xxxx. For example, 2
->, 2.0000 or 1.34 -> 1.3400, or 1.23474 - > 1.2374, or
1.23746->1.2375)
x1 =
x2 =
x3 =

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