Question

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.

Answer #1

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).

For the following function, determine the highest real root
of
f(x) = 2x3 – 11.7x2 + 17.7x - 5
by using (a) graphical methods, (b) fixed point iteration (three
iterations, x0 = 3) (Hint: Be certain that you develop a solution
that converges on the root), and (c) Newton-Raphson method (three
iterations, x0 = 3).
Perform an error check on each of your final root approximations
(e.g. for the last of the three iterations).

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...

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.)...

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...

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...

Use the fixed point iteration method to find a root for the
function g(x)=2^(-x) in the interval [0,1] with an error of
0.03%

Consider the function f(x,y) = ( x2 +
z2)ln(y)
a)Find the gradient of f.
b) Find the rate of change of f at the point (2, 1, 1) in the
direction of ?⃗ = 〈−2, 4, −4〉

17. I am using Newton’s method to ﬁnd the negative root of f(x)
= 3−x2.
(a) What would be a good guess for x1? Draw the line
tangent to f(x) at your x1 and explain why using
Newton’s method would lead to the negative root of the
function.
(b) What would be a bad guess for x1? Draw the line
tangent to f(x) at your x1 and explain why using
Newton’s method would not lead to the negative root of...

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