Question

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).

Answer #1

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).

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.

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...

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

Numerical Analysis: Apply the BFGS Method to minimize the
function f(x) = x12 - 2x1x2
+ 4x22 with the starting point
x0 = [-3,1]T. Thanks!

the function f(x) = ex - 2e-2x - 3/2 is
graphed at right. evidently, f(x) has a zero in the interval
(0,1).
(a) show that f(x) is increasing on (-infinity, infinity) (so
that no other zero of f exists.)
(b) use one iteration of Newton's method to estimate the zero,
starting with initial estimate x1 = 0.
(c) it appears from the graph that f(x) has an inflection point
at or near the zero of f. find the exact coordinates...

Consider a function f(x) =
2x3 − 11.7x2 +
17.7x − 5.
Identify the root of the given function after the third
iteration using the secant method. Use initial guesses
x–1 = 3 and x0 = 4.
CAN YOU PLZ SHOW ALL THE WORK. THANK YOU

f(x) = ex - 2x - 1 = 0 function's [1; 2] there's a
root within the closed range Show. Graphs f1 (x) = ex
and f2 (x) = 2x + 1 functions on an equivalent axis Indicate the
situation of the basis by drawing. Functional iteration by taking
x0 = 1.5 (sequential approaches) method with 5 decimal precision
(5D).

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