Question

Consider the function g (x) = 12x + 4 - cos x. Given g (x) =...

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 4.2 (page 229- Error bounds for Newtons Method) or intermediate value theorem to show that
x2 - b <0.0000003.
(c) Find a suitable function that you can perform fix piont iteration to find b. Calculate and compare the results of calculating the first three iterations of Newton's method and fix point iteration

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Consider the function g (x) = 12x + 4 - cos x. Given g (x) =...
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) =...
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)=ln⁡x+x2−3 a. Find the root of function ?(?)f(x) starting with ?0=1.0x0=1.0....
Newton's method: For a function ?(?)=ln?+?2−3f(x)=ln⁡x+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...
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
3.8/3.9 5. Use Newton's Method to approximate the zero(s) of the function. Continue the iterations until...
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.)...
If we want to minimize a function f(x) = e^(x^2) over R, then it is equivalent...
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).
Calculate two iterations of Newton's Method to approximate a zero of the function using the given...
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),...
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...
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 =...
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 =
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT