past calculus test question dealing with Newton's Method?

[pink butterfly]

Let f(x)=e^(x)+2x^(2)-2.

Note: If you do not wish to do the entire question, then please focus on doing parts d and e because I do not know how to and the textbook and classnotes do not cover this.

a) Write down the recurrence relation (formula) to use Newton's method in order to approximate the roots for THIS function.

b) This function has two roots. Find starting points, x1, for using Newton's method to approximate each one.
(I got x1= .5 and x1= -1)

c)Use Newton's method to one of those roots. Write down each successive approximation given by Newton's method until you come up with your final approximation that is accurate to 4 decimal places.
(I got .4579)

d) Write down the recurrence relation (formula) to use Newton's method in order to approximate the critical numbers for this function.

e) This function has one critical number. Use Newton's method to approximate the critical number. Write down each succesive approximation given by Newton's method until you come up with your final approximation that is accurate to 4 decimal places.

 

[θ βяιαη θ]

It seems that you know how to do this. Basically, choose starting points and iterate until the method converges towards a zero.

For the first 3, you are correct. Although, I think they wanted you to write:

x_n+1 = x_n - [e^(x_n) + 2(x_n)^2 - 2]/[e^(x_n) + 4(x_n)]

For the first one.

d)

We need to take the first and second derivative.

f(x) = e^x + 2x^2 - 2
f'(x) = e^x + 4x <== We need to zero this
f''(x) = e^x + 4 <== This is used for f'(x) to find zeroes

Then, the recursive function is:

x_n+1 = x_n - f(x_n)/f'(x_n)
==> x_n+1 = x_n - [e^(x_n) + 4(x_n)]/[e^(x_n) + 4]

e)

If we let x_0 = 0, we get:

x_1 = 0 - 1/(1 + 4) = -0.25
x_2 = -0.25 - [e^(-0.25) - 1]/[e^(-0.25) + 4] ≈ -0.203712

Continuing, we get that an approximate critical point is x ≈ -0.2039.

I hope this helps!