# Use Newton's method with the specified initial approximation x1 to find x3, the third approximation to the root of the given equation. (Round your answer to four decimal places.) What is x3?

##
(1/3)x^3+(1/2)x^2+1=0

x1 = -3

(1/3)x^3+(1/2)x^2+1=0

x1 = -3

Using N-R Iteration we get (to 4dp):

# x_2 = -2.4167 #

# x_3 = -2.1875 #

Continuing the iteration we get convergence to 8dp to the solution

Let

First let us look at the graphs:

graph{1/3x^3+1/2x^2+1 [-4, 4, -10, 10]}

We can see there is one solution in the interval

We can find the solution numerically, using Newton-Rhapson method

# \ \ \ \ \ \ \f(x) = 1/3x^3+1/2x^2+1 #

# :. f'(x) = x^2+x #

The Newton-Rhapson method uses the following iterative sequence

# { (x_1,=-3), ( x_(n+1), = x_n - f(x_n)/(f'(x_n)) ) :} #

Then using excel working to 8dp we can tabulate the iterations as follows:

We could equally use a modern scientific graphing calculator as most new calculators have an " *Ans* " button that allows the last calculated result to be used as the input of an iterated expression.

And we conclude that the solution is

By signing up, you agree to our Terms of Service and Privacy Policy

To find the third approximation (x_3) using Newton's method, you typically use the formula:

[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} ]

The initial approximation is (x_1). You would also need the function (f(x)) and its derivative (f'(x)). Since these are not provided in your question, I cannot give the specific (x_3) value. Could you provide the function (f(x)), its derivative (f'(x)), the initial approximation (x_1), and the previous approximations (x_1) and (x_2)? This information is needed to calculate (x_3) using Newton's method.

By signing up, you agree to our Terms of Service and Privacy Policy

- How do you find the linear approximation of #g(x)=sqrt(1+x)#?
- How do you estimate #(26.8)^(2/3)# using linear approximation?
- If a ball is thrown vertically upward from the ground with an initial velocity of 56 feet per second, for how long will the ball be going upward?
- How do you use linear approximation to the square root function to estimate square roots #sqrt 3.60#?
- How can I solve the mentioned problem?Please,help.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7