How do you use Newton's method to find the approximate solution to the equation #2x^5+3x=2#?

Answer 1

#x~~0.610#
(See below for Newton Method of approximation).

Noting that if #x=0# then #2x^5+3x < 2#
and if #x=1# then #2x^5+3x > 2#

we can start with "bracketing" values Low#=0# and High#=1#

At each iteration we evaluate the mid-point and adjust either the Low or High closing in the brackets about the solution value.

Here is what the first 10 iterations look like in a spreadsheet form:

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Answer 2

#x=0.610246# to 6dp

Let #f(x) = 2x^5+3x-2# Then our aim is to solve #f(x)=0#

First let us look at the graphs:
graph{2x^5+3x-2 [-10, 10, -5, 4.995]}

We can see there is one solution in the interval # 0 < x < 1 #.

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

# f(x) = 2x^5+3x-2 => f'(x) = 10x^4+3 #, and using the Newton-Rhapson method we use the following iterative sequence

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# { (x_0=1), ( x_(n+1)=x_n - f(x_n)/(f'(x_n)) ) :} #

Then using excel working to 6dp we can tabulate the iterations as follows:

And we conclude that the remaining solution is #x=0.610246# to 6dp

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Answer 3

To use Newton's method to find the approximate solution to the equation (2x^5 + 3x = 2), follow these steps:

  1. Choose an initial guess (x_0) close to the actual root of the equation.
  2. Use the formula (x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}) to iteratively refine the guess, where (f(x)) is the function representing the equation and (f'(x)) is its derivative.
  3. Repeat step 2 until the value of (x) converges to the desired level of accuracy.

In this case, the function (f(x) = 2x^5 + 3x - 2). Differentiate it to find its derivative, (f'(x)). (f'(x) = 10x^4 + 3).

Choose an initial guess, (x_0 = 0.5) for instance.

Then, apply the formula:

(x_{n+1} = x_n - \frac{2x_n^5 + 3x_n - 2}{10x_n^4 + 3}).

Repeat the process until you achieve the desired level of accuracy.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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