Use Newton’s Method with the initial approximation x1=2 to find the third approximation to the root. Round your answer to 5 decimal places. f(x)=−3x^5+2x^4+8x^3+2

To find the next approximation using Newton's Method, we need to follow these steps:

Given function f(x) = -3x^5 + 2x^4 + 8x^3 + 2 .

1. Find the derivative of the function f(x) :
f'(x) = \frac{d}{dx}(-3x^5 + 2x^4 + 8x^3 + 2)
f'(x) = -15x^4 + 8x^3 + 24x^2

2. Substitute the initial approximation x_1 = 2 into the formula for Newton's Method to get the second approximation:
x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}

x_2 = 2 - \frac{-3(2)^5 + 2(2)^4 + 8(2)^3 + 2}{-15(2)^4 + 8(2)^3 + 24(2)^2}
x_2 = 2 - \frac{-96 + 32 + 64 + 2}{-240 + 64 + 96}
x_2 = 2 - \frac{2}{-80}
x_2 = 2 + 0.025
x_2 = 2.025

Therefore, the third approximation to the root is x_3 = 2.025 .

\boxed{x_3 = 2.025}