Use Newton's method with the initial approximation $x _ { 1 } = 2$ to find $x _ { 2 }$ , the second approximation to a root of the equation $x ^ { 5 } - 34 = 0$ .
A) $\frac { 79 } { 40 }$

B) $\frac { 81 } { 40 }$
C)$\frac { 77 } { 40 }$
D) $\frac { 161 } { 80 }$
E) $\frac { 83 } { 40 }$
F) $\frac { 157 } { 80 }$

G)
$\frac { 163 } { 80 }$

H) $\frac { 159 } { 80 }$

Question

Use Newton's method with the initial approximation $x _ { 1 } = 2$ to find $x _ { 2 }$ , the second approximation to a root of the equation $x ^ { 5 } - 34 = 0$ .
A) $\frac { 79 } { 40 }$ 
 
B) $\frac { 81 } { 40 }$
C)$\frac { 77 } { 40 }$ 
D) $\frac { 161 } { 80 }$ 
E) $\frac { 83 } { 40 }$
F) $\frac { 157 } { 80 }$

G)
$\frac { 163 } { 80 }$

H) $\frac { 159 } { 80 }$

Quizplus · Accepted Answer

Newton's method formula is $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$. Here, $f(x) = x^5 - 34$, so $f'(x) = 5x^4$. With $x_1 = 2$, $f(2) = 2^5 - 34 = 2$, and $f'(2) = 5(2)^4 = 80$. Thus, $x_2 = 2 - \frac{2}{80} = 2 - \frac{1}{40} = \frac{79}{40}$.

Use Newton's Method with the Initial Approximation x1=2x _ { 1 } = 2x1​=2

Use Newton's Method with the Initial Approximation $x _ { 1 } = 2$