Using Newton's Method without a calculator

Apr 2010
156
0
How would you use Newton's Method without a calculator, if you are for example asked to to find a solution correct to 3 decimal places? I can only do it as a fraction, and my exam coming up soon and we not allowed to use calculators.
 

pickslides

MHF Helper
Sep 2008
5,237
1,625
Melbourne
post a question, I will show you how.
 
Apr 2010
156
0
Use Newton's method with the specified initial approximation x1 to find x3, the third approximation to the root of the given equation. (Give your answer to four decimal places):

5. x^3 + 2x - 4 = 0, x1 = 1
 

pickslides

MHF Helper
Sep 2008
5,237
1,625
Melbourne
\(\displaystyle f(x)=x^3 + 2x - 4 \implies f'(x)=3x^2 + 2\)

Newton's method \(\displaystyle x_{n+1} = x_n-\frac{f(x_n)}{f'(x_n)}\)

For \(\displaystyle x_1 = 1\)

\(\displaystyle x_{2} = x_1-\frac{f(x_1)}{f'(x_1)}=1-\frac{(1)^3 + 2(1) - 4)}{(3(1)^2 + 2}= 1-\frac{1+2-4}{3+2}= 1-\frac{-1}{5} = \frac{6}{5}\)

Your turn. Find \(\displaystyle x_2\)
 
Apr 2010
156
0
Yeah I got that, then I found x3 to be 932/790, but not sure of an efficient way of showing it to 4 decimal places without a calculator, besides using long division (which may indeed be very long lol)
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
The only way to show \(\displaystyle \frac{932}{789}\) "to four decimal places" is to divide.