Choose another quadratic surd in the range root 3 to root 23 and find four successive rational approximations,each of them accurate to within 10^-4 of the true value of the surd chosen. use the easiest starting point you can find.

Question

zehanz · Answer

Do you mean the use of Newton's method for approximation of (square) roots?

zehanz · Answer

In that case, suppose you want to find an approximation of, say $\sqrt 7$, consider the function 
$f(x)=x^2-7$. 
It's roots are $\pm\sqrt 7$.

We only need the positive root. Here is a drawing of the graph:

zehanz · Answer

Also remember that the equation of a line through a given point $P(x_p, y_p)$ and slope $m$is:

$y-y_p=m(x-x_p)$.

Newton's method works as follows: pick a point $P_1$ on the graph. The x-coordinate of $P_1$ is $x_1$.  The tangent line in this point has slope $m=f'(x_1)$.
We now have this line:

$y-f(x_1)=f'(x_1)(x-x_1)$.

The intersection of this line with the x-axis is the first approximation of our root.

anonymous · Answer

Nope sorry i shoudve specified. continued fractions

anonymous · Answer

@Compassionate

anonymous · Answer

@mathmate @Nnesha

anonymous · Answer

or not completely sure, is there any other way?
the homework is titles continued fractions so im assuming, ive never been taught newtons method, so it cant be that.

mathmate · Answer

@Maretch 
Have you learned how to do continued fractions for any number, or mainly for square-roots?