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.

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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.

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

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Nope sorry i shoudve specified. continued fractions
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.
@Maretch Have you learned how to do continued fractions for any number, or mainly for square-roots?

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