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anonymous

  • one year ago

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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  1. ZeHanz
    • one year ago
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    Do you mean the use of Newton's method for approximation of (square) roots?

  2. ZeHanz
    • one year ago
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    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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  3. ZeHanz
    • one year ago
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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.

  4. anonymous
    • one year ago
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    Nope sorry i shoudve specified. continued fractions

  5. anonymous
    • one year ago
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    @Compassionate

  6. anonymous
    • one year ago
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    @mathmate @Nnesha

  7. anonymous
    • one year ago
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    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.

  8. mathmate
    • one year ago
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    @Maretch Have you learned how to do continued fractions for any number, or mainly for square-roots?

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