Newton's Method

**dbakeg00** · November 10th 2010, 08:33 PM

I'm working on two problems that I need help on.

(1) The radius of a circle is to be measured and its area computed. If the radius can be measured to an accuracy of 0.001 in and the area must be accurate to 0.1in^2, estimate the maximum radius for which this process can be used.

(2) If pV=20 and p is measured as $5\pm0.02$ , estimate V.

Here are the formulas I have been using for these problems (out of Schaum's Calculus, 4th ed.):

$f(x+\Delta x) \sim f(x)+\dot f(x)*\Delta x)$

$\displaystyle x_{n+1}=x_n+\frac{f(x_n)}{f\prime(x_n)}$

I'm not requesting that someone work out the problems for me, just help me with where to start. Thanks

**CaptainBlack** · November 10th 2010, 10:48 PM

Originally Posted by dbakeg00

I'm working on two problems that I need help on.

(1) The radius of a circle is to be measured and its area computed. If the radius can be measured to an accuracy of 0.001 in and the area must be accurate to 0.1in^2, estimate the maximum radius for which this process can be used.

Let the radius be $$$ r$ and error in measuring it be $$$ e$ so the measured radius is $\hat{r}=r+e$ , then the computed area from the measumenet is:

$\hat{A}=\pi \hat{r}^2=\pi (r+e)^2=\pi r^2+ \pi (2r\; e+e^2)=A+ \pi (2r\; e+e^2)$

So the error in the area is $\pi (2r\; e+e^2)$ and if $e=0.001$ units and the error is the area is $0.1$ sq-units then:

$\pi (2\times 0.001 r+0.001^2)=0.1$

so:

$r=\frac{0.1}{2 \times 0.001}-\frac{0.001}{2}\approx 50$

Alternatively you use $A=f(r)=\pi r^2$ , then:

$A =f(r) \approx f(\hat{r}) - ef'(\hat{r})=\hat{A} -e f'(\hat{r})$

Then if $$$ u$ is the maximum permissible error in $$$ A$ you solve $u=|ef'(\hat{r})|$

Which is equivalent to the earlier method when the $e^2$ terms are ignored. This later method has the advantage that it will still work for more complicated functions (and in particular the function in the next question).

CB

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