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Exercise

A machinist is required to manufacture a circular metal disk with area $1000$ cm$^2$.

1. What radius produces such a disk?




2. If the machinist is allowed an error tolerance of $pm 5$ cm$^2$ in the area of the disk, how close to the ideal radius in part (1) must the machinist control the radius?




3. In terms of the $epsilon$, $delta$ definition of $lim limits_{x to a}{f(x)} = L$ what is $x$? What is $f(x)$? What value of $epsilon$ is given? What is the corresponding value of $delta$?

Solution

Note: I've decided to solve part (3) before part (2), as it helps set the stage of the solution of (2).

1. What radius produces such a disk?

$pi r_0^2 = 1000 implies r_0 = sqrt{frac{1000}{pi}} approx 17.8412$

3. In terms of the $epsilon$, $delta$ definition of $lim limits_{x to a}{f(x)} = L$ what is $x$? What is $f(x)$? What value of $epsilon$ is given? What is the corresponding value of $delta$?

$a = r_0 approx 17.8412$

$L = pi r_0^2 = 1000$

$x = r$

$f(x) = pi r^2$

$|r - r_0| < delta$

$|pi r^2 - pi r_0^2| < epsilon$

2. If the machinist is allowed an error tolerance of $pm 5$ cm$^2$ in the area of the disk, how close to the ideal radius in part (1) must the machinist control the radius?

$|pi r^2 - pi r_0^2| < epsilon$
$implies |pi r^2 - 1000| < 5$
$implies -5 < pi r^2 - 1000 < 5$
$implies 995 < pi r^2 < 1005$
$implies frac{995}{pi} < r^2 < frac{1005}{pi}$
$implies sqrt{frac{995}{pi}} < r < sqrt{frac{1005}{pi}}$
$implies 17.7966 < r < 17.8858$

$|r - r_0| < delta implies |r - 17.8412| < delta$

$|17.7966 - 17.8412| < delta implies 0.0446 < delta$

$|17.8858 - 17.8412| < delta implies 0.0446 < delta$

$delta = min(0.0446, 0.0446) = 0.0446$

Answer

1. What radius produces such a disk?

$$r_0 = 17.8412 text{ cm}$$

2. If the machinist is allowed an error tolerance of $pm 5$ cm$^2$ in the area of the disk, how close to the ideal radius in part (1) must the machinist control the radius?

$$delta = 0.0446 text{ cm}$$

3. In terms of the $epsilon$, $delta$ definition of $lim limits_{x to a}{f(x)} = L$ what is $x$? What is $f(x)$? What value of $epsilon$ is given? What is the corresponding value of $delta$?

$$|r - r_0| < delta$$

$$|pi r^2 - pi r_0^2| < epsilon$$

Request

Is my answer correct? If not, in what part of my solution did I make a mistake?

In part 3 you are asked about four items: $x$, $f(x)$, $epsilon$, and $delta$. Your answer omits these. To make it clear:

• The independent variable $x$ here is the radius $r$ of the disk


• The function $f$ maps radius to area, i.e., $rmapsto pi r^2$. In other words, $f(x)=pi x^2$.


• The given $epsilon$ is $5,text{cm}^2$
  


• The corresponding (or rather: a suitable) value of $delta$ is the $approx 0.0446,text{cm}$ you found

On a sidenote, we may observe that picking $delta$ such that $fracepsilondelta=f'(r_0)$ usually gives a good enough approximation. Indeed, for the problem at hand $f'(x)=2pi x$, so we quickly obtain $delta=frac{epsilon}{2pi r_0}approx 0.0446$.
Things become even easier with relative errors: Raising to $n$th power multiplies (small) relative errors by $n$. Hence for a desired relative output error of $frac 5{1000}=frac1{200}$, the relative input error should be bounded by $frac{1}{400}$, and indeed $frac1{400}cdot 17.8412approx 0.0446$.