Question

The side length of a square ice cube is claimed to be 1.0 inches, correct to within 0.001 inch. Use linear approximation to estimate the resulting error, measured in squared inches, in the surface area of the ice cube.

Answer 1

0.012

Explanation:

Linear approximation says that,

`f(x) \approx f(x_o)+f'(x_o)(x-x_o)`

For a cube the surface area is
`6x^2`.

So the side is 1.0 inch in, the surface area is
`=6*1^2 = 6` square inches.

In Linear approximation means you ignore the term
`x_o^2` , if
`x_o` is a small number, because then
`x_o^2` will be a very smalle number and that does not contribute much to the error.

So the surface area is approximately,

`6x^2=6x_o^2+12x_o(x-x_o)`

So here,
`x=1.001, x_o=0.001`

The error in the area is approximately,

`12 * 0.001=0.012`

So the error is 0.012.