Question

How do you estimate the quantity using Linear Approximation and find the error using a calculator of `1/(sqrt(95)) - 1/(sqrt(98))`?

Answer 1

Let `y = 1/sqrt(x)`. Then `y(100) = 1/sqrt(100) = 1/10`

`y = x^(-1/2) -> y' = -1/2x^(-3/2)`

We get that `y'(100) = -1/2(100)^(-3/2) = -1/(2(10^3)) = -1/2000`

Therefore the equation of the tangent at `x= 100` is

`y - 1/10 = -1/2000(x - 100)`

`y = -1/2000x + 1/20 + 1/10`

`y = -1/2000x + 3/20`

Therefore the linear approximation will be

`y(95) - y(98) = -1/2000(95) + 3/20 - (-1/2000(98) + 3/20)`

`y(95) - y(98) = 0.00150`

With a calculator we get `1/sqrt(95) - 1/sqrt(98) = 0.00158`

This means the percent error is

`"% error" = |(0.00150 - 0.00158)/0.00158| = 5%`

So our approximation is quite precise.

Hopefully this helps!