Sensitivity (Meta) Math Example 3

Follow the full solution, then compare it with the other examples linked below.

easy

For

f(x)=\sqrt{x}

, compute

f(100)

and

f(104)

. By what percentage does

f

change when

x

increases by 4%?

1
$f(100) = 10$ , $f(104) = \sqrt{104} \approx 10.198$ .
2
Change in $f$ : $0.198$ . Relative change: $\frac{0.198}{10} = 1.98\%$ .
3
Change in $x$ : $4\%$ . So a 4% increase in $x$ causes only about 2% increase in $f$ .

Answer

\text{4\% increase in }x \Rightarrow \approx 2\%\text{ increase in }f\;(\text{low sensitivity})

The square root function compresses changes: a 4% input change gives only a 2% output change. This follows from the derivative:

f'(x)/f(x) = \frac{1}{2x^{1/2}}/x^{1/2} = 1/(2x)

, giving half the relative change.

About Sensitivity (Meta)

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