Sensitivity (Meta) Math Example 1

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

easy

Compute

f(x) = x^3

x = 2

and

x = 2.1

. Find the sensitivity: by what percentage does

f

change when

x

changes by 5%?

1
$f(2) = 8$ , $f(2.1) = 9.261$ .
2
Change in $f$ : $9.261 - 8 = 1.261$ . Relative change in $f$ : $\frac{1.261}{8} \approx 15.8\%$ .
3
Change in $x$ : $\frac{0.1}{2} = 5\%$ .
4
Sensitivity: a 5% increase in $x$ causes about a 15.8% increase in $f$ . The function is sensitive — it amplifies errors by a factor of about 3.

Answer

\text{5\% change in }x \Rightarrow \approx 15.8\%\text{ change in }f

Sensitivity measures how much the output changes relative to the input. For

f(x) = x^3

, the exponent 3 amplifies percentage changes roughly threefold, which can be derived from the derivative:

f'(x)/f(x) \approx 3 \Delta x/x

About Sensitivity (Meta)

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