Sensitivity Math Example 2

Solution

1. 1
   At $x=0$: $F(0)=1$, $F(0.01)=e^{0.01}\approx1.01005$. $\Delta F/\Delta x\approx0.01005/0.01=1.005\approx F'(0)=e^0=1$.
2. 2
   At $x=5$: $F(5)=e^5\approx148.41$, $F(5.01)=e^{5.01}\approx149.90$. $\Delta F/\Delta x\approx1.49/0.01=149\approx F'(5)=e^5\approx148.4$.
3. 3
   Sensitivity at $x=5$ is about $148\times$ greater than at $x=0$. This explains why small errors in $x$ compound dramatically for large exponentials.