Problem $4$: Derivatives of Data with Errors $[1$ pt$]$

Aside from unequal spacing, another problem related to differentiating empirical data is that these data usually include measurement error. A shortcoming of numerical differentiation is that it tends to amplify errors in the data. See Figure $11$

Figure $1$: Illustration of how small data errors are amplified by numerical differentiation: $($a$).$ data with no error; $($b$).$ the resulting numerical differentiation of curve $($a$)$; $($c$).$ data modified to include slight errors; and $($d$)$ the resulting differentiation of curve $($c$)$ manifesting increased variability.

Fig. $11($a$)$ shows smooth, error$-$free data that when numerically differentiated yield a smooth result $($Fig$.1.$b$)).$ In contrast, Fig. $1.$c$)$ uses the same data, but with alternating points raised and lowered slightly. This minor modification is barely apparent from Fig. $1($c$).$ However, the resulting effect in Fig. $1($d$)$ is significant.

To address this issue, one of the the primary approaches for determining derivatives for imprecise data is to use curve fitting to fit a smooth and differentiable function to the data and then calculate derivatives using the fitted function. For instance, the stress vs$.$ strain data in Homework $4$ Problem $4$ presents a noisy data set. Review the solution for this problem $($provided on Canvas$),$ and compute the value of derivative at $x=3\mathrm{.}2$ using the fitted $5-$th order polynomial. $[0\mathrm{.}4$ pts