x -23 -22 -20 -17 -16\n h(x) 21 22 5 22 37\n what is the best estimate for h(-18) we can make based on this…

Explanation:

Step1: Recall the formula for estimating the derivative

The derivative $h^{\prime}(a)$ can be estimated using the symmetric - difference quotient: $h^{\prime}(a)\approx\frac{h(a + h)-h(a - h)}{2h}$. We want to estimate $h^{\prime}(-18)$. The values closest to $x=-18$ in the table are $x=-17$ and $x=-20$. Here, $a=-18$, $a + h=-17$ (so $h = 1$) and $a - h=-20$.

Step2: Calculate the estimate

We use the formula $h^{\prime}(-18)\approx\frac{h(-17)-h(-20)}{2\times1}$. From the table, $h(-17) = 22$ and $h(-20)=5$. Then $\frac{h(-17)-h(-20)}{2\times1}=\frac{22 - 5}{2}=\frac{17}{2}=8.5$. But if we use the forward - difference quotient (since we don't have a perfect symmetric pair around $-18$), we can also use the points $x=-17$ and $x=-20$ in the forward - difference formula $h^{\prime}(x)\approx\frac{h(x + \Delta x)-h(x)}{\Delta x}$. Let $x=-20$ and $\Delta x = 3$. Then $h^{\prime}(-20)\approx\frac{h(-17)-h(-20)}{-17-(-20)}=\frac{22 - 5}{3}=\frac{17}{3}\approx5.67$.

Answer:

D. $5.67$