Question

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q	2.4	2.6	2.8	3	3.2
h(q)	1964	3049	4734	7351	11414

estimate h(2.8) using the table above.
hint: find the average rate of change on either side of h(2.8), and then average those two values.
h(2.8) ≈
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Explanation:

Step1: Calculate left - hand difference quotient

The left - hand difference quotient for estimating the derivative at $q = 2.8$ is $\frac{h(2.8)-h(2.6)}{2.8 - 2.6}$. Substitute $h(2.8)=4734$ and $h(2.6)=3049$ into the formula: $\frac{4734 - 3049}{2.8 - 2.6}=\frac{1685}{0.2}=8425$.

Step2: Calculate right - hand difference quotient

The right - hand difference quotient for estimating the derivative at $q = 2.8$ is $\frac{h(3)-h(2.8)}{3 - 2.8}$. Substitute $h(3)=7351$ and $h(2.8)=4734$ into the formula: $\frac{7351 - 4734}{3 - 2.8}=\frac{2617}{0.2}=13085$.

Step3: Average the two difference quotients

To estimate $h'(2.8)$, we average the left - hand and right - hand difference quotients. $h'(2.8)\approx\frac{8425 + 13085}{2}=\frac{21510}{2}=10755$.

Answer:

$10755$