Questions: Given the function g(x)=-x^2+8x+24, determine the average rate of change of the function over the interval 2 ≤ x ≤ 10.

Given the function g(x)=-x^2+8x+24, determine the average rate of change of the function over the interval 2 ≤ x ≤ 10.
Transcript text: Given the function $g(x)=-x^{2}+8 x+24$, determine the average rate of change of the function over the interval $2 \leq x \leq 10$.
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Solution

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Solution Steps

Step 1: Identify the function and interval

The given function is g(x)=−x2+8x+24 g(x) = -x^{2} + 8x + 24 , and the interval over which the average rate of change is to be calculated is 2≤x≤10 2 \leq x \leq 10 .

Step 2: Calculate g(2) g(2)

Substitute x=2 x = 2 into the function: g(2)=−(2)2+8(2)+24=−4+16+24=36. g(2) = -(2)^{2} + 8(2) + 24 = -4 + 16 + 24 = 36.

Step 3: Calculate g(10) g(10)

Substitute x=10 x = 10 into the function: g(10)=−(10)2+8(10)+24=−100+80+24=4. g(10) = -(10)^{2} + 8(10) + 24 = -100 + 80 + 24 = 4.

Step 4: Compute the average rate of change

The average rate of change is given by: Average rate of change=g(10)−g(2)10−2=4−368=−328=−4. \text{Average rate of change} = \frac{g(10) - g(2)}{10 - 2} = \frac{4 - 36}{8} = \frac{-32}{8} = -4.

Final Answer

−4\boxed{-4}

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