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.

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

Solution

Solution Steps

Step 1: Identify the function and interval

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

Step 2: Calculate $ g(2) $

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

Step 3: Calculate $ g(10) $

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

Step 4: Compute the average rate of change

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

Final Answer

$\boxed{-4}$

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