Question

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given the function $g(x) = x^2 - 5x - 2$, determine the average rate of change of the function over the interval $1 \leq x \leq 9$.
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Explanation:

Step1: Recall the average rate of change formula

The average rate of change of a function \( g(x) \) over the interval \([a, b]\) is given by \(\frac{g(b) - g(a)}{b - a}\). Here, \( a = 1 \) and \( b = 9 \).

Step2: Calculate \( g(1) \)

Substitute \( x = 1 \) into \( g(x)=x^{2}-5x - 2 \):
\( g(1)=(1)^{2}-5(1)-2 = 1 - 5 - 2=-6 \)

Step3: Calculate \( g(9) \)

Substitute \( x = 9 \) into \( g(x)=x^{2}-5x - 2 \):
\( g(9)=(9)^{2}-5(9)-2 = 81 - 45 - 2 = 34 \)

Step4: Apply the average rate of change formula

Using \( a = 1 \), \( b = 9 \), \( g(1)=-6 \), and \( g(9)=34 \):
\(\frac{g(9)-g(1)}{9 - 1}=\frac{34-(-6)}{8}=\frac{40}{8} = 5\)

Answer:

\( 5 \)