Question

given ( f(x) = 17 - x^2 ), what is the average rate of change in ( f(x) ) over the interval (1, 5)? (\frac{1}{4}) (-6) (1) (-\frac{1}{2})

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) over the interval \([a, b]\) is given by \(\frac{f(b) - f(a)}{b - a}\). Here, \( a = 1 \) and \( b = 5 \).

Step2: Calculate \( f(1) \)

Substitute \( x = 1 \) into \( f(x)=17 - x^{2} \):
\( f(1)=17-(1)^{2}=17 - 1 = 16 \)

Step3: Calculate \( f(5) \)

Substitute \( x = 5 \) into \( f(x)=17 - x^{2} \):
\( f(5)=17-(5)^{2}=17 - 25=-8 \)

Step4: Calculate the average rate of change

Using the formula \(\frac{f(5)-f(1)}{5 - 1}\), substitute \( f(5)=-8 \) and \( f(1)=16 \):
\(\frac{-8 - 16}{5 - 1}=\frac{-24}{4}=-6\)

Answer:

\(-6\) (corresponding to the option with value \(-6\))