Question

f(x) = x^3 - 9x
what is the average rate of change of f over the interval 1, 6?
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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 \( b = 6 \), and \( f(x)=x^{3}-9x \).

Step2: Calculate \( f(6) \)

Substitute \( x = 6 \) into \( f(x) \):
\( f(6)=6^{3}-9\times6 \)
\( = 216 - 54 \)
\( = 162 \)

Step3: Calculate \( f(1) \)

Substitute \( x = 1 \) into \( f(x) \):
\( f(1)=1^{3}-9\times1 \)
\( = 1 - 9 \)
\( = -8 \)

Step4: Calculate the average rate of change

Using the formula \(\frac{f(6)-f(1)}{6 - 1}\), substitute the values of \( f(6) \) and \( f(1) \):
\(\frac{162-(-8)}{6 - 1}=\frac{162 + 8}{5}=\frac{170}{5}=34\)

Answer:

34