Question

find the average rate of change of ( f(x)=2x^{3}-19 ) over the interval ( -3,1 ). write your answer as an integer, fraction, or decimal rounded to the nearest tenth. simplify any fractions.

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a=-3\), \(b = 1\), and \(f(x)=2x^{3}-19\).

Step2: Calculate \(f(-3)\) and \(f(1)\)

• For \(x=-3\):

\(f(-3)=2(-3)^{3}-19=2\times(-27)-19=-54 - 19=-73\).

• For \(x = 1\):

\(f(1)=2(1)^{3}-19=2\times1-19=2 - 19=-17\).

Step3: Substitute into the average - rate - of - change formula

\(\frac{f(1)-f(-3)}{1-(-3)}=\frac{-17-(-73)}{1 + 3}=\frac{-17 + 73}{4}=\frac{56}{4}\).

Answer:

\(14\)