Question

find the average rate of change of ( f(x)=-14sqrt{x + 2} ) over the interval (12,16). 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 average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 12\), \(b=16\), and \(f(x)=-14\sqrt{x + 2}\).

Step2: Calculate \(f(12)\) and \(f(16)\)

• For \(x = 12\): \(f(12)=-14\sqrt{12 + 2}=-14\sqrt{14}\approx-14\times3.742=-52.388\)
• For \(x = 16\): \(f(16)=-14\sqrt{16+ 2}=-14\sqrt{18}=-14\times3\sqrt{2}\approx-14\times4.243=-59.402\)

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

\(\frac{f(16)-f(12)}{16 - 12}=\frac{-59.402-(-52.388)}{4}=\frac{-59.402 + 52.388}{4}=\frac{-7.014}{4}=-1.7535\approx - 1.8\)

Answer:

\(-1.8\)