find the average rate of change of ( f(x)=-14sqrt{x + 2} ) over the interval (12,16). write your answer as…

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)