how do the average rates of change for the pair of functions compare over the given interval?\n\n$f(x)=3x^{2}…

how do the average rates of change for the pair of functions compare over the given interval?\n\n$f(x)=3x^{2}$\n$g(x)=12x^{2}$\n$-6\\leq x\\leq -4$\n\nthe average rate of change of $f(x)$ over $-6\\leq x\\leq -4$ is $\\square$. the average rate of change of $g(x)$ over $-6\\leq x\\leq -4$ is $\\square$. the average rate of change of $g(x)$ is $\\square$ times that of $f(x)$.\n\n(simplify your answers. type integers or decimals.)

how do the average rates of change for the pair of functions compare over the given interval?\n\n$f(x)=3x^{2}$\n$g(x)=12x^{2}$\n$-6\\leq x\\leq -4$\n\nthe average rate of change of $f(x)$ over $-6\\leq x\\leq -4$ is $\\square$. the average rate of change of $g(x)$ over $-6\\leq x\\leq -4$ is $\\square$. the average rate of change of $g(x)$ is $\\square$ times that of $f(x)$.\n\n(simplify your answers. type integers or decimals.)

Answer

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}).

Step2: Calculate the average rate of change for (f(x)=3x^{2}) over ([-6,-4])

Here (a=-6), (b = - 4). First, find (f(-6)) and (f(-4)): (f(-6)=3\times(-6)^{2}=3\times36 = 108) (f(-4)=3\times(-4)^{2}=3\times16=48) Then, (\frac{f(-4)-f(-6)}{-4-(-6)}=\frac{48 - 108}{-4 + 6}=\frac{-60}{2}=-30)

Step3: Calculate the average rate of change for (g(x)=12x^{2}) over ([-6,-4])

Here (a=-6), (b=-4) First, find (g(-6)) and (g(-4)): (g(-6)=12\times(-6)^{2}=12\times36 = 432) (g(-4)=12\times(-4)^{2}=12\times16 = 192) Then, (\frac{g(-4)-g(-6)}{-4-(-6)}=\frac{192-432}{-4 + 6}=\frac{-240}{2}=-120)

Step4: Find the ratio of the average rate of change of (g(x)) to (f(x))

Let (A_{f}) be the average rate of change of (f(x)) and (A_{g}) be the average rate of change of (g(x)). (\frac{A_{g}}{A_{f}}=\frac{-120}{-30}=4)

Answer:

The average rate of change of (f(x)) over (-6\leq x\leq - 4) is (-30). The average rate of change of (g(x)) over (-6\leq x\leq - 4) is (-120). The average rate of change of (g(x)) is (4) times that of (f(x)).