Question

Find the Average Rate of Change (AROC) of y = 4(1/2)^x over the interval [-2, 0].

a) 6
b) -6
c) 4
d) -4

Answer 1

• b) -6

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Find the value of the function at the endpoints of the given interval

• x = - 2 ⇒ y = 4(1/2)⁻² = 4(2)² = 4(4) = 16
• x = 0 ⇒ y = 4(1/2)⁰ = 4(1) = 4

Average rate of change is

• Change in y / change in x =
• (4 - 16)/(0 - (-2)) =
• - 12 / 2 =
• - 6

Correct choice is B.

Answer 2

b) -6

Explanation:

The average rate of change of function f(x) over the interval a ≤ x ≤ b is given by:

`(f(b)-f(a))/(b-a)`

Given the interval is [-2, 0]:

• a = -2
• b = 0

Substitute the endpoints of the interval into the function and solve:

`\begin{aligned}x=-2 \implies f(-2)&=4\left((1)/(2)\right)^(-2)\\& = 4(4)\\&=16\end{aligned}`

`\begin{aligned}x=0 \implies f(0)&=4\left((1)/(2)\right)^(0)\\& = 4(1)\\&=4\end{aligned}`

Therefore:

`\begin{aligned}\implies(f(b)-f(a))/(b-a)&=(f(0)-f(-2))/(0-(-2))\\\\&=(4-16)/(0+2)\\\\&=(-12)/(2)\\\\&=-6\end{aligned}`