Question

Which of the given functions have an average rate of change of -12 on the interval [0, 2]?

Answer 1

Answer:

The correct answers are:

1)  $f(x)=-5^x$

2) $f(x)=7-12x$

3) $f(x)=-3(3^x)$

Step-by-step explanation:

The rate of change of a function in the interval [a,b] is calculated by:

$Rate\ of\ change=\dfrac{f(b)-f(a)}{b-a}$

We have: [a,b]=[0,2]

1)

$f(x)=-5^x$

Hence, we have:

$f(2)=-5^2\\\\\\f(2)=-25$

and

$f(0)=-5^0\\\\\\f(0)=-1$

Hence, the rate of change in the interval [0,2] is:

$=\dfrac{-25-(-1)}{2-0}\\\\\\=\dfrac{-24}{2}\\\\\\=-12$

Hence, option: (1) is correct.

2)

$f(x)=7-12x$

We know that for any linear function of the type y=mx+c

The rate of change in any interval =m

Here we have: m= -12

Hence, Rate of change= -12

Option: 2 is correct.

3)

$f(x)=-3(3^x)$

$f(2)=-3\times 3^2\\\\\\f(2)=-27$

and

$f(0)=-3\times 3^0\\\\\\f(0)=-3$

Hence, Rate of change is:

$=\dfrac{-27-(-3)}{2-0}\\\\\\=\dfrac{-24}{2}\\\\\\=-12$

Hence, option: 3 is correct.

4)

$f(x)=-12^x$

$f(2)=-12^2=-144$

$f(0)=-12^0=-1$

Hence, rate of change is:

$=\dfrac{-144-(-1)}{2-0}\\\\\\=\dfrac{-143}{2}$

Option: 4 is incorrect.

5)

$f(x)=-\dfrac{1}{12}x+9$

It is linear function.

$Hence\ ,\ rate\ of\ change=\dfrac{-1}{12}$

Hence, option: 5 is incorrect.

6)

$f(x)=12x-\dfrac{2}{3}$

Again it is a linear function with rate of change: 12

Hence, option: 6 is incorrect.

Answer 2

Answer:

The answer is f(x) = 7 - 12x

Step-by-step explanation:

f(x) = 7 - 12x has average rate -12 on the interval [0 , 2] because:

f(0) = 7 - 12(0) = 7

f(1) = 7 - 12(1) = -5

f(2) = 7 - 12(2) = -17

The difference between 7 , -5 , -17 is constant and equal -12

∴ The average rate of change is -12