Question

Which exponential function has the greatest average rate of change over the interval [0,2]

Answer 1

Step 1

Given;

Step 2

A) The equation of the graph will be;

`\begin{gathered} y=ab^x \\ x=1,y=6 \\ 6=ab^1---(1) \\ x=0 \\ y=4 \\ 4=ab^0-----(2) \\ a=4 \\ \end{gathered}`
`\begin{gathered} From\text{ equation 1} \\ 6=4b \\ b=(6)/(4)=(3)/(2) \\ The\text{ function is} \\ y=4(1.5)^x \end{gathered}`

At the interval [0,2]

`\begin{gathered} y_1=4(1.5)^0 \\ y_1=4 \\ y_2=4(1.5)^2 \\ y_2=9 \\ Average\text{ rate of change=}(f(b)-f(a))/(b-a)=(9-4)/(2-0)=2.5 \end{gathered}`

B)

`\begin{gathered} 1.5=a(2)^0 \\ a=1.5 \\ y=1.5(2)^x \\ f(b)=6 \\ f(a)=1.5 \end{gathered}`
`Average\text{ rate of change=}(6-1.5)/(2-0)=2.25`

C)

`\begin{gathered} f(x)=3(1.6)^x \\ f(b)=7.68 \\ f(a)=3 \\ Average\text{ rate of change=}(7.68-3)/(2-0)=2.34 \end{gathered}`

D)

`Average\text{ rate of change=}((25)/(2)-8)/(2-0)=2.25`

Answer;

`Option\text{ A}`