Question

For the function f(x)=2x^2+1, find the average rate of change from x=2 to x=5

Answer 1

`\large\bf{\underline{\underline{\mathfrak{Question}:}}}`

For the function f(x)=2x^2+1, find the average rate of change from x=2 to x=5

`\large\bf{\underline{\underline{\mathfrak{Solution}:}}}`

Given that,

`\small\rm{f(x)=2x^2+1}`

If x = 2 or x = 5

Let's assume that,

`:{\Longrightarrow{\small{\rm{a_1=2}}}}`

`:{\Longrightarrow{\small{\rm{a_2=5}}}}`

By putting these values in the function we get,

`\small\bold{In\:case\:x=a_1}`

`:{\Longrightarrow{\small{\rm{f(a_1)=2(2)^2+1}}}}`

`:{\Longrightarrow{\small{\rm{f(a_1)=2(4)+1}}}}`

`:{\Longrightarrow{\small{\rm{f(a_1)=8+1}}}}`

`:{\Longrightarrow{\small{\rm{f(a_1)=9}}}}`

Next,

`\small\bold{In\:case\:x=a_2}`

`:{\Longrightarrow{\small{\rm{f(a_2)=2(5)^2+1}}}}`

`:{\Longrightarrow{\small{\rm{f(a_2)=2(25)+1}}}}`

`:{\Longrightarrow{\small{\rm{f(a_2)=50+1}}}}`

`:{\Longrightarrow{\small{\rm{f(a_2)=51}}}}`

Now,

To find the average rate of change concept used:

`\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{\boxed{\large{\rm{(f(a_2)-f(a_1))/(a_2-a_1)}}}}`

By putting the values; we get,

`:{\Longrightarrow{\small{\rm{(51-9)/(5-2)}}}}`

`:{\Longrightarrow{\small{\rm{(42)/(3)}}}}`

`:{\Longrightarrow{\small{\rm{(14)/(1)}}}}`

`{\therefore{\small{\rm{14\:is\:the\:average\:rate\:of\:change.}}}}`

Answer 2

14

Explanation:

In order to find the average rate of change, we can apply the formula:

`\displaystyle{\text{Average rate of change} = (f(b)-f(a))/(b-a)}`

Let b = 5, a = 2. Thus, substitute in the values:

`\displaystyle{\text{Average rate of change} = (f(5)-f(2))/(5-2)}\\\\\displaystyle{\text{Average rate of change} = (\left(2(5)^2+1\right)-\left(2(2)^2+1\right))/(3)}\\\\\displaystyle{\text{Average rate of change} = (51-9)/(3)}\\\\\displaystyle{\text{Average rate of change} = (42)/(3)}\\\\\displaystyle{\text{Average rate of change} = 14}`

Hence, the average rate of change is 14.