Question

Consider the function p(x)= x^2 + 6x + 5
What is the average rate of change from x = 2 to x = 3

Answer 1

Answer:

11.

Step-by-step explanation:

We have been given a function $p(x)= x^2+6x+5$ and we are asked to find the the average rate of change from x = 2 to x = 3.

We will use average rate of change formula to solve our given problem.

$\text{Average rate of change}=\frac{f(b)-f(a)}{b-a}$

Upon substituting our given values in above formula we will get,

$\text{Average rate of change}=\frac{p(3)-p(2)}{3-2}$

Let us find p(3) by substituting x=3 is our given formula.

$p(3)=3^2+6*3+5$

$p(3)=9+18+5$

$p(3)=32$

Let us find p(2) by substituting x=2 is our given formula.

$p(2)=2^2+6*2+5$

$p(2)=4+12+5$

$p(2)=21$

Let us substitute values of our function at x = 2 and x = 3 in average rate of change formula.

$\text{Average rate of change}=\frac{32-21}{3-2}$

$\text{Average rate of change}=\frac{11}{1}$

$\text{Average rate of change}=11$

Therefore, the average rate of change for our given function from x=2 to x=3 is 11.

Answer 2

Answer:

= 11

Step-by-step explanation:

The  average rate of change of f(x) over an interval between 2 points (a ,f(a)) and (b ,f(b)) is the slope of the secant line  connecting the 2 points.

Therefore;

= (f(b)- f(a))/(b-a)

a = 2, b= 3

f(2) = 2² + 6(2) +5

f(2) =21

f(3) = 3² + 6(3) + 5

f(3) = 32

Thus; the rate of change will be;

= (32-21)/(3-2)

= 11

This means that the average of all the slopes of lines tangent to the graph of f(x) between (2,21) and (3 ,32) is 11.