Question

F

Answer 1

Answer:

The average rate of change of $f(x) = x^2 +6x+13$  is 12

The average rate of change of $f(x) =- x^2 +6x+13$ is 10

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.

We can calculate the average rate of change between the 2 points  by

$\frac{f(b) - f(a)}{b -a}$-------------------(1)

(1) The average rate of change of the function $f(x) = x^2 +6x+13$  over  the interval 1 ≤ x ≤ 5

f(a) = f(1)

$f(1) = (1)^2 +6(1) + 13$

f(1) =1+6+13

f(a) =  20---------------------(2)

f(b) = f(5)

$f(5) = (5)^2 +6(5)+13$

f(5) = 25 +30 +13

f(5) = 68-----------------------(3)

The average rate of change between (1 ,20) and (5 ,68 ) is

Substituting eq(2) and(3) in (1)

=$\frac{f(5) - f(1)}{5-1}$

=$\frac{68 -20}{5-1}$

= $\frac{48}{4}$

=12

This means that the average of all the slopes of lines tangent to the graph of f(x) between  (1 ,20) and (5 ,68 ) is 12

(2) The average rate of change of the function  $f(x) = -x^2 +6x+13$ over  the interval -1 ≤ x ≤ 5

f(a)  = f(-1)

$f(1) = (-1)^2 +6(-1) + 13$

f(1) =1-6+13

f(1) = 8---------------------(4)

f(b) = f(5)

$f(5) = (5)^2 +6(5)+13$

f(5) = 25 +30 +13

f(5) = 68-----------------------(5)

The average rate of change between (-1 ,8) and (5 ,68 ) is

Equation (1) becomes

$\frac{f(5) - f(-1)}{5-(-1)}$

On substituting the values

=$\frac{68 - 8}{5-(-1)}$

=$\frac{60}{5+1}$

=$\frac{60}{6}$

= 10

This means that the average of all the slopes of lines tangent to the graph of f(x) between  (-1 ,8) and (5 ,68 ) is 10