Find the average rate of change of the function on the interval specified for real number

Question

h(x) = 3x+4 on [2,2+h]

j(x) = 3x^3 on [1,1+h]

Answer 1

Hi Brenden w.,

The average rate of change is: Δh/Δx = (h_f - h_i)/(x_f - x_i).

For h(x) = 3x + 4 on [2, 2+h], (where x_i equals 2 and x_f equals 2+h):

h_i(2) = 3(2) + 4 = 10

h_f(2+h) = 3(2+h) + 4 = 3h + 10

For Δh/Δx = (h_f - h_i)/(x_f - x_i):

(3h + 10 - 10)/(2 + h - 2) = 3h/h = 3 (the average rate of change).

The average rate of change is: Δj/Δx = (j_f - j_i)/(x_f - x_i):

For j(x) = 3x³ on [1, 1+h]:

j_i(1) = 3(1)³ = 3

j_f(1+h) = 3(1+h)³ = 3h³ +9h² + 9h + 3

For Δj/Δx = (j_f - j_i)/(x_f - x_i):

(3h³ +9h² + 9h + 3 -3)/(1+h-1) = (3h³ +9h² + 9h)/h = 3h² + 9h +9 (the average rate of change).

I hope this helps, Joe.

Answer 2

h(2+h) = 3(2+h)+4 = 6 + 3h + 4 = 3h + 10

h(2) = 10

AROC = {(3h+10) - 10 }/ {(2+h - 2)} = 3h/h = 3

j(1+h) = 3(1+h)^3

j(1) = 3

AROC = [3(1+h)^3 - 3]/(1+h-1)

= 3 [( 1+h)^3 - 1 ]/h

= (h^3 + 3h^2 +3h +1 - 1 )/ h

= h^2 + 3h + 3