# "[Note, the average rate of change is defined over a closed" #
# "interval, for example, over the interval" \ [ -3, 5 ]. \ \ "It is not" #
# "defined over an open interval, such as the interval" \ (-7, 10)." #
# "When looking at the definition of this, as below," #
# "this necessity will be clear. (I'll assume you meant closed" #
# "intervals in your question.) ]" #
# "Recall the definition of the average rate of change of a" #
# "function, over a closed interval:" #
# "average rate of change of" \ \ f(x), \ "over the interval" \ \ [ a, b ] \ = #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad { f(b) - f(a) }/{ b - a }. #
# "So, for our examples here:" #
# "1) average rate of change of" \ f(x) = x^3 + 1," #
# \qquad \qquad "over the interval" \ [ 2, 3 ] = #
# \qquad \qquad \qquad { f(3) - f(2) }/{ 3 - 2} \ = \ { [ 3^3 + 1 ] - [ 2^3 + 1 ] }/{ 3 - 2 } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ = \ { [ 28 ] - [ 9 ] }/{ 1 } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ = \ 28 - 9 \ = \ 19. #
# "2) average rate of change of" \ f(x) = x^3 + 1,#
# \qquad \qquad "over the interval" \ [ -1, 1 ] = #
# \qquad \qquad \qquad { f(1) - f(-1) }/{ 1 - (-1) } \ = \ { [ 1^3 + 1 ] - [ (-1)^3 + 1 ] }/{ 1 - (-1) } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ = \ { [ 2 ] - [ -1 +1 ] }/{ 1 + 1 } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ = \ { 2 - 0 }/2 \ = \ 2/2 \ = \ 1. #
# "So, summarizing:" #
# "1) average rate of change of" \ f(x) = x^3 + 1, "over" \ [ 2, 3 ] \qquad \ = 19. #
# "2) average rate of change of" \ f(x) = x^3 + 1, "over" \ [ -1, 1 ] = 1. #
