If a straight line passes through two points #(x_1, y_1)# and #(x_2, y_2)# where #x_2 > x_1#. then the run is #(x_2 - x_1)#, the rise is #(y_2 - y_1)# and the slope #m# of the line is defined as:
#m = (Delta y)/(Delta x) = # run / rise #= (y_2 - y_1)/(x_2 - x_1) #
If instead of a straight line, we have a function #f(x)# which is continuous and otherwise well-behaved over the interval #[x_1, x_2]# and #f(x_1) = y_1# and #f(x_2) = y_2# then the average slope of #f(x)# over the interval #[x_1, x_2]# is also #m = (y_2 - y_1)/(x_2 - x_1)#
If you are familiar with the terminology, we are basically evaluating the integral of the derivative of #f(x)# over the interval #[x_1, x_2]#, then dividing by the length of the interval. This is like adding up the slopes at each point and dividing by the number of measurements to get the average.
