On a cross-country journey the distance travelled at a given time by a wagon caravan is given by the equation:
p(t) = t^5 + 2t^3 - t^2 + 12
Starting at t = -1, one of the wagons starts moving on its own at a constant rate, and rejoins the caravan at t = 1. What is the equation that this lone wagon follows during this time interval?
How far did the caravan go between t = -1 and t = +1?
p(-1) = -1 -2 -1 +12 = 8
p(1) = +1 +2 -1 +12 = 14
14-8 = 6 or 3 miles (or furlongs or whatever) per hour (or minute or light year or whatever) which is constant slope of equation of form
p = m t + b
so
form is p = 3 t + b
when t = -1, p = 8
so
8 = 3 (-1) + b
11 = b
so
p = 3 t + 11