2. (10 pts) A company introduces a new product for which the number of units sold is S(t) = 200 5- 9 2+t. where is the time in months. Show that there is a month in the first year when the rate of change of the 450 number of units sold is equal to 7 units per month.The Mean Value Theorem. Let f be a function which satisfies the following hypotheses: 1. f is continuous on the closed interval [a, &] Note we no longer need 2. f is differentiable on the open interval (a, b) flaj = f( b) Then there is a number c in (a, b) such that f(b)-fla) b-a or, equivalently. /(b) - no) = S'(c)(b - a).mother at the start of the trip and when you arrive at your destination in 1.75 hours later. Your mother is a mathematician, and she immediately complains about your speeding. How did she know that you had to violate the speed limit at least once? Let S be the position function (it is continuous and and letv be the velocity function differentiable everywhere s ( 0) = 0 and s( 1. 75 ) = 142 The average velocity = s(1.75 )-5(0) 142-0 2 81.1 1.75 - 0 1.75 your average velocity over the course of your trip was about 81 mph By MVT your mother knew your instantaneous rate of change was about 81 mph at least once on your drive home The Mean Value Theorem