17. (i) name a result from class that implies that one cannot accelerate from driving 70 miles/hour to 80…
17. (i) name a result from class that implies that one cannot accelerate from driving 70 miles/hour to 80 miles/hour without at some point driving 75 miles/hour. (ii) name a result from class that implies that if your average velocity was 75 miles/hour on your drive from manhattan to topeka, then your instantaneous velocity was 75 miles/hour at some point during your drive.
Answer
Brief Explanations:
(i) The Intermediate - Value Theorem in calculus states that if a function (in this case, the velocity function which is assumed to be continuous) takes on two values, it must take on every value in between. Since velocity is a continuous function (in normal driving scenarios), when accelerating from 70 mph to 80 mph, it must pass through 75 mph. (ii) The Mean - Value Theorem in calculus says that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the open interval where the instantaneous rate of change (instantaneous velocity) is equal to the average rate of change (average velocity). So, if the average velocity from Manhattan to Topeka is 75 mph, there is a point where the instantaneous velocity is 75 mph.
Answer:
(i) Intermediate - Value Theorem (ii) Mean - Value Theorem