Name a result from class that implies that one cannont accelerate from driving 60 miles/hour to 70 miles/hours without at some point 65 miles/hour
The correct answer and explanation is:
The result from class that implies one cannot accelerate from 60 miles per hour to 70 miles per hour without at some point reaching 65 miles per hour is known as the Intermediate Value Theorem (IVT).
The IVT is a fundamental concept in calculus, stating that for any continuous function f(x)f(x), if f(a)f(a) and f(b)f(b) are values of the function at two points aa and bb, and f(a)≠f(b)f(a) \neq f(b), then for any value LL between f(a)f(a) and f(b)f(b), there exists some cc in the interval [a,b][a, b] such that f(c)=Lf(c) = L. This theorem applies to real-world situations like the one described with speed.
In the case of driving from 60 miles per hour to 70 miles per hour, the speed function is continuous. The car is accelerating smoothly without jumps or interruptions in speed. The initial speed is 60 mph, and the final speed is 70 mph. According to the IVT, since 65 mph lies between 60 mph and 70 mph, there must be some point during the acceleration where the car’s speed reaches exactly 65 mph.
This result holds because the speed of the car changes continuously; it cannot “skip” over any intermediate speed. So, if a driver accelerates from 60 to 70 mph, at some point they will pass through every speed in between, including 65 mph. This is a practical demonstration of the Intermediate Value Theorem in a real-world context, where the behavior of continuous functions can be observed in motion.