Question

1. (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.

Explanation:

Step1: Recall relevant calculus results

In calculus, the Intermediate - Value Theorem and the Mean - Value Theorem are relevant here.

Step2: Identify the theorem for part (i)

The Intermediate - Value Theorem for continuous functions states that if a function \(y = f(x)\) is continuous on a closed interval \([a,b]\), and \(k\) is a number between \(f(a)\) and \(f(b)\), then there exists at least one number \(c\in(a,b)\) such that \(f(c)=k\). In the context of velocity, if velocity is a continuous function (which is a reasonable assumption in most real - world driving scenarios), to go from a velocity of 70 miles/hour to 80 miles/hour, since 75 miles/hour is between 70 and 80, there must be a point where the velocity is 75 miles/hour.

Step3: Identify the theorem for part (ii)

The Mean - Value Theorem states that if \(y = f(x)\) is continuous on the closed interval \([a,b]\) and differentiable on the open interval \((a,b)\), then there exists at least one number \(c\in(a,b)\) such that \(f^{\prime}(c)=\frac{f(b)-f(a)}{b - a}\). In the context of velocity, if the position function \(s(t)\) is continuous on \([a,b]\) (the time interval of the drive) and differentiable on \((a,b)\), and the average velocity \(\frac{s(b)-s(a)}{b - a}=75\) miles/hour, then there is a point \(t = c\) where the instantaneous velocity \(s^{\prime}(c) = 75\) miles/hour.

Answer:

(i) Intermediate - Value Theorem
(ii) Mean - Value Theorem