Q. 77 - Calculus

Question

Explain in practical terms what the Extreme Value Theorem says about each continuous function defined in Exercises $75-77$. Then explain in practical terms what the Intermediate Value Theorem says in each situation.  

The number of gallons of gas in Phil’s new station wagon $t$ days after he bought it is given by the function $g\left(t\right)$. When he purchased the station wagon one year ago, the tank had $19$ gallons of gas in it. Today he ran out of gas. 

Step-by-Step Solution

Answer

The Extreme Value Theorem tells us that there is some time in the past one year when Phew's wagon contained the most gas and at some time it contained the least gas.

The Intermediate Value Theorem tells us that for every number of gallons of gas $K$ between $0$ and $19$ gallons, there is some time $t$ in the past one year for which $g\left(t\right)=K$.  

Step 1. Given Information

Amount of gas in the tank one year ago $=19$ gallons

Amount of gas in the tank now $=$empty

$g\left(t\right)$ is the function that denotes the number of gallons of gas in Phil’s new station wagon $t$ days after he bought it.

We need to find what the Extreme Value Theorem and Intermediate Value Theorem says in this situation.  

Step 2. Extreme Value Theorem

The function $g\left(t\right)$ should be continuous because amount of gas changes continuously over time and cannot jump from one value to another.

The Extreme Value Theorem tells us that there is some time in the past one year when Phew's wagon contained the most gas and at some time it contained the least gas.

Step 3. Intermediate Value Theorem

The Intermediate Value Theorem tells us that for every number of gallons of gas $K$ between $0$ and $19$ gallons, there is some time $t$ in the past one year for which $g\left(t\right)=K$.