IVT Activity

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Recap of Theorem

In class, we studied the Intermediate Value Theorem:

IVT If $f$ is a continuous function on the interval $[a,b]$ , then $f$ takes every value between $f(a)$ and $f(b)$ . In other words, given any value $k$ between $f(a)$ and $f(b)$ , there is some input $c$ in $(a,b)$ such that $f(c) = k$ .

This theorem has an important corollary concerning the existence of roots of functions. Remember a root is a point $x=c$ such that $f(c) =0$ .

If $f$ is a continuous function on the interval $[a,b]$ , and if $f$ takes opposite signs at the endpoints $a$ and $b$ , then $f$ has a root on the interval $[a,b]$ .

In this module we will prove the surprising fact that there exists a pair of points directly across from each other on the equator of the earth which have the same exact temperature.

Video Prep for Activity

The following video will prepare you for the rest of the activity.

Application

As stated in the video, our goal is to show that there is a pair of antipodal points on the equator with the same temperature. We will break this down into individual steps:

Temperature Function

Let $T(t)$ be the function that measures the temperature at every point on the equator. It inputs points on the equator (in terms of an angle parameter $t$ ) and outputs the temperature at that point.

What types of discontinuity can the function $T(t)$ have? (Select all that apply.)

Removable Discontinuity Jump Discontinuity Infinite Discontinuity There are no possible discontinuities

A removable discontinuity for $T(t)$ would imply the temperature fails to exist at some point.
A jump discontinuity for $T(t)$ would imply that the temperature jumps between two infinitesimally close points on the equator.
An infinite discontinuity would means it gets infinitely hot or cold at some point on the equator.

If the value of $T(0)=80$ (degrees Fahrenheit), then the value of $T(2\pi ) = \answer {80}$ .

Difference Function

We are trying to find two antipodal points on the equator with the same temperature, i.e. we are looking for a point $t$ such that $T(t) = T(t+\pi )$ . To apply the above corollary though, we want to make it so that we are looking for a root of a function instead.

Define a new function $D(t)$ which measures the difference between the temperature at $t$ and the temperature at its antipodal point, i.e. $D(t) = T(t) - T(t+\pi ).$

The function $D(t)$ isis not continuous.

Notice $D(t)$ is the difference of two continuous functions.

A root of $D(t)$ is a point $t$ with $D(t) = \answer {0}$ .

What does it mean if $D(t) = 0$ (i.e. if we are at a root of $D(t)$ )?

The temperature at location $t$ is equal to $0$ . The temperature at location $t+\pi$ is equal to $0$ . The temperatures at locations $t$ and $t+\pi$ are equal. The temperatures at locations $t$ and $t+\pi$ are not equal.

Sign Condition from Corollary

If Quito $(t=0)$ and its antipodal point $(t=\pi )$ have the same temperature then we are done. Indeed, we have found a pair of antipodal points on the equator with the same exact temperature. Let’s therefore assume they have different temperatures. Moreover, let’s assume that the temperature at Quito is higher than at its antipode, i.e. $T(0) > T(\pi )$ .

Under this assumption what can you say about the value $D(0)$ ?

$D(0)$ is positive $D(0)$ is negative $D(0) = 0$

Under this assumption what can you say about the value $D(\pi )$ ?

$D(\pi )$ is positive $D(\pi )$ is negative $D(\pi ) = 0$

Remember $T(2\pi ) = T(0)$ .

How would these signs change if we had made the opposite assumption that the temperature at Quito is lower than at its antipode?

There would be no change. The signs would flip for each. Both signs would be positive. Both signs would be negative.

Notice we have now examined all possible cases: either $T(0) = T(\pi )$ , $T(0)>T(\pi )$ (which was the original assumption), or $T(0) < T(\pi )$ (which was the assumption in the last question).

Applying Corollary

We are now ready to apply the above corollary to the function $D(t)$ . We have already checked that $D(t)$ is continuous and argued that a root of this function corresponds to a pair of antipodal points with the same exact temperature.

To which interval should we apply the corollary?

$[0,\pi ]$ $[\pi /2,3\pi /2]$ $[\pi /2,2\pi ]$ $[0,2\pi ]$

Explain why we can now use the corollary to argue that there are a pair of points on the equator with the same temperature.

If $T(0) \neq T(\pi )$ , then $D(0)$ and $D(\pi )$ have opposite signs. Since $D(t)$ is continuous, we can use the corollary on the interval $[0,\pi ]$ to get a point where $D(t)=0$ , which is a point where $T(t) = T(t+\pi )$ , which is what we wanted.

Does the corollary tell us which pair of points on the equator have the same temperature?

Yes No

Recall that the Intermediate Value Theorem is an existence theorem. It says that there exists an input $c \in (a,b)$ such that $f(c) = k$ , but it does not tell us where on the interval $(a,b)$ this input $c$ occurs. Likewise, we do not know where on the equator this pair of antipodal points live, we have only proved the existence of such a pair of points.