T.2.a) Running an exponential through two data points

Here are the two points and and the line function going through them.

Add to this plot the plot of the exponential function
       
that goes through the same two points.

Answer:

Call the exponential function that goes through these two points:

is a line function that goes through:

So is given by:

But
       ,
so is given by:

This is the exponential function
       
with    and .
Here comes the plot:

Lookin' fine.

T.2.b.i)   and exponential data analysis

Information relating the variable and the function may consist of nothing more than some measurements indicating the observed value of for various values of .  A quick plot of the data usually reveals to the alert eye whether the functional relationship of in terms of is of the form
       .
If the plot of the data looks like this:

Then you are happy fitting these data with a line:

Not bad.
Sometimes the points do not line up in a straight line:

This looks suspiciously exponential.
How do you better eyeball an exponential relationship?
How do you come up with a compromise exponential function
       
whose plot runs through or near the data points above?

Answer:

Here are the data points in the form :

The following instruction keeps the first slots fixed but replaces the corresponding second slot by its base logarithm:

Now plot the logdata points:

These points nearly line up in a perfect straight line.

This is a dead give-away that the data display a strong exponential relationship.

Let's agree that
       
is a compromise exponential function whose plot does a good job of going with the flow of these data points.
is a compromise line function that flows with:

So is given by:

But
       ,
so is given by:

This is the exponential function
       
with    and .
Here comes the plot:

No problem at all.

T.2.b.ii)

If there were an entry in the exponential data table in part i), what would you bet that would be?

Answer:

Go with the exponential fit and plug in :

T.2.b.iii)

If there were an entry in the exponential data table in part i), what would you bet that would be?

Answer:

Go with the exponential fit and plug in :