G.1.a)

One version of the Race Track Principle:

Horses:

If two horses start a race at the same point, then the faster horse is always ahead.

Functions:

If and for , then
         for .

Put
        and
and look at:

Use the Race Track Principle and what you see above to explain why
         for
provided .

Heavy Tip:

You are given that.
This tells you that
       
so
       

G.1.b)

Another version of the Race Track Principle:

Horses:

If two horses start at the same point and they run at nearly the same speed for the whole race, then they run very close together all the way and are likely to end the race in a photo finish.

Functions:

If and
     is nearly the same as for
then
     is nearly the same as for .

Now look at:

Both functions start their race at together:

Here's a plot of the instantaneous growth rates of both functions for :

Use this plot and the Race Track Principle to determine so that and are nearly the same for .
Check yourself with a plot.

G.1.c.i)

Here's another version of the Race Track Principle:

Horses:

If two horses end the race in a dead heat, then the faster horse was behind all the time except at the finish line.

Functions:

If and
     for
then
     ........... for  .

Delete and fill in the blank above.

G.1.c.ii)

Go with and and look at:

Remembering that spends all its life oscillating between and , use the Race Track Principle and what you see above to explain why
         for .

G.1.d.i)

Another version of the Race Track Principle:

Horses:

If two horses run at exactly the same speed for the whole race, and they are tied at one point of the race, then they are tied throughout the race.

Functions:

Suppose is one point with and and
     for all 's with ;  
then
     ................. for .

Delete and fill in the blank above.

G.1.d.ii)

Look at this:

Explain why this gives a calculus explanation of the identity
         for all the 's.

G.1.e.i)

Here's yet another version of the Race Track Principle:

Horses:

If two horses run at exactly the same speed and at one point of the race one horse is lengths ahead, then the same horse was lengths ahead throughout the race.

Functions:

Suppose is one point and
      
and
     for all 's;
then
     ..............  
for all 's.
Fill in the blank above.

G.1.e.ii)

Look at this:

Explain why this gives a calculus explanation of the identity
         for all the 's.