Section Summary: 11.2

1. Definitions

   None to speak of.

2. Theorems

   None.

3. Properties/Tricks/Hints/Etc.

   It is important to distinguish between the derivatives with respect to the parameter t and with respect to x: in general

   

   One reflects the time rate of change in y; the other indicates the tangential rate of change in the curve of y=f(x).

4. Summary

   Suppose that we can write our parametric curves as y=F(x):

   

   For example,

   

   Then, using the chain rule,

   

   or

   

   Problem #40, p. 688 shows that this is often possible: ``If f' is continuous and for , show that the parametric curve x=f(t), y=g(t), can be put in the form y=F(x).''

   f' must be of fixed sign (positive or negative) on [a,b], as it is continuous and never zero (by the intermediate value theorem).

   Thus f is strictly monotonic (either increasing or decreasing) on the interval, and hence one-to-one on [a,b].

   Therefore exists and is continuous on [f(a),f(b)], so

   

   and

   

   That is,