- Your homework is returned (and was a little tricky). Graded
- 4, p. 567
- 16
- 32
- Next week we'll start polar coordinates. Please read 10.3 and 10.4 in advance.
- We start with the the chain rule, and the derivative
dy/dx. We must express this derivative in terms of time, which
is the actual independent variable in parametric curves:
- Tangent lines and Arc Length
-
Suppose a particle is following the parametric curve
- the point is
and
- the slope is
evaluated at
.
- Examples:
- #1, p. 675
- #3 (we need a point and a slope, dy/dx)
- #20 (where is dy/dx=0 or
?)
- #41
in time.
The tangent line to the curve at
We can compute how far the particle has travelled during the interval
easily
using the dirt formula, 
(in its modified
form 
).
In this case, the rate is just the speed. So we compute the integral
This is actually just a re-expression of the arc length formula:
and
.
But arc length may be different from the distance the particle travelled: a particle can revisit many sections of the curve y(x) -- so once again we need to be careful to distinguish between the independent variable of interest (whether x or t).
Let's calculate how far a particle travelled when parameterized by
and 
, with 
What happens to our integral if we double the time interval?
- the point is