can you break your "y-dependence"? (that is, can
you handle functions of other variables, such as
x(y)? How about x(f) -- would you panic?)
Quiz tomorrow over related rates
Section 4.1: linear approximations and applications
linearizations
rely on the fact that as we zoom in on a function, the
approximation of the tangent line gets better and better....
We're approximating -- we're going to make errors. That's okay!
Errors are related to the second derivative: if the function is straight
as it passes a point, then the tangent line will be a perfect approximation; if
it's curving, then the tangent line approximation will tend to get worse as the
function curves away. The speed with which it curves away is related to the
size of the second derivative:
on the interval on which the approximation is used, and h is
the distance from x=a at the point of tangency.
Examples:
#4, p. 179
#8, p. 179
#18, p. 180
Questions over the Directed reading handout, Section 4.1 (html)(pdf)?