The formal definition for partial derivatives. It's
in the blue box on p641.
Example 5 illustrates how, in a particular context,
keeping track of units may help clarify the meaning of partial derivatives
in that context.
For 14.2, your primary goal should be to make sure you
understand Examples 2 and 3. You should become sufficiently fluent with
differentiation that you can find partial derivatives algebraically without
much difficulty.
In 14.3, pay special attention to:
The notion of local linearity. Roughly speaking,
for a differentiable function z = f(x,y), when we zoom in on a particular
point the graph eventually becomes very nearly linear.
Remember that "differentiable
at a point" does not mean
the same thing as "has partial derivatives at that point". The precise
definition of differentiability is covered in 14.8. As we mentioned
in class, it means that the linear approximation formula
does give a good linear approximation.
The formula for the tangent plane to z=f(x,y) at
the point where (x,y) = (a,b).
The tangent-plane formula as a formula for linear
approximation or local linearization.
Local linearity or linearization for more than two
variables
In 14.3, less important than the rest: pp 655 - 656 (about
differentials). If you're seeing this language/notation in other courses,
read up on it here. Otherwise don't worry about it, at least for now.