Questions as you prepare for the quiz? See me in office hours or send
me e-mail or post on the Q&A folder
in the FirstClass conference. To ask about a particular problem
from the text, give page or section number, give problem number,
and make your question as specific as you can. In the Q&A
conference, answer each other's questions as you can.
Suggestions for Practice Problems (don't
attempt to work all of them):
13.4 p634: some from 1-5, 6, 7, 12, 13,
some from 15-17, 18 or 19
Interpreting partial derivatives for
real-world functions
Finding partial derivatives
algebraically
Using contour diagrams or function graphs
to decide signs of partial derivatives
Using contour diagrams to estimate partial
derivatives
Using sections for fixed x or y to estimate
partial derivatives
Finding equation for plane tangent to
surface z = f(x,y) at point where (x,y) = (a,b)
Definition for directional
derivative
Notation used for directional
derivative
Interpreting directional derivatives for
real-world functions
Finding directional derivatives
algebraically (use formula mentioned a few bullets
down)
Using contour diagrams or function graphs
to decide sign of directional derivative
Using sections corresponding to given
direction vector to get an "eyeball" estimate for directional
derivative
Definition of gradient
Geometric properties of gradient (what
direction does it point? what information does its magnitude
give us?)
Directional derivative for f at point P in
direction of unit vector u is same as dot product of
grad f(P) with direction vector u
For v a non-unit vector, directional
derivative in direction of v is understood to be
directional derivative in direction of unit vector with same
direction as v
Using geometric characterization of
gradient to find directions of no change, direction of greatest
increase for f, direction of greatest decrease, and rates of
greatest increase and decrease
Notations used for higher-order partial
derivatives
Not included but good to know: using
contour diagrams to decide sign of second-order partials; using
sections for fixed x or y to decide sign of second-order
partials
Linear approximation L(x,y) for function
f(x,y) near point (a,b)
Quadratic approximation Q(x,y) for function
f(x,y) near point (a,b) [or second-order Taylor polynomial
for f with basepoint (a,b)]
Criterion sufficient to guarantee
second-order mixed partials for f(x,y) are equal (and ways we
use it)
Evaluating determinants for square
matrices
Algebraic characterization of cross
product; using it to find cross products
Geometric characterization of cross
product; using it to find simple cross products such as
i X j; using it to find areas of triangles and
parallelograms
Algebraic properties of cross product (see
the box on p631)
Finding equation for the plane through
three points (not on a line)