MAT 267 Test 2 -- Review
11.1 Understand the three ways of representing two variable functions (algebraic, numerical
and graphical) Be able to find the domain of two and three variable functions and give
the result in set notation and graph. Know how to graph level curves for two variable
functions.
11.3 Know how to find partial derivatives of two and three variable functions algebraically
and graphically. Be able to describe the meaning of the partial derivative for application
problems.
11.4 Be able to find the equation of the tangent plane to a surface at a given point. Know how
to use it to find the linear approximation (local linearization) of a two variable function.
Understand how to use differentials to estimate errors.
11.5 Know both versions of the chain rule: Case I. z = f (x(t),y(t)) and Case II: z = f
(x(s,t),y(s,t)). Be able to find implicit derivatives for two and three variable functions
11.6 Know how to find the directional derivative of a two variable function. Understand the
gradient, the meaning of its direction and its length. Be able to estimate the gradient
graphically. Be able to use the gradient of three variable functions to find the equation of
a tangent plane to a surface at a given point.
11.7 Know how to find the critical points of a two variable function. Be able to use the
second derivative test to classify the critical points and know its conclusion regarding
local minimums and maximums. To find absolute minimums and maximums on a given
region, first find the critical points inside the region, then express the boundaries as one
variable function(s) and find the critical points for those. Evaluate all of those points for
the function you are maximizing (minimizing). The largest value will be the absolute
maximum and the smallest value will be the absolute minimum.
12.1 Understand the definition of the double integrals. Understand how to divide a rectangular
region into subrectangles, and how to calculate double Riemann sums. Know how to find
the average value of a two variable function over a rectangular region. Be able to
calculate double integrals using iteration. Understand the statement of Fubini’s theorem.
12.2 Know how to find double integrals over type I regions (vertical slices) and over type II
regions (horizontal slices). Remember that you cannot simply interchange the integral
terms in these cases. IF a region can be interpreted both as type I and type II then you can
use this fact to change the order in the integration. In cases like these be able to rewrite
the region as the other type, and be able to change accordingly the end points of the
integral terms.
12.3 Know how to change a double integral into polar coordinates. Remember that you
always have to multiply by the additional term r in this case.