STUDY GUIDE FOR MIDTERM 3
1. There will be no graphing of surfaces requirement on the midterm. However, when working
through some of the homework assignments or examples in the Openstax text, take a look at
the equation z = f(x,y) and the illustrated surface, if applicable. For your personal enrichment
and your own education, you want to develop some idea of how these functions look like.
In particular, we did talk a little bit about traces of surfaces, which are the intersections with
these surfaces and the X-Y plane or the X-Z plane or the Y-Z plane, or any plane parallel to the
aforementioned. The intersections of the surface with these various planes help you reconstruct
a 3D image, just as an MRI scan of slices of your brain leads to a full 3D picture of your brain.
The intersection of the surface z =f(x,y) with a plane that is parallel to the X-Y plane (z = c being
it equation) but projected onto the X-Y plane (the floor) has the special name of “level curve”. It
tells you that the trace above it (at z =c) is a path on the surface along which the height does not
change. In a topographic map context where you can consider the surface as a terrain or
landscape, these level curves would be referred to as “height contour lines”.
2. You need to know/memorize the definition and be able to explain the “formula” if applicable for
the following (no proofs):
a) The limit of a function of two variables; bring out the meaning of the disk with
radius δ, the meaning of ε, and the necessity to check all paths approaching the
reference point (a,b)
b) Continuity of a function of two variables.
c) The partial derivatives AND the (directional) derivative in any other direction
besides the X- and Y-direction. Again, for the partial and directional derivative alike
you have to be able to write down and explain both their definition and the formula
to calculate them.
d) The two- and three-dimensional gradient 𝛻𝑓(𝑥, 𝑦) and 𝛻𝐹(𝑥, 𝑦, 𝑧).
e) The critical points of a function of two variables.
f) The relative or local maximum or minimum on an open region R in the X-Y plane and
the absolute maximum or minimum on a closed region R in the X-Y plane.
3. You need to know/memorize the laws of limits and, if applicable, indicate how you used them in
the evaluation of a particular limit (not a likely question).
4. Study the PROOF or derivation of the equation for the tangent plane to an implicitly defined
surface F(x,y,x) = 0 and an explicitly defined surface f(x,y) = 0.
5. You have to be able to calculate/evaluate existing limits of f(x,y) by applying the limit laws or be
able to show that certain limits do not exist because of path-dependent results. You also have to
be able to establish continuity for a function of two variables.
6. You have to be able to calculate/evaluate partial derivatives of f(x,y), and correctly apply the
chain rule for functions of multiple variables (as the HW assignments show, these derivatives
could illustrate physical or real-world situations, such as rates-of-change of a cylinder’s volume
with radius, thus becoming a mini “word-problem”).
7. You have to be able to calculate the gradient of a function f(x,y), as well as the directional
derivative, and find the direction of steepest ascent/descent and the direction of no change
(related to level curves).
8.
You have to be able to find the equation of a tangent plane to a surface F(x,y,z) = 0 and to
surface z= f(x,y).
9. You have to be able to give the (linear) approximation to a function f(x,y) = f(a+Δx, b + Δy) based
on the tangent plane equation L(x,y), as well as the approximation to the change Δz = f(x,y) –
f(a,b) by calculating dz = L(x,y) – f(a,b).
10. You have to be able to find critical points for a function f(x,y) defined on an open region R in the
X-Y plane, and then decide whether they correspond to local extrema or absolute extrema.
Likewise, be able to discuss absolute extrema on a closed region R in the X-Y plane. The secondderivative test will be PROVIDED to you on the exam.