MATH 23 – Midterm 1 Spring Semester 2007
Duration: 50 minutes
Instructions: Answer all questions, without the use of notes, books or calculators. Partial credit
will be awarded for correct work, unless otherwise specified. The total number of points is 100.
1. (20 pts) Given the function f(x, y ) = 9x2+y2−1
(a) Draw at least 2 cross-sections of f(x, y)with xfixed.
(b) Draw at least 2 contours.
(c) SKETCH the surface in a manner consistent with what you found above.
2. (18 pts) A plane p(x, y )is parallel to the vectors ~v1=~
i−~
j−~
kand ~v2= 2
~
i+~
j.
(a) Find the normal of this plane.
(b) Find the equation of p(x, y )if the plane goes through the point (0,2,−1).
3. (20 pts) Above the point (0,1) in the xy -plane, the plan tangent to a function f(x, y)is
p(x, y) = 5 + x−3y.
a) What is f(0,1)?
b) What is the gradient of f(x, y)at (0,1)?
c) What is the directional derivative of f(x, y)at (0,1) in the direction ~v =−
~
i−~
j?
d) If xand yare functions of time x(t) = sin tand y(t) = e2t, compute df
dt at t= 0.
4. (18 pts) Consider the function f(x, y ) = x3−3x+y2−y.
(a) Find and classify all the critical points of f(x, y ).
(b) Does this function have a global minimum over D={all x≥1and all y≥1}? If so,
find it, if not explain why.
5. (24 pts) Answer the following questions in no more than two lines of text. No computations
are required.
(a) What is the normal of the plane tangent to the surface g(x, y , z) = 0 at a point (x0, y0, z0)
on the surface?
(b) If you are told that the point (x0, y0)maximizes f(x, y)subject to g(x, y )=0, what can
you say about the partial derivatives or gradients of these functions at (x0, y0)?
(c) At a given point, the gradient of f(x, y )is ∇f=~
i+~
j. In what direction would you
have to move if you wanted to maintain a constant value of f(x, y)?
(d) If B(s, r )is the price of burritos, sthe price of salsa and rthe price of rice, what is the
meaning of ∂B
∂r ?
(e) If you know that limx→0f(x, mx) = limx→0f(x, k x2) = 2, what can you conclude about
the continuity of f(x, y)at the origin?
(f) If ~a ·(~
b×~c) = 0, what can you conclude about the vectors ~a,~
band ~c?
1