MATH 23 – Midterm 1 Fall 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 70.
1. (13 pts: 7, 6) f(x, y) = p16 −4x2−y2.
(a) Draw a contour map of fshowing at least three level curves. Remember to label your
axes and level curves.
(b) Find a vector function (or parametric equations) that represents the intersection curve
of the graph z=f(x, y)and the plane x= 1.
2. (15 pts: 2, 2, 5, 3, 3) Consider the plane Π:2x−y+ 3z= 0 and the vector ~v =<2,−2,4>.
(a) Find a normal vector ~n to the plane Π.
(b) Does the plane πpass through the origin? Why?
(c) Find proj~n~v, the vector projection of ~v onto ~n from part (a). (If you cannot solve part (a),
use ~n =<1,0,3>.)
(d) Find the distance between the point (2,−2,4) and the plane Π.
(e) What can you say about the direction of ~v −proj~n~v ?
3. (15 pts: 5 each) In a contour map of the function f(x, y), the point (0,2) lies on the level curve
f(x, y) = 5. We also know that fx(0,2) = −3and fy(0,2) = 4.
(a) Find the direction in which f(x, y)increases fastest at (0,2), and find the maximum rate
of increase.
(b) Find one tangent vector to the level curve f(x, y ) = 5 at (0,2).
(c) Find an equation of the tangent plane to the graph z=f(x, y )at the point above (0,2).
4. (15 pts: 10, 5) Consider the function f(x, y) = x−x2−y2.
(a) Find and classify all critical points of f(x, y ).
(b) Find the absolute maximum and absolute minimum values of f(x, y )over
D={(x, y)|x2+y2≤1}.
5. (12 pts: 3 each) Answer the following questions in no more than two lines of text.
(a) Is it possible for a function fto have fx(x, y ) = 3x2−yand fy(x, y) = x3−1as partial
derivatives? Explain why.
(b) Write down a vector function ~r(t)(or parametric equations) for a space curve whose
curvature is zero everywhere.
(c) If B(s, r )is the price of burritos, sthe price of beans and rthe price of rice, what is the
meaning of ∂B/∂s ?
(d) If you know that limx→0f(x, mx) = limx→0f(x, k x2) = 2, what can you conclude about
lim(x,y)→(0,0) f(x, y)?
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