MATH 23 – Practice 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 50.
1. (10 pts) Given the function z2=x2+y2−1
a) Draw cross-sections of f(x, y)with xfixed.
b) Draw at least 3 contours.
c) Noting the symmetry in xand y,SKETCH the surface in a manner consistent with
what you found above.
2. (9 pts) Given the vector ~
N=~
i−2~
j+ 3~
k,
a) Find the equation of the plane perpendicular to ~
Nand going through p= (0,2,4).
b) Decompose the vector ~v =~
j−~
kinto two parts ~a and ~
bsuch that ~a is parallel to ~
N,~
b
is perpendicular to ~
Nand ~v =~a +~
b.
c) Find a vector perpendicular to both ~
Nand ~v.
3. (10 pts) Consider the function f(x, y ) = e2xsin y.
a) What is the tangent plane to f(x, y)above (0, π/4)?
b) In which direction does f(x, y)decreases the fastest at (0, π/4)?
c) What is the maximum rate of increase of f(x, y)at that point?
d) If xand ydepend on another variable u:x(u) = u3and y(u) = π/4 + 3u, compute df
du
at u= 0.
4. (9 pts) Consider the function f(x, y ) = (x2−4)(y2−1).
a) Find all the critical points of f(x, y).
b) Select a critical point and determine if it is a maximum, minimum or saddle-point.
c) Over the domain x≥1and y≥2, does this function have a global minimum? If so
find it, if not, why not?.
5. (12 pts) Answer the following questions in no more than two lines of text (much less is
actually needed if you are right on point). No computations are required.
a) Is it possible to have a function which is differentiable but not continuous? How about
a continuous function which is not differentiable?
b) If you want to minimize f(x, y)subject to g(x, y) = c, what is the Lagrangian function
you should use?
c) Describe in words the level surfaces of g(x, y, z) = (x+ 2y−3z)3.
d) Explain in words the meaning of the directional derivative of f(x, y) = x/y at the
point (2,3) in the direction −
~
i−2~
j.
e) Give one possible use of the quadratic expansion of a function.
f) How long is a vector which is the sum of two perpendicular unit vectors?
1