MATH 23 – Midterm 1 Spring Semester 2008
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 points: 5 each) Given two vectors ~u =~
i−2~
j+ 3~
kand ~v =~
j+ 2~
k.
(a) Find an equation of the plane which is parallel to both ~u and ~v and goes through the
point (2,5,3).
(b) Decompose ~u into two vectors ~a and ~
bsuch that ~u =~a +~
b, with ~a parallel to ~v and ~
b
perpendicular to ~v.
2. (15 points: 5 each)
(a) Find parametric equations that represent the curve of intersection of the cylinder
x2+y2= 9 and the plane y+z= 1.
(b) Find the arc length of the helix ~r(t) =<sin 3t, 4t, cos 3t >,0≤t≤2.
(c) Find parametric equations for the tangent line to the helix in part (b) at the point (0,0,1).
3. (15 points total) Consider the function f(x, y ) = px2+ 4y2−4.
(a) (5 points) Draw a contour map of fshowing at least 3 level curves. Remember to label
your axes and level curves.
(b) (2 points) Draw 2 vertical traces of the graph z=f(x, y ), one with x= 0 and the other
with y= 0.
(c) (3 points) Sketch the graph z=f(x, y )showing your level curves and traces in parts
(a) and (b).
(d) (5 points) Calculate fx(1,1) and fy(1,1).
4. (10 points: 2 each) Answer the following questions in no more than two lines of text.
(a) A vector function ~r(t)represents a space curve. If we know that
d~r
dt
= 1 for all t, what
is the geometric significance of the parameter tother than time?
(b) Is it true that if ~u ×~v = 0 then either ~u =~
0or ~v =~
0? Explain why.
(c) How can you show that lim
(x,y)→(a,b)f(x, y)does not exist.
(d) Give an example of a function f(x, y )and a point (a.b)such that fx(a, b)and fy(a, b)
both exist but fis not even continuous at (a, b). You may describe your example using
formulas, pictures or words.
(e) What is the length of the sum of two perpendicular unit vectors?
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