MATH 23 – Final exam Spring Semester 2007
Duration: 180 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) Answer the following questions in no more than two lines of text or formulas (much less is
usually needed if you are right on point).
(a) How do you verify that two vectors ~v and ~w are perpendicular?
(b) If ~c =~a ×~
b, what can you say about the length and direction of ~c?
(c) Give two properties of the gradient ∇fof a function f(x, y, z).
(d) Sketch, name or describe a function that is discontinuous at the origin and a function that is
continuous but not differentiable at the origin.
(e) If a contour of f(x, y)is given by y2+x3= 1, find a point where f(x, y) = f(0,1) (other than
(0,1), of course).
(f) How would you verify whether a vector field ~
Fis a gradient field (conservative) or not?
(g) If Sis a surface and ~
Fa vector field, when can you use the divergence theorem directly to
calculate the flux of ~
Fthrough S?
(h) Sketch or describe in words a vector field with a positive curl everywhere (a formula is not
sufficient)4
(i) Give a parametrization of the line going from (−1,−1,3) to (0,1,4).
(j) Which of the following are vectors?
(i) A velocity field (~u) (ii) ~a ·~
b
(iii) The divergence of a velocity field (div ~
F) (iv) ~a ×~
b
(v) The curl of a three dimensional velocity field (curl ~
F) (vi) The gradient of a function (∇f)
2. (9 pts) Given the implicit function z2+ 9x2+y2/4 = 1
(a) Draw at least 2 cross-sections of this surface by keeping xfixed (specify the value of x).
(b) Draw at least 2 contours.
(c) SKETCH the surface in a manner consistent with what you found above.
3. (8 pts) Consider the following three points in space m1= (−1,0,2),m2= (1,4,2) and m3= (0,2,1)
(a) Find the vectors ~v1going from m1to m2and ~v2going from m1to m3.
(b) Using ~v1and ~v2, find the equation of the plane going through these three points.
4. (9 pts) Above the point (−1,2) in the xy-plane, the plane tangent to the function f(x, y) = xy is
labeled p(x, y).
(a) What is p(−1,2)?
(b) What is the equation of p(x, y)?
(c) If xand yare functions of time x(t) = −t2and y(t) = 2 cos(t−1), use the chain rule to compute
df
dt at t= 1.
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