Sample Final Exam
Problems from this sample final (given in Fall 2002) are assigned as
homework problem as listed on the syllabus. (E.g., S4a on the
syllabus refers to problem 4(a) on this final.)
- Find all critical points of the function
f (x, y) = x2 + y2 + xy2 . Then classify each critical point as a local maximum, local
minimum, or saddle point.
- The temperature at the point (x, y, z) in space is given by
T(x, y, z) = xyz3. T is measured in degrees Celsius (
oC); x, y, and z are measured in kilometers (km). At a certain
instant of time, a space ship is at the point (2,3,1) and headed
toward the point (3,4,3) at a rate of 5 kilometers per second (5
km/s). At what rate is the temperature experienced by the spaceship
increasing at that instant? Your answer should be in terms of
degrees Celsius per second (
- (a) Let M be the surface parametrized by
f(s, t) = (t, t2, s3).
Find the equation of the tangent plane to this surface at the
point (2,4,1). Express your answer in the form
Ax + By + Cz = D.
(b) Find a parametrization for the plane in
4x - y = 4; that is, find an equation of the form
x = x0 + sa + tb.
- Let S be the solid bounded above by the sphere
x2 + y2 + z2 = 16 and below by the paraboloid
z = (x2 + y2)/6.
(a) Parametrize the solid S using cylindrical coordinates.
F(x, y, z) = (x + yz, sin(x9z6), cos(x7y8)). Use
the Divergence Theorem to find the flux of
F through the
boundary of S. Use the outward pointing normal.
F(x, y, z) = (- y, x, sin(x7y8z9)) and let M be the
surface given by
z = x2 + y2 - 9, z < 0.
(a) Give a parametrization of the boundary of M as a curve in
(b) Using any valid method, evaluate the integral of
curl F over the surface M, where M has the outward pointing
- Let M be the surface
x2 + y2 + z2 = 4, z > 0. Using the
outward-pointing normal, find the flux through M for the vector
F(x, y, z) = (y, x, z).
f(x, y) = (y2, x + y) and let g be a function from
R such that the matrix of partial derivatives
of g is given by
[sin(x2), cos(xy)]. Let h be the
h = gof.
(a) Find the matrix of partial derivatives for h at the point (2,3).
(b) Assuming that
h(2, 3) = 4, use the result of part (a) to find
the equation of the tangent plane to the graph of h at the point
(2,3). Write your answer in the form
z = Ax + By + C.
f (x, y) = (x + y)2 and let R be the region in the plane
bounded by the lines x + y = 0, x + y = 1,
2x - y = 0, 2x - y = 1. Use the change of variables u = x + y,
v = 2x - y to evaluate
the double integral of f over R.
- Let M be the level surface
g(x, y, z) = 3 where
g(x, y, z) = x2 + y2/4 + z2/9.
(a) Find the equation of the tangent plane to the surface M at
the point (1,2,3).
(b) Compute the line integral of the vector field
F(x, y, z) = (2x, y/2, 2z/9) over the curve C, where C is parametrized by
x(t) = (cos4t2, sin4t2, t + 1),
0 < t < (2)1/2.