Math 427L Exam 1 Practice Problems

Math 427L Exam 1 Practice Problems
(1) Find the equation for the line passing through (0, 1, 1) and (0, 1, 0).
(2) Suppose ~v = i + 2j − k and w
~ = 3i + j.
(a) Compute ~v · w
~ and ~v × w.
~
(b) Find the cosine of the angle between the two vectors.
(c) Determine the area of the parallelogram defined by the two vectors.
(3) Suppose that ~u, ~v , w
~ are all unit vectors that are orthogonal to each other. If ~a =
α~u + β~v + γ w
~ show that
α = ~a · ~u,
β = ~a · ~v ,
γ = ~a · w
~
(4) Convert (3, π/6, −4) from cylindrical to Cartesian coordinates and to spherical coordinates.


1 1 1


(5) Suppose A is the matrix  x y z . This matrix is known as a Vandermonde matrix.
x2 y 2 z 2
(a) Determine |A|, the determinant of A.
(b) BONUS: Show that |A| 6= 0 if (and only if) x, y, z are all different. (HINT: Try to
factor the polynomial |A|)
(6) Find a unit vector parallel to both planes 8x + y + z = 1 and x − y − z = 0.
(7) Describe the graph of f (x, y) = 3x2 + y 2 .
(8) Sketch the level curves of f (x, y) = x2 − 9y 2 for c = 0, −1.
(9) Suppose f (u, v) = (cos u, v + sin u) and g(x, y, z) = (x2 + πy 2 , xz). Compute D(f ◦ g) at
(0, 1, 1) using the chain rule.
(10) Let u = e−x−y and v = exy , and let z = uv. Compute the partial derivatives ∂z/∂x and
∂z/∂y.
(11) Suppose z =
holds:
f (x−y)
y
where f is differentiable and y 6= 0. Show that the following identity
z+y
∂z
∂z
+y
= 0.
∂x
∂y
(12) Let ~r(t) = (t cos(πt), t sin(πt), t) be a path in R3 . Where will the the tangent line to ~r
at t = 1 intersect the xy plane?
(13) Let f (x, y) = (1−x2 −y 2 )1/2 . Determine the equation of the tangent plane at ( √13 , √13 , √13 ).
(14) Determine if the limit exists
r
x + y lim
x−y
(x,y)→(0,0)
1
2
(15) In which direction is the directional derivative of f (x, y) = (x2 − y 2 )/(x2 + y 2 ) at (1, 1)
equal to zero?
(16) Suppose f (x, y) = xy/(x2 + y 2 ) if (x, y) 6= (0, 0) and f (0, 0) = 0. Show that the partial
derivatives of f exist (you’ll have to use the definition) but explain why the function is
not differentiable.
(17) At time t = 0 a particle is ejected from the surface x2 + 2y 2 + 3z 2 = 6 at the point
(1, 1, 1) in a direction normal to the surface at a speed of 10 units per second. At what
time does it cross the sphere x2 + y 2 + z 2 = 103? At least set up the equation to solve
for t, you can use software to solve it.
(18) The height of a certain mountain is described by the function
h(x, y) = 2.6 − 0.0003y 2 − 0.0007x2 .
At (x, y) = (−2, 4) in what direction is the steepest upward path?
2
(19) Find the second-order Taylor polynomial for f (x, y) = y 2 e−x at (1, 1).
(20) Suppose that u(x, y) and v(x, y) have continuous mixed partial derivatives and satisfy
the so-called Cauchy-Riemann equations:
∂v
∂u
∂v
∂u
=
=−
∂x
∂y
∂y
∂x
.
Show that both u and v are harmonic.