Math Answers

1. Use traces to sketch the graph of f(x, y) = p
16 − x
2 − 16y
2
.
2. For each of the following, identify the surface and sketch its graph.
a) x = 4y
2 + z
2 − 4z + 4.
b) x
2 + 4y
2 + z
2 − 2x = 0
3. a) What does the equation x
2 + y
2 = 1 represent as a curve in R
2
?
b) What does it represent as a surface in R
3
?
4. Find an equation of the tangent plane and normal to the surface at the specified point.
a) z = 4x
2 − y
2 + 2y; (−1, 2, 4)
b) z = y ln x; (1, 4, 0)
5. Find an equation for the tangent plane and find symmetric equations for the normal line
to the surface z = arctan( y
x
) at the point (1, 1,
π
4
).
6. Sketch the region bounded above by the cone z =
p
x
2 + y
2 and below by the sphere
x
2 + y
2 + z
2 = z
7. Find the angle of inclination θ of the tangent plane to the surface 3x
2 + 2y
2 − z = 15 at
the point (2, 2, 5).

1. Find the directional derivative of each of the functions at the given point in the direction
of the vector v.
a) f(x, y) = 1 + 2x
√y, (3, 4), v = h4, −3i.
b) g(x, y, z) = x arctan( y
z
), (1, 2, −2), r = h1, 1, −1i
2. Find the maximum rate of change of f = sin(xy) at (1, 0) and the direction in which it
occurs.
3. Find all points at which the direction of fastest change of the function
f(x, y) = x
2 + y
2 − 2x − 4y is h1, 1i
4. The temperature T in a metal ball is inversely proportional to the distance from the
center of the ball, which we take to be the origin. The temperature at the point (1, 2, 2)
is 120◦
.
a) Find the rate of change of of T at (1, 2, 2) in the direction toward the point (2, 1, 3).
b) Show that at any point in the ball the direction of greatest increase in temperature
is given by a vector that points toward the origin.

6. Find the absolute minimum of f(x, y) = x
2 + y
2 + 2y − 1 on D = {(x, y)|x
2 +
y
2
4 ≤ 1}
7. Use Lagrange multipliers to find the maximum and minimum values of the function
subject to the given constraint(s).
a) f(x, y, z) = xyz; x
2 + 2y
2 + 3z
2 = 6
b) f(x, y, z) = yz + xy; xy = 1, y2 + z
2 = 1
8. Find the first four terms in the Taylor series expansion of the function:
f(x, y) = e
−y
2
cos(x + y) about the point (0, 0).

1. Write a complete Cayley Table for D6, the dihedral group of order 6.
2. Prove that if G is a group with property that the square of every element is the identity, then G is
abelian.
3. Construct the Cayley table for the group generated by g and h, where g and h satisfy the relations
g
3 = h
2 = e and gh = hg2
.
4. Let H and K be subgroups of a group G such that gcd(|H|, |K|) = 1. Apply Lagrange’s theorem to
show that |H ∩ K| = 1.
5. Consider the group Z12 and the subgroup H =< [4] >= {[0], [4], [8]}. Are the following pairs of elements
related under ∼H? Justify your answer.
(a) [3], [11],
(b) [3], [7],
(c) [5], [11],
(d) [6], [9],
(e) find all left cosets of H in G. Are they different from the right cosets?

1. Write a complete Cayley Table for D6, the dihedral group of order 6.
2. Prove that if G is a group with property that the square of every element is the identity, then G is
abelian.
3. Construct the Cayley table for the group generated by g and h, where g and h satisfy the relations
g
3 = h
2 = e and gh = hg2
.
4. Let H and K be subgroups of a group G such that gcd(|H|, |K|) = 1. Apply Lagrange’s theorem to
show that |H ∩ K| = 1.
5. Consider the group Z12 and the subgroup H =< [4] >= {[0], [4], [8]}. Are the following pairs of elements
related under ∼H? Justify your answer.
(a) [3], [11],
(b) [3], [7],
(c) [5], [11],
(d) [6], [9],
(e) find all left cosets of H in G. Are they different from the right cosets?

Evaluate the following integrals
1. ∫
C
z
2
z
2 + 2z + 2
where C is the circle |z| = 2.
2. ∫
C
Ln(1−z)dz, where C is the parallelogram with vertices ±i, ±(1+i).
3. ∫ 2π
0
1
1 + 3 cos θ
dθ.
4. ∫
C
z
2
z
4 − 1
dz. where C is the circle |z + 1| = 1
5. ∫
C
e
z
π − i
dz. where C is the unit circle

1. Find the indicated derivatives:
a) u =
x+y
y+z
, x = p + r + t, y = p − r + t, z = p + r − t;
∂u
∂r .
b) y
5 + x
2
y
3 = 1 + yex
2
;
dy
dx .
c) ln(x + yz) = 1 + xy2
z
3
; ∂z/∂y
2. Let f(x, y) = x
2
y + x
3
y
2 and suppose you dont know what φ(t) = (x(t), y(t)) is, but you
know φ(2) = (1, 1), dx
dt (2) = 3, and dy
dt (2) = 1. Find the derivative of f(φ(t)) when t = 2.
3. Show that the following functions are functionally dependent and find a relation connect￾ing them:
f(x, y, z) = x + y + z, g(x, y, z) = x
2 + y
2 + z
2
, h(x, y, z) = xy + yz + xz
4. Find the local maxima, minima, and saddles of the functions h(x, y) = (2x−x
2
)(2y −y
2
).
5. Find the largest volume of a box with an open top, and surface area 100m2
.
6. Find the absolute minimum of f(x, y) = x
2 + y
2 + 2y − 1 on D = {(x, y)|x
2 +
y
2
4 ≤ 1}