MA1024
Calculus IV
Name
Final Exam, A Term, 2016
Section:
Show all work needed to reach your answers.
1. (30 points)
xy (a) Let F (x, y, z) = sin
. Please compute Fz (x, y, z) and Fz (1, π, 4)
z
Fz (x, y, z) =
Fz (1, π, 4) =
(b) Please find an equation
of the tangent plane to the surface
xy √
F (x, y, z) = sin
= 2/2 at the point (1, π, 4).
z
Equation of Tangent Plane:
2. (20 points) Please compute the following limits, if it exist.
lim
(x, y) → (0, 0)
y 6= −x3
sin 2x − 2x + y
x3 + y
.
Limit:
3. (40 points) Consider the vector function x(t) = h3 cos t, 3 sin t, 4ti. Please compute the
velocity vector v(t), the unit tangent vector T (t), and the arc length s(3π) for the curve
traced out by this vector function between t = 0 amd t = 3π. Next, please find the
tangential component of acceleration, aT (t) (a scalar). Finally, if a particle following the
curve traced out by this vector function flies off the curve at t = 3π and thereafter moves
along a line, please find a vector function that traces out this line.
v(t) =
T (t) =
s(3π) =
aT (t) =
Vector Function Tracing Out Tangent Line:
4. (60 points)
(a) Sketch or describe the region (domain) of integration for
Z
2
−3
Z
y2
(x2 + y) dx dy .
0
Region:
(b) Using the most appropriate coordinates, please set up but do not evaluate the triple
iterated integral of x2 + y 2 + z 2 over the cylinder given by x2 + y 2 ≤ 2, −2 ≤ z ≤ 3.
Triple Iterated Integral:
(c) Please set up triple iterated integrals for the mass of a hemipherical region (domain)
inside the sphere x2 + y 2 + z 2 = 4 and above the x, y-plane when the density is given by
δ(x, y, z) = 1 + z 2 . First write the triple iterated integral in cylindrical coordinates, then
the triple iterated integral in spherical coordinates. Do Not Evaluate!
Cylindrical:
Spherical:
5. (16 points) Please find an equation of the plane which also passes through the origin and
is parallel to both the vectors h1, 0, −1i and h2, −1, 1i.
Equation of Plane:
6. (14 points) Please complete the following table:
Table
Sphere
Plane
x2
rectangular
+ y2 + z2 = 4
z =x+y
cylindrical
spherical
θ = π/3
7. (10 points) Captain Ralph is in trouble near the sunny side of Mercury and notices that
the hull of his ship is beginning to melt. The temperature in his vicinity is given by
T = e−x + e−2y + e3z .
If he is at the point (1, 1, 1), what direction should he proceed in order to cool fastest (find
a vector pointing in the appropriate direction)?
Direction:
8. (10 points) Suppose the F : R3 → R is a differentiable function, and suppose that no partial
derivative is not zero. Consider the surface in R3 defined implicitly by F (x, y, z) = 0. Please
∂y ∂x
= 1. Hint: implicit differentiation.
prove or disprove that
∂x ∂y