Math 2210-1
Name:
Test 1
ST#
Show all your work in a neat and organized manner. Write your answers and solutions in
the space provided. Please box your answers. You may use a calculator.
List of Equations
~u · ~v
~v
|~v |2
~
Work = F~′ · D
~
r
(t)
T~ (t) = ′
|~r (t)|
|T~ ′ (t)|
|T~ ′ (t)|
= ′
κ=
|~v (t)|
|~r (t)|
|x′ y ′′ − y ′ x′′ |
κ=
[(x′ )2 + (y ′ )2 ]3/2
|y ′′ |
κ=
[1 + (y ′ )2 ]3/2
~
dT
~
= κN
ds
2
d2 s ~
ds
~
~a = 2 T +
κN
dt
dt
~
~a = aT T~ + aN N
|Ax0 + By0 + Cz0 − D|
√
L=
A2 + B 2 + C 2
~r′ · ~r′′
aT = T~ · ~a =
|~r′ |
|~r′ × ~r′′ |
aN = |T~ × ~a| =
|~r′
′
′′
|~r × ~r |
κ=
|~r′ |3
~ = T~ × N
~
B
x = ρ sin φ cos θ
y = ρ sin φ sin θ
z=ρ
pcos φ
ρ = x2 + y 2 + z 2
y
tan θ =
x
z
cos φ = p
2
x + y2 + z 2
Proj~v ~u =
Math 2210-1
Test 1
1. (3pts) Give the Cartesian equation of the curve parameterized by x = t − 5, y =
√
7t, 0 ≤ t ≤ 7.
dy
d2 y
2. (6pts) Find
of the curve given by x = 5t2 − 3t + 1, y = 2t5 − 1 without eliminating the
and
dx
dx2
parameter.
3. (8pts) Say whether the given quantity is a vector or a scalar.
(a) ~u · ~v
(b) ~u × ~v
(c) Distance
(d) |~v |
(e) Speed.
(f) Acceleration.
(g) c~u where c is a real number.
(h) Magnetic Field.
4. (3pts) A company sells two products, and their revenue function is given by R(x, y, p, q) where x and
y represent the number of product one and product two sold, respectively, and p and q represent the
price at which product one and product two are sold, respectively. Give a description of what the
function R(25, 100, p, 9) represents.
5. (4pts) Which of the points A = (0, 5, 3), B = (2, 7, 0), C = (−5, −3, 7) is closest to the xz-plane? Which
one lies on the xy-plane? Explain your reasoning.
6. (2pts) Sketch the graph of the equation x2 + y 2 = z.
7. (5pts) Find the angle between −4~i + 3~j − ~k and 2~i − 7~j + 4~k.
8. (5pts) Find the equation of the plane perpendicular to the curve ~r(t) = (8t2 −4t+3)~i+(sin t−4t)~j−cos t~k
at the point t = π3 .
9. (4pts) Match the following functions with their graphs below.
(a) z = x3 − sin y
2
(b) z = −e−x −y
1
(c) z = 2
x + y2
2
(d) z = −y 2
(A)
(C)
(B)
(E)
(D)
10. (10pts) What is the curvature of x = 7 sin 3t, y = 7 cos 3t, z = 14t at t =
π
9?
11. (5pts) A weight of 12 pounds is suspended by three wires with resulting tensions 4~i − 11~j + 3~k, −5~i +
3~j + ~k, and a~i + b~j + c~k. Determine a, b, and c if ~k points straight up.
12. (7pts) Find the equation of the line through the point (3,5,-2) which is perpendicular to the plane
x−6
y+7
z+9
containing the lines x = 2 + 7t, y = 3 − t, z = −3 + 4t and
=
=
.
2
4
−2
13. (6pts) Change the equation r2 + 2z 2 = 4 to spherical coordinates.