Math 2210-1
Review 1
√
√
1. Give the Cartesian equation of the curve given parametrically by x = 3 t − 3, y = 2 4 − t; 3 ≤ t ≤ 4.
2. Find
dy
dx
and
d2 y
dx2
of the curve given parametrically by x = 2θ2 ; y = 5 sin θ; θ 6= 0.
3. Sketch the graph of the equation y = 2 in 3-dimensional space. Describe it.
4. Which of the points A = (2, 1, 3), B = (3, 0, 0), C = (3, −5, 7) is closest to the yz-plane? Which one
lies on the xz-plane? Which one is farthest from the xy-plane? In each case, how do you know?
5. Perform the following operations on the given vectors:
~a = 2~i + 3~k
~b = ~i + 3~j
~c = ~i − 3~j − 2~k
(a) kak
(b) ~b + ~c
(c) 2~c
(d) ~a · ~c
(e) ~b × ~a
6. Represent the vector 4~i − 3~j as a vector parallel to the vector ~v = 9~i − 13~j and a vector perpendicular
to ~v .
7. Below is the path ζ of a housefly. Assume that the housefly travels at constant speed. On the figure,
sketch the velocity vectors and acceleration vectors at the points A, B and C. Please label them
accordingly. Explain why you have drawn them as you have.
11
00
A
B 0011
C
11
00
8. The acceleration vector of a particle is given by ~a(t) = t~j. If ~v (0) = ~i + 2~j and ~r(0) = ~0, what are the
velocity and position vectors for the particle?
~ k, aT , and aN for ~r(t) = 1 t3~i + 1 t2~j at t = 1.
9. Find T~ , N,
3
2
10. Find the point on the curve y = ex where the curviture is a maximum.
11. A weight of 30 pounds is suspended by three wires with resulting tensions 3~i + 4~j − ~k, −8~i − 2~j + 10~k,
and a~i + b~j + c~k. Determine a, b, and c assuming that ~k points straight up.
12. Find the equation of the plane through (1,1,2), (0,0,1), and (-2,-3,0).
13. Find the symmetric equations of the line through (2,-4,5) that is parallel to the plane 3x + y − 2x = 5
y−5
z−1
x+8
=
=
.
and perpendicular to the line
2
3
−1
14. Find the velocity, acceleration and speed of ~r(t) = sin 2t~i + cos 3t~j + cos 4t~k at t =
π
2.
15. Find the curvature, unit tangent vector, principal normal, binormal, at , and aN of ~r = sin 3t~i+cos 3t~j+~k
at t = π9 .
16. Sketch the graph of x2 + z 2 = 9 in three space.
17. Sketch the graph of 3x2 − 4y 2 = 12z in three space.
18. Change the equation x2 + y 2 + 4z 2 = 10 to spherical coordinates.
19. Change the equation ρ = 2 cos φ to cylindrical coordinates.
20. Change the equation r2 cos 2θ = z to Cartesian coordinates.