Parametric Curves and Polar Coordinates
Is each polar coordinate unique?
No - the same point can correspond to different Polar coordinates.
What does x^2 + y^2 = 1 represent in R^2 and R^3?
R^2: Unit circle
R^3: Infinite cylinder
What is another name for the cross sections of a surface?
Trace
Let P be a point with Cartesian coordinates (4, 4, −1). Find its Cylindrical coordinates.
(4sqrt(2), pi/4, -1)
Let C be a curve given by r(t) = ⟨2cost, 2sint, t⟩,t ∈ R. Sketch C.
Helix :3
Describe the following Polar curve in Cartesian coordinates.
r^2sin(2θ) = 1
y= 1/2x
A line
Identify −x^2 + y^2 − z^2 = 1
Hyperboloid of two sheets
Describe the surface z = x^2 − y^2 in Cylindrical coordinates.
z = r^2cos(2θ)
Find the point on the curve r(t) = ⟨2cost, 2sin t, e^t ⟩ for 0 ≤ t ≤ pi, where the tangent line is parallel to the plane sqrt(3)x + y = 1.
<sqrt(3), 1, e^(pi/6)>
Sketch the graph of the following curve:
x = sin(t)
y = (sin(t))^2
y = x^2, D: [-1, 1] since -1 ≤ sin(t) ≤ 1
Unit sphere: Empty shell, x^2 + y^2 + z^2 = 1
Unit ball: Filled shell, x^2 + y^2 + z^2 ≤ 1
Identify 3x^2 - y^2 - 3z^2 = 0
Cone
Describe the surface ρcosθ = 1 into Cartesian coordinates
z = 1
The position function of a particle is given by r(t) = ⟨t^2, 5t, t^2 − 16⟩. When is the speed a minimum?
Minimum speed at t = 4.
|v(4)| = sqrt(153)
Find all the points where the tangent line of the polar curve r = 1 + cosθ is either vertical or horizontal.
BONUS 400 POINTS: Find the area enclosed by the curve r = 3cosθ but outside of the curve 1 + cosθ.
1) Horizontal: (3/4, 3sqrt(3)/4), (3/4, -3sqrt(3)/4), (0, 0)
Vertical: (2, 0), (-1/4, sqrt(3)/4), (-1/4, -sqrt(3)/4)\
2) pi
Find the linear equation of the plane that passes through the points P(0, 1, 1), Q(1, 0, 1) and R(1, 1, 0). Calculate the distance between this plane and the plane given by x + y + z = 8.
Plane: x + y + z = 2
Distance: 2sqrt(3)
Identify x^2 + y^2 + 2y - z^2 = 0
Hyperboloid of one sheet
If θ = c corresponds to a line and r = c corresponds to a circle in Polar coordinates, what do they represent in Cylindrical coordinates?
θ = c: Plane that makes an angle c with the positive x-axis
r = c: Cylinder of radius c
Find a parametrization of the curve given by the intersection of z = sqrt(x^2 + y^2) and z = 1 − y.
Parameter: t
x = t
y = -1/2(t^2 - 1)
z = 1 + 1/2(t^2 - 1)
Let C be a curve given by x(t) = t^2, y(t) = t^3 - 3t
a) Find all the points of self-intersection of C
b) Find all the tangent lines at the points from (a).
c) Find all the points on C where the tangent line is horizontal or vertical.
d) Determine where the curve is concave upward or downward.
e) BONUS 300 POINTS: Using information from parts (a)-(d), sketch C.
f) BONUS 200 POINTS: Find the area of the region enclosed by the curve C.
a) (3, 0)
b) y = sqrt(3)(x - 3), y = -sqrt(3)(x - 3)
c) Horizontal tangent: (1, 2) for t = -1, (1, -2) for t = -2
Vertical tangent: (0, 0) for t = 0
d) Concave upward when t > 0, Concave downward when t < 0
e) :3
f) 24sqrt(3)/5
Find the symmetric equations of the line that is the intersection of the planes x + y + z = 1 and x + 2y + 2z = 1.
x = 1, y/-1 = z/1
Let C be a parametric curve given by r(t) = ⟨cost, (cos(t))^2, cos(t)sin(t)⟩.
Show that this curve lies on the surface of a sphere and a cone.
Sphere: x^2 + (y - 1)^2 + z^2 = 1
Cone: y^2 + z^2 = x^2
What does p = c and φ = c represent in Spherical cooridnates?
p = c: Sphere of radius c
φ = c: Cone where points make an angle c with the positive z-axis
Consider parametric curves r1(t) = ⟨t, 1 − t, 3 + t^2⟩ and r2(t) = ⟨3 − t,t − 2,t^2⟩. Find the point of intersection of these curves and find the angle between the curves at the point of intersection.
Point of intersection: (1, 0, 4)
Angle: arccos(1/sqrt(3))