The
parametric straight line paths of two objects are given.
a) do the objects crash (at the same location at the same
time)? If so, at what time?
b) do the paths of the objects intersect (at the same point at
different times)? If so, how close do they get to each other?
c) do the objects and paths miss each other? If so, how close
do the objects get to each other and how close do their paths get...
3. (5 points) (a): Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.$$ x=e^{-t} \cos t, \quad y=e^{-t} \sin t, \quad z=e^{-t} ; \quad(1,0,1) $$(b): Find the unit tangent vector \(\mathbf{T}\), the principal unit normal \(\mathbf{N}\), and the curvature \(\kappa\) for the space curve,$$ \mathbf{r}(t)=<3 3="" 4="" sin="" cos="" t="">$$
The parametric equations
x = x1 +
(x2 −
x1)t, y
= y1 +
(y2 −
y1)t
where
0 ≤ t ≤ 1
describe the line segment that joins the points
P1(x1,
y1)
and
P2(x2,
y2).
Use a graphing device to draw the triangle with vertices
A(1, 1), B(4, 3), C(1, 6). Find the
parametrization, including endpoints, and sketch to check. (Enter
your answers as a comma-separated list of equations. Let x
and y be in terms of t.)
1. Given parametric equations below, find the values of t where
the the parametric curve has a horizontal and vertical
tangents.
a) x=t^2 - t, y= t^2 + t
b) x= e^(t/10)cos(t), y= e^(t/10)sin(t)
2. Find the arc length of the graph of the parametric equations
on the given intervals.
a) x= 4t+2, y = 1-3t , −1 ≤ t ≤ 1
b) x= e^(t/10)cos(t), y= e^(t/10)sin(t), 0 ≤ t ≤ 2π
1. Given parametric equations below, find dy/dx , equations of
the tangent and normal line at the given point.
(a) x = t^2 , y = t^3−3t at t = 1
(b) x = cos(t), y = sin(2t) at t = π/4
2. ) Given parametric equations below, find d^2y/dx^2 and
determine the intervals on which the graph of the curve is concave
up or concave down.
(a) x = t^2 , y = t^3−3t
(b) x = cos(t), y...
Question: (a) Find parametric equations for the line of
intersection of the planes given by 3x − 2y + z = 1 and 2x + y − 3z
= 3.
(b) Find the equation of the plane orthogonal to both of these
planes and passing through the point (−2, 1, 1).